# migrations

dir: embeddings_queue
filename: 00001-embeddings.sqlite.sql
hash: d3755dfd232be8e8301f4d7fcfb3a486
sql:
CREATE TABLE embeddings_queue (
    seq_id INTEGER PRIMARY KEY,
    created_at TIMESTAMP NOT NULL DEFAULT CURRENT_TIMESTAMP,
    operation INTEGER NOT NULL,
    topic TEXT NOT NULL,
    id TEXT NOT NULL,
    vector BLOB,
    encoding TEXT,
    metadata TEXT
);


dir: sysdb
filename: 00001-collections.sqlite.sql
hash: 38352d725ad1c16074fac420b22b4633
sql:
CREATE TABLE collections (
    id TEXT PRIMARY KEY,
    name TEXT NOT NULL,
    topic TEXT NOT NULL,
    UNIQUE (name)
);

CREATE TABLE collection_metadata (
    collection_id TEXT REFERENCES collections(id) ON DELETE CASCADE,
    key TEXT NOT NULL,
    str_value TEXT,
    int_value INTEGER,
    float_value REAL,
    PRIMARY KEY (collection_id, key)
);


dir: sysdb
filename: 00002-segments.sqlite.sql
hash: 2913cb6a503055a95f625448037e8912
sql:
CREATE TABLE segments (
    id TEXT PRIMARY KEY,
    type TEXT NOT NULL,
    scope TEXT NOT NULL,
    topic TEXT,
    collection TEXT REFERENCES collection(id)
);

CREATE TABLE segment_metadata (
    segment_id TEXT  REFERENCES segments(id) ON DELETE CASCADE,
    key TEXT NOT NULL,
    str_value TEXT,
    int_value INTEGER,
    float_value REAL,
    PRIMARY KEY (segment_id, key)
);


dir: sysdb
filename: 00004-tenants-databases.sqlite.sql
hash: 048867ce8fcdefe4023c7110e4433591
sql:
CREATE TABLE IF NOT EXISTS tenants (
    id TEXT PRIMARY KEY,
    UNIQUE (id)
);

CREATE TABLE IF NOT EXISTS databases (
    id TEXT PRIMARY KEY, -- unique globally
    name TEXT NOT NULL, -- unique per tenant
    tenant_id TEXT NOT NULL REFERENCES tenants(id) ON DELETE CASCADE,
    UNIQUE (tenant_id, name) -- Ensure that a tenant has only one database with a given name
);

CREATE TABLE IF NOT EXISTS collections_tmp (
    id TEXT PRIMARY KEY, -- unique globally
    name TEXT NOT NULL, -- unique per database
    topic TEXT NOT NULL,
    dimension INTEGER,
    database_id TEXT NOT NULL REFERENCES databases(id) ON DELETE CASCADE,
    UNIQUE (name, database_id)
);

-- Create default tenant and database
INSERT OR REPLACE INTO tenants (id) VALUES ('default_tenant'); -- The default tenant id is 'default_tenant' others are UUIDs
INSERT OR REPLACE INTO databases (id, name, tenant_id) VALUES ('00000000-0000-0000-0000-000000000000', 'default_database', 'default_tenant');

INSERT OR REPLACE INTO collections_tmp (id, name, topic, dimension, database_id)
    SELECT id, name, topic, dimension, '00000000-0000-0000-0000-000000000000' FROM collections;
DROP TABLE collections;
ALTER TABLE collections_tmp RENAME TO collections;


dir: sysdb
filename: 00005-remove-topic.sqlite.sql
hash: b1367c826b8fba5f96f27befdc1d42d2
sql:
-- Remove the topic column from the Collections and Segments tables

ALTER TABLE collections DROP COLUMN topic;
ALTER TABLE segments DROP COLUMN topic;


dir: sysdb
filename: 00006-collection-segment-metadata.sqlite.sql
hash: 4eea7468935bf25d4604a0fed2366116
sql:
-- SQLite does not support adding check with alter table, as a result, adding a check
-- involve creating a new table and copying the data over. It is over kill with adding
-- a boolean type column. The application write to the table needs to ensure the data
-- integrity.
ALTER TABLE collection_metadata ADD COLUMN bool_value INTEGER;
ALTER TABLE segment_metadata ADD COLUMN bool_value INTEGER;


dir: sysdb
filename: 00007-collection-config.sqlite.sql
hash: 1c7e63bba346a42a18b6ab7f1c989bed
sql:
-- Stores collection configuration dictionaries.
ALTER TABLE collections ADD COLUMN config_json_str TEXT;


dir: sysdb
filename: 00008-maintenance-log.sqlite.sql
hash: 0a0e7e93111a01789addf64961c6127c
sql:
-- Records when database maintenance operations are performed.
-- At time of creation, this table is only used to record vacuum operations.
CREATE TABLE maintenance_log (
  id INT PRIMARY KEY,
  timestamp INT NOT NULL,
  operation TEXT NOT NULL
);


dir: sysdb
filename: 00009-segment-collection-not-null.sqlite.sql
hash: 054355aef9e63702bf54ea29e61563f1
sql:
-- This makes segments.collection non-nullable.
CREATE TABLE segments_temp (
    id TEXT PRIMARY KEY,
    type TEXT NOT NULL,
    scope TEXT NOT NULL,
    collection TEXT REFERENCES collection(id) NOT NULL
);

INSERT INTO segments_temp SELECT * FROM segments;
DROP TABLE segments;
ALTER TABLE segments_temp RENAME TO segments;


dir: metadb
filename: 00001-embedding-metadata.sqlite.sql
hash: 2b4cf52c4bb2676e21d6860a4409f856
sql:
CREATE TABLE embeddings (
    id INTEGER PRIMARY KEY,
    segment_id TEXT NOT NULL,
    embedding_id TEXT NOT NULL,
    seq_id BLOB NOT NULL,
    created_at TIMESTAMP NOT NULL DEFAULT CURRENT_TIMESTAMP,
    UNIQUE (segment_id, embedding_id)
);

CREATE TABLE embedding_metadata (
    id INTEGER REFERENCES embeddings(id),
    key TEXT NOT NULL,
    string_value TEXT,
    int_value INTEGER,
    float_value REAL,
    PRIMARY KEY (id, key)
);

CREATE TABLE max_seq_id (
    segment_id TEXT PRIMARY KEY,
    seq_id BLOB NOT NULL
);

CREATE VIRTUAL TABLE embedding_fulltext USING fts5(id, string_value);


dir: metadb
filename: 00002-embedding-metadata.sqlite.sql
hash: 12a570f7121b3a8ce750a2a7c36da20f
sql:
-- SQLite does not support adding check with alter table, as a result, adding a check
-- involve creating a new table and copying the data over. It is over kill with adding
-- a boolean type column. The application write to the table needs to ensure the data
-- integrity.
ALTER TABLE embedding_metadata ADD COLUMN bool_value INTEGER


dir: metadb
filename: 00003-full-text-tokenize.sqlite.sql
hash: f97ad6334aeaa8f419f01110b648b97a
sql:
CREATE VIRTUAL TABLE embedding_fulltext_search USING fts5(string_value, tokenize='trigram');
INSERT INTO embedding_fulltext_search (rowid, string_value) SELECT rowid, string_value FROM embedding_metadata;
DROP TABLE embedding_fulltext;


dir: metadb
filename: 00004-metadata-indices.sqlite.sql
hash: fb36603a45ee2cd0254cef3ef86585e8
sql:
CREATE INDEX IF NOT EXISTS embedding_metadata_int_value ON embedding_metadata (key, int_value) WHERE int_value IS NOT NULL;
CREATE INDEX IF NOT EXISTS embedding_metadata_float_value ON embedding_metadata (key, float_value) WHERE float_value IS NOT NULL;
CREATE INDEX IF NOT EXISTS embedding_metadata_string_value ON embedding_metadata (key, string_value) WHERE string_value IS NOT NULL;


# embeddings_queue

created_at: 2025-01-09 14:43:47
topic: persistent://default/default/155c55a6-dc21-4efd-bb5d-885bafc8bd13
id: e45bf032-978a-4827-b827-a129fad4346a
encoding: FLOAT32
metadata:
{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nElementary Mathematics:\nCore Skills (Book 1)\nMathematics core concepts that enhance computational fluency\nfor TMSCA and Math Olympiad\nBy: Ice Venkatesh\nAuthor and Founder of Vibba LLC\nPage 1 of 113\n\nELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nPublished in 2024 by Vibba LLC\nhttps://Vibba.com\nCopyright \u00a9 2024 by Ice Venkatesh\n@Vibba LLC All rights reserved. No part of this publication may be reproduced, stored in a retrieval\nsystem, or transmitted in any form or by any means, electronic, mechanical, photocopying, or\notherwise, without the publisher's prior permission.\nEditor: Arti Venkatesh\nPage 2 of 113\n\nELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nI am dedicating my first book to my Mathematics Teacher,\nMr. Doug Bell\nMathematics Teacher\nTHE VILLAGE SCHOOL\nA NORD ANGLIA EDUCATION SCHOOL\nHouston TX USA\nPage 3 of 113"}


created_at: 2025-01-09 14:43:47
topic: persistent://default/default/155c55a6-dc21-4efd-bb5d-885bafc8bd13
id: e213b470-4e72-4450-8f38-7e96bcb0e6f2
encoding: FLOAT32
metadata:
{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nI am dedicating my first book to my Mathematics Teacher,\nMr. Doug Bell\nMathematics Teacher\nTHE VILLAGE SCHOOL\nA NORD ANGLIA EDUCATION SCHOOL\nHouston TX USA\nPage 3 of 113\n\nELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nAcknowledgments\nThis book would not have been possible without the incredible support\nfrom Village School, whose belief in my vision fueled my journey. I am\ndeeply grateful to my mom, Arti Venkatesh, for her pivotal role in\ntypesetting the initial draft and for always encouraging me to chase my\ndreams.\nA special acknowledgment to those who have inspired and empowered\nme to embrace the wonders of mathematics and explore the limitless\npossibilities it holds. You sparked my passion for mathematical\nadventures, and for that, I am forever thankful.\nTo my friends at Village School\u2014you are my constant source of\ninspiration, reminding me every day of the power of curiosity,\nperseverance, and friendship. Together, we are changing the way the\nworld sees math!\nPage 4 of 113"}


created_at: 2025-01-09 14:43:47
topic: persistent://default/default/155c55a6-dc21-4efd-bb5d-885bafc8bd13
id: 32beb6a0-65ff-4fe8-84ac-53932e79129c
encoding: FLOAT32
metadata:
{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nTable of content\nForeword by Bill Delbrugge .......................................................................................... 8\nForeword by Cindy Proske ........................................................................................... 9\nIntroduction ................................................................................................................. 10\nImportance of this Book .............................................................................................. 14\nHow to use this Book series ....................................................................................... 16\nPath to Expert ............................................................................................................. 17\nSection 1 .................................................................................................................... 18\n1-A1: Addition using Mental MATH ........................................................................ 18\n1-S1: Subtraction Method 1 ................................................................................... 20"}


created_at: 2025-01-09 14:43:47
topic: persistent://default/default/155c55a6-dc21-4efd-bb5d-885bafc8bd13
id: a36f00e6-20a1-4075-8a24-b9b8cbbf83cf
encoding: FLOAT32
metadata:
{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "1-A1: Addition using Mental MATH ........................................................................ 18\n1-S1: Subtraction Method 1 ................................................................................... 20\n1-S2: Subtraction Method 2 ................................................................................... 22\n1-P1: Percentage Trick .......................................................................................... 25\nSection 1 Order these Steps ................................................................................... 27\nPractice Quiz: .......................................................................................................... 28\nMath Wizards: Shakuntala Devi .............................................................................. 29\nSection 2 .................................................................................................................... 30\n2-T1: Table of 9 Trick ............................................................................................. 30\n2-M1: Multiply by 99 ............................................................................................... 32"}


created_at: 2025-01-09 14:43:47
topic: persistent://default/default/155c55a6-dc21-4efd-bb5d-885bafc8bd13
id: c08b557f-e4b1-4924-b7da-74f2aa691d85
encoding: FLOAT32
metadata:
{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "2-T1: Table of 9 Trick ............................................................................................. 30\n2-M1: Multiply by 99 ............................................................................................... 32\n2-D1: Dividing by 9 ................................................................................................. 34\nSection 2 Order these Steps ................................................................................... 36\nSection 2 Quiz ......................................................................................................... 37\nMath Wizards: Zerah Colburn ................................................................................. 38\nSection 3 .................................................................................................................... 39\n3-M1: Two-Digit Multiplication (FOIL) ..................................................................... 39\n3-M2: Multiply by 12 ............................................................................................... 41\n3-M3: Add to 10 Tens Digit Trick ............................................................................ 43"}


created_at: 2025-01-09 14:43:47
topic: persistent://default/default/155c55a6-dc21-4efd-bb5d-885bafc8bd13
id: 7f6c5031-836f-4697-ae82-5bf58cbd1c0d
encoding: FLOAT32
metadata:
{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "3-M2: Multiply by 12 ............................................................................................... 41\n3-M3: Add to 10 Tens Digit Trick ............................................................................ 43\n3-M4: Add to 10 Units Digit Trick ............................................................................ 45\nSection 3 Order These Steps .................................................................................. 47\nSection 3 Quiz ......................................................................................................... 48\nMath Wizards: \u00c9variste Galois ................................................................................ 49\nSection 4 .................................................................................................................... 50\n4-M1: Multiplying by 11 ........................................................................................... 50\n4-M2: Multiplying by 111 ......................................................................................... 52\nPage 5 of 113"}


created_at: 2025-01-09 14:43:47
topic: persistent://default/default/155c55a6-dc21-4efd-bb5d-885bafc8bd13
id: aad9ce38-56d9-4c7d-9c84-435baaf16a4b
encoding: FLOAT32
metadata:
{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nSection 4 Order these Steps ................................................................................... 54\nSection 4 Quiz ......................................................................................................... 54\nMath Wizards: Srinivasa Ramanujan....................................................................... 55\nSection 5 ..................................................................................................................... 56\n5-M1: Multiply by 25 ................................................................................................ 56\n5-M2: Multiply by 50 ................................................................................................ 58\n5-M3: Multiply by 75 ................................................................................................ 60\n5-M4: Multiply two numbers ending with 5 ............................................................... 62\n5-M5: Multiplying by 15 ............................................................................................ 64\nSection 5 Order these Steps ................................................................................... 66"}


created_at: 2025-01-09 14:43:47
topic: persistent://default/default/155c55a6-dc21-4efd-bb5d-885bafc8bd13
id: 5d093ca5-b52a-478f-ac6c-24d6a09f68db
encoding: FLOAT32
metadata:
{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "5-M5: Multiplying by 15 ............................................................................................ 64\nSection 5 Order these Steps ................................................................................... 66\nMath Wizards: Alexander Craig Aitken .................................................................... 68\nSection 6 ..................................................................................................................... 69\n6-M1: Multiplying two numbers both over 100 ......................................................... 69\n6-M2: Multiplying two numbers both under 100 ....................................................... 71\n6-M3: Multiplying two numbers over/under 100 ....................................................... 73\nSection 6 Order these Steps ................................................................................... 75\nSection 6 Quiz ......................................................................................................... 76\nMath Wizards: Thomas Fuller .................................................................................. 77"}


created_at: 2025-01-09 14:43:47
topic: persistent://default/default/155c55a6-dc21-4efd-bb5d-885bafc8bd13
id: 065e16f8-a1ed-4948-b6a9-69cb0b9d8683
encoding: FLOAT32
metadata:
{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "Section 6 Quiz ......................................................................................................... 76\nMath Wizards: Thomas Fuller .................................................................................. 77\nSection 7 ..................................................................................................................... 78\n7-M1: Multiplying Two digits number by 101 ........................................................... 78\n7-M2: Multiplying Three digits number by 101 ......................................................... 79\n7-M3: Multiplying Two digits number by 1001 ......................................................... 81\n7-M4: Multiplying Three digits number by 1001 ....................................................... 83\n7-M5: Multiplying four digits number by 1001 .......................................................... 84\nSection 7 Order these Steps ................................................................................... 86\nSection 7 Quiz ......................................................................................................... 87\nMath Wizards: Carl Friedrich Gauss ........................................................................ 88"}


created_at: 2025-01-09 14:43:47
topic: persistent://default/default/155c55a6-dc21-4efd-bb5d-885bafc8bd13
id: 5095406e-deaa-4751-8df9-624dbd97d29c
encoding: FLOAT32
metadata:
{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "Section 7 Quiz ......................................................................................................... 87\nMath Wizards: Carl Friedrich Gauss ........................................................................ 88\nSection 8 ..................................................................................................................... 89\n8-M1: Multiply A number with an Equal Number of 9's ............................................ 89\n8-M2: Multiply the number of digits less than the number of 9s ............................... 91\nSection 8 Order these Steps ................................................................................... 93\nSection 8 Quiz ......................................................................................................... 94\nMath Wizards: George Parker Bidder ...................................................................... 95\nSection 9 ..................................................................................................................... 96\n9-A1: Sum of even numbers between 1 and 100 .................................................... 96\nPage 6 of 113"}


created_at: 2025-01-09 14:43:47
topic: persistent://default/default/155c55a6-dc21-4efd-bb5d-885bafc8bd13
id: ea953942-f32d-4218-8f3b-2fdc7c5209e6
encoding: FLOAT32
metadata:
{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\n9-A2: Sum of odd numbers between 1 and 100 ...................................................... 98\nSection 9 Order these Steps ................................................................................. 100\nSection 9 Quiz ....................................................................................................... 100\nMath Wizards: Aryabhata ...................................................................................... 101\nAnswer Sheet ........................................................................................................... 102\nBibliography .............................................................................................................. 111\nUseful Links .............................................................................................................. 111\nAbout The Author ..................................................................................................... 113\nPage 7 of 113"}


created_at: 2025-01-09 14:43:47
topic: persistent://default/default/155c55a6-dc21-4efd-bb5d-885bafc8bd13
id: 1fed90bd-abe1-4383-b98d-87a07a0245f5
encoding: FLOAT32
metadata:
{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nForeword by Bill Delbrugge\nIce Venkatesh has truly inspired us all with her fantastic new math book. Ice has\nmasterfully created a book that breaks down difficult concepts into small steps,\nhelping scholars find success.\nIce has a passion for mathematics and is dedicated to making it accessible to all,\nwhich is truly remarkable. As Benjamin Franklin once said, \"An investment in\nknowledge pays the best interest.\" Through this book, Ice has made a powerful\ninvestment, one that will surely inspire countless minds to explore, understand,\nand appreciate the beauty of math.\nIce, your ability to simplify complex concepts and present them in a way that\nsparks curiosity and understanding is a gift. This book is more than just a\nresource; it is a guide that helps learners at all levels unlock the true potential of\nmathematics. You have transformed math from something that may seem\nintimidating into a tool that empowers people, encouraging them to think critically\nand solve problems in new and innovative ways.\nYour work will undoubtedly have a lasting impact, providing both students and\neducators with the knowledge and confidence they need to succeed. Thank you"}


created_at: 2025-01-09 14:43:47
topic: persistent://default/default/155c55a6-dc21-4efd-bb5d-885bafc8bd13
id: eaf49aab-176d-4c44-8a2c-a7eeb21307d4
encoding: FLOAT32
metadata:
{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "and solve problems in new and innovative ways.\nYour work will undoubtedly have a lasting impact, providing both students and\neducators with the knowledge and confidence they need to succeed. Thank you\nfor sharing your love for math with the world and for making this investment in the\nminds of future generations.\nEducationally Yours,\nBill Delbrugge\nHead of School, The Village School\nPage 8 of 113"}


created_at: 2025-01-09 14:43:47
topic: persistent://default/default/155c55a6-dc21-4efd-bb5d-885bafc8bd13
id: cb655963-8024-4b84-957f-be7cf31eea93
encoding: FLOAT32
metadata:
{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nForeword by Cindy Proske\nHaving started the Village Middle School Math Team in 1996, it warms my heart\nto see this team not only thriving but producing students like Ice Venkatesh who\nis willing to share her tips and tricks with other students wanting to participate in\nmath competitions.\nThe hard work and perseverance, it takes to be successful in math competitions\nis not an easy task. With Ice\u2019s help, other students will be able to have a roadmap\nto guide them to their success. The math and science team not only helps\nstudents solve complex problems, but it gives them a place to form friendships as\nwell as learn about teamwork, perseverance, and the thrill of competition. These\nskills last a lifetime for these competitors and increase their understanding and\nskills in the classroom.\nI am so proud that Ice Venkatesh will help others embrace the beauty of numbers\nand the excitement of the challenge. What YOU Do Matters, Ice!\nBest Wishes,\nCindy Proske\nDirector of Middle School, The Village School\nPage 9 of 113"}


created_at: 2025-01-09 14:43:47
topic: persistent://default/default/155c55a6-dc21-4efd-bb5d-885bafc8bd13
id: 64d47c04-fc75-4681-b7bc-5566698f7ecd
encoding: FLOAT32
metadata:
{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nIntroduction\nThis is a great book for Middle School students who want to excel and enjoy mathematics.\nThis book will help to improve your problem-solving and critical thinking skills.\nTo make learning math more engaging and accessible, various innovative methods have\nbeen developed, including Vedic Mathematics, Singapore Mathematics, and Mental\nMathematics. These approaches are taught in numerous countries around the world, each\nwith unique features and benefits. Core Skills is an extremely easy-to-learn book, filled with\nsuch well-known concepts. Let us understand a little bit more about these concepts.\nPage 10 of 113"}


created_at: 2025-01-09 14:43:47
topic: persistent://default/default/155c55a6-dc21-4efd-bb5d-885bafc8bd13
id: f45a6984-f527-4147-9a09-84be9537115d
encoding: FLOAT32
metadata:
{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nVedic Mathematics\nVedic Mathematics is an ancient system of Indian mathematics that consists of a set of\ntechniques or \"sutras\" (aphorisms) to solve mathematical problems in a more straightforward\nand faster way. These techniques are derived from ancient Indian texts known as the Vedas,\nspecifically the \"Atharvaveda\". The system was popularized in the 20th century by Jagadguru\nSwami Bharati Krishna Tirthaji Maharaj, who authored a book titled \"Vedic Mathematics\" in\n1965, which compiled 16 sutras (formulas) and 13 sub-sutras (sub-formulas) that cover a\nwide range of mathematical concepts.\nKey Features of Vedic Mathematics\nSimplicity and Speed: The methods in Vedic Mathematics are designed to be simple and\nenable quick calculations. The concepts are very efficient compared to conventional\nmethods, especially for algebra, geometry, and higher levels.\nVersatility: The techniques can be applied to a variety of mathematical problems, including\naddition, subtraction, multiplication, division, square roots, cube roots, algebraic equations,\nand complex calculations.\nMental Calculation: Vedic Mathematics emphasizes mental math techniques, encouraging"}


created_at: 2025-01-09 14:43:47
topic: persistent://default/default/155c55a6-dc21-4efd-bb5d-885bafc8bd13
id: 2133490f-bb2b-4196-9454-cc141eb295ce
encoding: FLOAT32
metadata:
{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "addition, subtraction, multiplication, division, square roots, cube roots, algebraic equations,\nand complex calculations.\nMental Calculation: Vedic Mathematics emphasizes mental math techniques, encouraging\nthe user to perform calculations in their head without the need for paper or calculators.\nFlexibility: There are often multiple ways to solve a problem using Vedic methods, providing\nflexibility, and encouraging creative approaches.\nPattern Recognition: Vedic Mathematics often involves recognizing patterns and using\nthem to simplify complex calculations. For instance, certain multiplication techniques involve\nspecific patterns that reduce the complexity of the calculation.\nVedic Math Example:\nOne Vedic Mathematics technique is called \u201cEkadhikena Purvena,\u201d which means \u201cone more\nthan the previous one.\u201d This is particularly useful for finding squares of numbers ending in 5.\nFor example, to find the square of 15:\nTake the number before 5, which is 1.\nAdd 1 to it, so 1 + 1 = 2.\nMultiply this sum by the original number before 5, so 2 x 1 = 2.\nWrite 25 at the end. The result is 225.\nSo, 152=225.\nPage 11 of 113"}


created_at: 2025-01-09 14:43:47
topic: persistent://default/default/155c55a6-dc21-4efd-bb5d-885bafc8bd13
id: 278e8131-09df-4f49-ae59-d6dcd1bde800
encoding: FLOAT32
metadata:
{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nSingapore Mathematics\nSingapore Mathematics, similar to Vedic Mathematics, focuses on efficiency and\nunderstanding and is a teaching method that originates from the national mathematics\ncurriculum of Singapore. It is recognized for its clear and structured approach which aims to\ndevelop student\u2019s mathematical abilities and problem-solving skills. Here is a brief overview:\nConceptual Understanding: Singapore Math emphasizes a deep understanding of core\nmathematical concepts rather than rote memorization of procedures.\nCPA Approach: It implies the Concrete-Pictorial-Abstract (CPA) approach, where students\nfirst use tangible objects to understand a concept (Concrete), then represent these concepts\nwith pictures or diagrams (Pictorial), and finally use mathematical symbols to express the\nconcepts (Abstract).\nProblem Solving: Students are taught to solve problems in several ways, using model\ndrawing, number bonds, and mental calculations.\nMastery: The curriculum is designed for students to achieve mastery of a topic before\nmoving on to more complex ideas.\nBoth Vedic and Singapore Mathematics encourage mental calculation and a deeper"}


created_at: 2025-01-09 14:43:47
topic: persistent://default/default/155c55a6-dc21-4efd-bb5d-885bafc8bd13
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "drawing, number bonds, and mental calculations.\nMastery: The curriculum is designed for students to achieve mastery of a topic before\nmoving on to more complex ideas.\nBoth Vedic and Singapore Mathematics encourage mental calculation and a deeper\nunderstanding of mathematical principles, but they differ in their origins, cultural context, and\nspecific techniques used for teaching and learning mathematics. Singapore Math is adopted\nin various countries for its effectiveness in teaching mathematics to young learners.\nSingapore Math Example:\nSingapore Mathematics often uses model drawing to solve word problems. For instance, if a\nproblem states that John has 20 apples and gives away 8, how many does he have left? A\nSingapore Math approach would be to draw a bar model:\nDraw a bar and split it into two parts.\nLabel one part with 20 (total apples) and the other part with 8 (apples given away).\nSubtract the second part from the first to find the answer.\nJohn has 20\u22128=12\napples left.\nPage 12 of 113"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nMental Mathematics:\nMental Mathematics, also known as mental math or mental arithmetic, is the ability to perform\nmathematical calculations in your head without the use of paper, pencil, or a calculator. It\ninvolves using various techniques and strategies to solve math problems quickly and\naccurately. Here are some key aspects of Mental Mathematics:\nNumber Sense: Developing a strong understanding of numbers and their relationships.\nMental Strategies: Using techniques like breaking down problems, rounding numbers, and\nusing known facts to simplify calculations.\nPractice: Regular practice to improve speed and accuracy in mental calculations.\nApplications: Useful in everyday situations like calculating tips, making quick estimates,\nand solving problems on the spot.\nMental Mathematics is similar to Vedic and Singapore Mathematics and it emphasizes\nmental agility and efficient problem-solving. It is a valuable skill that can enhance overall\nmathematical understanding and confidence.\nMental Math Example:\nA mental math strategy for multiplication is to break numbers into easier chunks. For\nexample, to multiply 14 by 5:\nMultiply 10 by 5 to get 50.\nMultiply 4 by 5 to get 20."}


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topic: persistent://default/default/155c55a6-dc21-4efd-bb5d-885bafc8bd13
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "mathematical understanding and confidence.\nMental Math Example:\nA mental math strategy for multiplication is to break numbers into easier chunks. For\nexample, to multiply 14 by 5:\nMultiply 10 by 5 to get 50.\nMultiply 4 by 5 to get 20.\nAdd the two results together: 50 + 20 = 70.\nSo, 14 \u00d7 5 = 70\nPage 13 of 113"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nImportance of this Book\nMath is indeed a foundational subject with a wide array of benefits. It helps develop critical\nthinking and problem-solving skills, which are essential for academic success but also for\nnavigating everyday life. Here are some ways math skills are applied in daily life:\n1. Critical Thinking: Math teaches logical reasoning and the ability to approach problems\nmethodically.\n2. Problem-Solving: Mathematical problems often require identifying patterns, making\nconnections, and applying known methods to new situations.\n3. Attention to Detail: Solving math problems accurately requires careful attention to detail\nand precision.\n4. Perseverance: Working through challenging problems builds resilience and\npersistence, valuable traits in life.\n5. Practical value: By incorporating math into everyday activities and other academic\nsubjects, kids can see its practical value and develop a stronger appreciation for the\nsubject.\nWho will benefit from this book?\nThis book is designed to benefit a wide range of students, from those who are just beginning\nto learn foundational math concepts to those who have developed a passion for mathematics"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "subject.\nWho will benefit from this book?\nThis book is designed to benefit a wide range of students, from those who are just beginning\nto learn foundational math concepts to those who have developed a passion for mathematics\nand are looking to prepare for math competitions. It provides a comprehensive and structured\napproach to learning that caters to the diverse needs and abilities of young learners.\nFor Beginners\nFor children at the initial stages of their mathematical journey, this book offers an engaging\nand accessible introduction to essential math concepts such as Vedic Mathematics,\nSingapore Math, and Mental Math.\nFor Advanced Learners\nFor students who have already developed a strong interest in mathematics, this book serves\nas a valuable resource to deepen their understanding and enhance their skills. Advanced\nlearners will find challenging problems and sophisticated techniques that push the\nboundaries of their knowledge and abilities.\nPreparing for Math Competitions\nThis book is an indispensable tool for students aspiring to excel in math competitions. It\nincludes a variety of problem-solving strategies, tips, and tricks that are commonly used in"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "Preparing for Math Competitions\nThis book is an indispensable tool for students aspiring to excel in math competitions. It\nincludes a variety of problem-solving strategies, tips, and tricks that are commonly used in\ncompetitive math. The exercises and problems are designed to simulate the types of\nquestions encountered in competitions, providing students with ample practice and\nexposure.\nPage 14 of 113"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nHow to Use Elementary Mathematics: Core Skills Book I\nElementary Mathematics: Core Skills Book I serves as an essential guide, introducing\nfundamental tricks that lay the groundwork for advanced mathematical understanding. The\nbook is meticulously structured to ensure that children can grasp the underlying concepts\nand also learn how to effectively solve mathematical problems.\nThe book is divided into sections, with each section dedicated to explaining concepts that\nare either similar or help solve similar types of problems. This organization allows for a\ncoherent and logical progression through various mathematical topics, making it easier for\nstudents to build on their knowledge as they advance.\nEach section encompasses multiple tricks, each designed to simplify and elucidate complex\nmathematical problems.\nTo reinforce learning, every trick is accompanied by two examples. These examples provide\na visual and practical understanding of the trick, demonstrating its application and\neffectiveness.\nTo ensure that students can apply the concepts independently, each trick is followed by five\npractice problems. These problems are carefully crafted to reinforce the trick and provide"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "effectiveness.\nTo ensure that students can apply the concepts independently, each trick is followed by five\npractice problems. These problems are carefully crafted to reinforce the trick and provide\nample practice.\nAt the end of each section, there is a comprehensive quiz covering all the tricks taught within\nthat section. This quiz serves as an assessment tool, allowing students to review and\nconsolidate their understanding of the material.\nAdditional Resources\nAnswer sheets for all practice problems and section quizzes are provided at the end of the\nbook. For further convenience and support, these answer sheets are also available on the\nVibba website (Vibba.com). This dual availability ensures that students and educators have\neasy access to the answers, facilitating a smooth learning and teaching experience.\nIn conclusion, Elementary Mathematics: Core Skills Book I is an invaluable resource for\nyoung learners. Its structured approach, clear explanations, and ample practice opportunities\nensure that students develop a solid foundation in mathematics. By mastering the tricks and\nconcepts presented in this book, students will be well-prepared for more advanced\nmathematical studies and problem-solving challenges.\nPage 15 of 113"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nHow to use this Book series\nThis Elementary Mathematics book focuses on Core skills and tricks required to get started.\nThis book is our first book to get started on the mathematics tricks series. Total of 5 books\nseries.\nIntermediate Competent Advanced\nCore Skills Expert Skills\nTricks Focus area Skills Skills Skills\n(4th Grade) (8th Grade)\n(5th Grade) (6th Grade) (7th Grade)\nAddition\nSubtraction\nMultiplication\nDivision\nPower\nSquare Root\nTable\nFraction\nIntegers\nComplement\nAverage\nFactors\nUnion\nSet Theory\nQuadratic\nMidpoint\nTriangular\nBase\nSequence\nConversion\nLogical Reasoning\nDiagonals\nEach book will give an easy step-by-step approach so that you can understand and\nmemorize how to solve these Mathematics tricks. On average each book will have 25 ~ 35\ntricks covering various Mathematics topics with many different and easy-to-master tricks from\nVedic Math, Singapore Math, and Mental Math areas.\nPage 16 of 113"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nPath to Expert\nThis Elementary Mathematics Book: Core Skills is the starting point for a kid to excel in\nMath, one of the key components for STEM (Science, Technology, Engineering, and\nMathematics) education. Besides this physical book, you can access our Cloud-based\nWebsite (https://Vibba.com) which provides in-depth online classes for kids who want to\nexcel in math with all these tricks. These tricks will help kids get a head start in TMSCA,\nAMC, Math Counts, and Texas Math League and excel in their educational journey.\nPage 17 of 113\n\nELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nSection 1\n1-A1: Addition using Mental MATH\nWhen:\nAdding complex two or three-digit numbers\nHow:\n1. Add units digits.\n2. Add tens digits plus the remainder if any\n3. Add hundreds digits plus any remainder if any\nVisual Flow:\nStart\nProblem: 576 + 687 =?\nStep 1: Add units digits\n6 + 7 = 13\nWrite down 3, carry over 1\nStep 2: Add ten digits\n7 + 8 = 15\nStep 3: Add carry 1\n15 + 1 = 16\nWrite down 6, carry over 1\nStep 4: Add Hundreds Digits\n5 + 6 = 11\nPage 18 of 113"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nStep 5: Add carry 1\n11 + 1 = 12\nWrite down 12\nAnswer: 576 + 687 = 1263\nEnd\nFigure 1: 1-A1: Addition using Mental MATH Visual Flow\nExample and Practice Questions\nExample 1 Practice Questions\nProblem: 576 + 687 =?\nStep 1: 6 + 7 = 13\nStep 2: 7 + 8 = 15 + 1 = 16\n1. 457 + 232 =\nStep 3: 5 + 6 = 11 + 1 = 12\nAnswer: 576 + 687 = 1263\n2. 869 + 542 =\n3. 385 + 809 =\nExample 2\nProblem: 576 + 687 =? 4. 215 + 975 =\nStep 1: 6 + 7 = 13\n5. 748 + 932 =\nStep 2: 7 + 8 = 15 + 1 = 16\nStep 3: 5 + 6 = 11 + 1 = 12\nAnswer: 576 + 687 = 1263\nTable 1-A1: Addition using Mental Math\nPage 19 of 113\n\nELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\n1-S1: Subtraction Method 1\nWhen:\nSubtracting from a number that is a multiple of 10\nHow:\n1. Subtract the last digit from 10.\n2. Subtract all other digits from 9.\nVisual Flow:\nStart\nProblem: 1000 - 374\nStep 1: Subtract Units Digit\n10 - 4 = 6\nWrite down 6\nStep 2: Subtract Tens Digit\n9 - 7 = 2\nWrite down 2\nStep 3: Subtract Hundreds Digit\n9 - 3 = 6\nWrite down 6\nAnswer: 1000 - 374 = 626\nEnd\nFigure 2: 1-S1: Subtraction Method 1 Visual Flow\nPage 20 of 113"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nExample and Practice Questions\nExample 1 Practice Questions\nProblem: 1000 - 374\nStep 1: 10 \u2013 4 = 6\nStep 2: 9 \u2013 7 = 2\nStep 3: 9 \u2013 3 = 6 1. 1000 - 785 =\nAnswer: 1000 \u2013 374 = 626\n2. 1000 - 395 =\nExample 2 3. 10000 - 4093 =\nProblem: 10000 \u2013 5748 =?\n4. 10000 - 8237 =\nStep 1: 10 \u2013 8 = 2\nStep 2: 9 \u2013 4 = 5\n5. 10000 - 9833 =\nStep 3: 9 \u2013 7 = 2\nStep 4: 9 \u2013 5 = 4\nAnswer: 10000 \u2013 5748 = 4252\nTable 2:1-S1: Subtraction Method 1\nPage 21 of 113\n\nELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\n1-S2: Subtraction Method 2\nWhen:\nNeed to find the difference between the two numbers.\nHow:\n1. Take the difference of two units digits.\n2. Subtract that from ten.\n3. Take the difference between the tens digits.\n4. Subtract that from 9.\n5. Take the difference between the hundreds digits.\n6. Subtract that from 9.\n7. Keep continuing till the second to last digit\n8. On the last digit, find the difference between the two numbers\n9. Subtract one from that difference\nVisual Flow:\nStart\nProblem: 914 \u2013 325 =?\nStep 1: Find the difference between\nunits digits\n5 - 4 = 1\nStep 2: Subtract the result of step 1\nfrom 10\n10 - 1 = 9\nStep 3: Find the difference between\ntens digits\n2\u2013 1 = 1\nPage 22 of 113"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nStep 4: Subtract the result of step 3\nfrom 9\n9 \u2013 1 = 8\nStep 5: Find the difference between\nhundreds digits\n9 \u2013 3 = 6\nStep 6: Subtract the result of step 5\nfrom 9\n6 \u2013 1 = 5\nAnswer: 914 \u2013 325 = 589\nEnd\nFigure 3: 1-S2: Subtraction Method 2 Visual Flow\nPage 23 of 113\n\nELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nExample and Practice Questions\nExample 1 Practice Questions\nProblem: 743 - 469 =\nStep 1: 9 \u2013 3 = 6\nStep 2: 10 - 6 = 4\nStep 3: 6 \u2013 4 = 2\nStep 4: 9 \u2013 2 = 7\nStep 5: 7 \u2013 4 = 3 (this is a special case 1. 785 \u2013 234 =\nif the top digit is greater than the\nbottom) 2. 395 \u2013 224 =\nStep 6: 3 \u2013 1 = 2\n3. 893 - 228 =\nAnswer: 743 \u2013 469 = 274\nExample 2 4. 658 - 249 =\nProblem: 934 \u2013 325 =\n5. 833 \u2013 287 =\nStep 1: 5 \u2013 4 = 1\nStep 2: 10 \u2013 1 = 9\nStep 3: 3 \u2013 2 = 1\nStep 4: 1 \u2013 1 = 0\nStep 5: 9 \u2013 3 = 6\nAnswer: 934 \u2013 325 = 609\nTable 3:1-S2: Subtraction Method 2\nPage 24 of 113"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\n1-P1: Percentage Trick\nWhen:\nTrying to find the percentage of a number\nHow:\n1. Put the percent on a simpler number (ending in 0 or 5 preferably)\n2. Round up or down if units digit is not 0.\n3. Then find the percentage using a new number.\n4. Subtract the percentage of difference when rounding.\n5. 45% of 90\n6. 90% of 45\n7. (100-10) % of 45\n8. 100% of 45=45\n9. 10% of 45=4.5\n10. 45-4.5=40.5\n11. 45% of 90 = 40.5\nVisual Flow:\nStart\nProblem: 15% of 95\nStep 1: Find 95% of 15\nStep 2: (100 \u2013 5)% = 15\nStep 3: Find 100% of 15\n15\nStep 4: 5% of 15\n.75\nPage 25 of 113"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nStep 5: Subtract Step 5 from Step 3\n15 - 0.75 = 14.25\nAnswer: 15% of 95 = 14.25\nEnd\nFigure 4: 1-P1: Percentage Trick Visual Flow\nExample and Practice Questions\nExample 1 Practice Questions\nProblem: 64% of 50=?\nStep 1: 50% of 64\nStep 2: (100 - 50) % of 64\nStep 3: 100% of 64 = 64\nStep 4: 50% of 64 = 32\n1. 72% of 40 =\nStep 5: 64 \u2013 32 = 32\nAnswer: 64% of 50 = 32\n2. 60% of 80 =\n3. 56% of 95 =\nExample 2\nProblem: 15% of 95=? 4. 89% of 90 =\nStep 1: 95% of 15\n5. 48% of 64 =\nStep 2: (100-5) % of 15\nStep 3: 100% of 15 = 15\nStep 4: 5% of 15 = 0.75\nStep 5: 15 - 0.75 = 14.25\nAnswer: 15% of 95 = 14.25\nTable 4:1-P1: Percentage Trick\nPage 26 of 113"}


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topic: persistent://default/default/155c55a6-dc21-4efd-bb5d-885bafc8bd13
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nSection 1 Order these Steps\nArrange the steps for the following tricks. Please write down the step number in the left\ncolumn to solve this math trick.\n1. Adding two complex two or three-digit numbers:\nStep Numbers Actions\nStep: Add tens digits and put the remainder under the\nhundreds answer\nStep: Add the first hundred digits\nStep: Add the unit digits but put any remainder under the\ntens digit answer\n2. The minuend is a multiple of ten:\nStep Numbers Actions\nStep: Subtract the last digit from 9\nStep: Subtract the last digit from 10\n3. Need to find the difference of two numbers:\nStep Numbers Actions\nStep: Take the difference between the hundred\u2019s digits\nStep: Subtract that from 9\nStep: Take the difference of two units\u2019 digits\nStep: Subtract that from ten\nStep: Take the difference between the ten\u2019s digits\nStep: Subtract that from 9\nStep: Keep continuing\u2026\n4. Trying to find the percentage of a number:\nStep Numbers Actions\nStep: Subtract the last digit from 9\nStep: Subtract the last digit from 10\nPage 27 of 113"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nPractice Quiz:\nThe following quiz comprises 28 questions designed to assess your understanding of the\ntricks taught in Section 1. Begin by identifying the appropriate trick to apply for each\nproblem, and then follow the steps to solve it. Answers are available at the end of the book.\nQuiz\n1. 10000 \u2013 5789 = 15. 63% of 200=\n2. 691 + 418 = 16. 10000 \u2013 8123 =\n3. 85% of 75 = 17. 902 \u2013 388 =\n4. 1000 \u2013 459 = 18. 1658 \u2013 279 =\n5. 52% of 130 = 19. 374 + 586 =\n6. 1000 \u2013 378 = 20. 942 \u2013 378 =\n7. 274 + 963 = 21. 746 \u2013 415 =\n8. 77% of 150 22. 55% of 80 =\n9. 582 + 649 = 23. 10000 \u2013 4890 =\n10. 10000 \u2013 2465 = 24. 48% of 100 =\n11. 849 \u2013 473 = 25. 857 + 671 =\n12. 48% of 85 = 26. 531 \u2013 456 =\n13. 537 + 286 = 27. 1000 \u2013 634 =\n14. 382 \u2013 239 = 28. 391 + 806 =\nTable 5: Section 1 Quiz\nPage 28 of 113"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nMath Wizards: Shakuntala Devi\nShakuntala Devi: That is Incalculable!\nShakuntala Devi, born in 1939, is celebrated for her incredible aptitude in mental\narithmetic, earning her the title of \"Human Calculator.\" Her mathematical\nprowess was first showcased at the age of five, and she quickly became\nrenowned for her ability to perform complex calculations with astonishing speed.\nDevi's talents brought her international recognition, and she performed at various\nprestigious events, impressing audiences with her seemingly supernatural\ncalculation skills.\nOne of Devi's most notable achievements occurred in 1976, when The New York\nTimes reported her remarkable ability to add and multiply large numbers in\nrecord time. She successfully added the numbers 25,842, 111,201,721,\n370,247,830, and 55,511,315, and then multiplied the sum by 9,878, arriving at\na correct total of 5,559,369,456,432 in under twenty seconds. This feat\nhighlighted her extraordinary mental agility and earned her a place among the\nworld's top mathematicians.\nDespite her impressive feats, some of Devi\u2019s records have been met with\nskepticism due to inadequate documentation and the challenging nature of"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "world's top mathematicians.\nDespite her impressive feats, some of Devi\u2019s records have been met with\nskepticism due to inadequate documentation and the challenging nature of\nverifying such rapid calculations. Her most famous claim involved multiplying two\nthirteen-digit numbers in just twenty seconds during a 1980 demonstration at\nImperial College, London. Although her answer of\n18,947,668,177,995,426,773,730 was correct, the conditions of the test have\nbeen debated, with some experts questioning the feasibility of replicating the feat\nunder strict conditions.\nRegardless of the controversies, Shakuntala Devi's legacy remains significant.\nHer ability to perform rapid calculations and her contributions to the field of\nmathematics, including her authorship of a book on mental arithmetic,\nunderscore her exceptional talent. Devi's work continues to inspire and captivate\nmathematicians and enthusiasts around the world, affirming her place as one of\nhistory\u2019s greatest mathematical prodigies.\n\"Mathematics is the most beautiful and most powerful creation\nof the human spirit.\" \u2013 Stefan Banach\nPage 29 of 113"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nSection 2\n2-T1: Table of 9 Trick\nWhen:\nFinding the product of a nine and a number\nHow:\n1. Take the non-nine number and subtract one.\n2. This is your tens digit for the answer.\n3. Subtract the new number from nine.\n4. This is your units digit for the answer.\nVisual Flow:\nStart\nProblem: 9 x 4 =?\nStep 1: Subtract 1 from the second\nnumber\n4 \u2013 1 = 3\nStep 2: Subtract Result from 9\n9 \u2013 3 = 6\nWrite down 6 (units digit)\nStep 3: Combine Results\n36\nAnswer: 9 x 4 =36\nEnd\nFigure 5: 2-T1: Table of 9 Trick\nPage 30 of 113\n\nELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nExample and Practice Questions\nExample 1 Practice Questions\nProblem: 9 x 4 =?\nStep1: 4 \u2013 1 = 3 1. 9 x 3 =\nStep2: 9 \u2013 3 = 6\nAnswer: 9 x 4 = 36 (30 + 6) 2. 9 x 5 =\n3. 9 x 7 =\nExample 2\nProblem: 9 x 12 =? 4. 9 x 13 =\nStep1: 12 \u2013 1 = 11\nStep2: 9 \u2013 11 = -2 5. 9 x 14 =\nAnswer: 9 x 12 = 108 (110 - 2)\nTable 6:2-T1: Table of 9 Trick\nPage 31 of 113"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\n2-M1: Multiply by 99\nWhen:\nMultiplying any two-digit number with 99\nHow:\n1. Subtract 1 from the non-99 number; this is the first two digits of the answer.\n2. Subtract the tens digit of the non-99 number from 9; this is the tens digit of the\nanswer.\n3. Subtract the unit digit of the difference from step 1 from 9; this is the unit digit of the\nanswer.\nVisual Flow:\nStart\nProblem: 68 x 99 =?\nStep 1: Subtract 1 from the non-99\nnumber\n68 \u2013 1 = 67\nWrite down 67 (part of the final\nanswer)\nStep 2: Subtract each digit from 9\n9 \u2013 6 = 3\n9 \u2013 7 =2\nWrite down the remaining part of the\nanswer\nStep 3: Combine Results\n6732\nAnswer: 68 x 99 = 6732\nEnd\nFigure 6:2-M1: Multiply by 99\nPage 32 of 113\n\nELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nExample and Practice Questions\nExample 1 Practice Questions\nProblem: 68 x 99 =?\nStep1: 68 \u2013 1 = 67\n1. 47 x 99 =\nStep2: 9 \u2013 6 = 3\nStep3: 9 \u2013 7 = 2\n2. 53 x 99 =\nAnswer: 68 x 99 = 6732\n3. 76 x 99 =\nExample 2\nProblem: 87 x 99 =? 4. 82 x 99 =\nStep1: 87 \u2013 1 = 86\n5. 94 x 99 =\nStep2: 9 \u2013 8 = 1\nStep3: 9 \u2013 6 = 3\nAnswer: 87 x 99 = 8613\nTable 7:2-M1: Multiply by 99\nPage 33 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\n2-D1: Dividing by 9\nWhen:\nFinding the remainder of a number when the divisor is 9.\nHow:\n1. Add all the digits together.\n2. Divide the sum by 9.\n3. That is the remainder.\nVisual Flow:\nStart\nProblem: 865 / 9 =?\nStep 1: Add digits of the first number\n8 + 6 + 5 = 19\nStep 2: Divide the Sum by 9\n19 / 9 = 2\nWith remainder 1\nAnswer: 865 / 9 = 96 Remainder = 1\nEnd\nFigure 7:2-D1: Dividing by 9\nPage 34 of 113\n\nELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nExample and Practice Questions\nExample 1 Practice Questions\nProblem: 865 / 9 =?\nStep1: 8 + 6 + 5 = 19 1. 573 / 9; r=\nStep2: 19 / 9 = 2 r1\nAnswer: Remainder=1 2. 829 / 9; r=\n3. 999 / 9; r=\nExample 2\nProblem: 2468 / 9 =? 4. 1546 / 9; r=\nStep1: 2 + 4 + 6 + 8 = 20\nStep2: 20 / 9 = 2 r2 5. 4312 / 9; r=\nAnswer: Remainder = 2\nTable 8:2-D1: Dividing by 9\nPage 35 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nSection 2 Order these Steps\nArrange the steps for the following tricks. Please write down the step number in the left\ncolumn to solve this math trick.\n1. Table of 9 Trick:\nStep Numbers Actions\nStep: This is your tens digit for the answer\nStep: Subtract the new number from nine\nStep: Take the non-nine number and subtract one\nStep: This is your units digit for the answer\n2. Multiply by 99:\nStep Numbers Actions\nStep: Subtract the units digit of the difference from step 1 from 9; this\nis the units digit of the answer.\nStep: Subtract the tens digit of the non-99 number from 9; this is the\ntens digit of the answer\nStep: Subtract 1 from the non 99 number; this is the first two digits of\nthe answer\n3. Dividing by 9:\nStep Numbers Actions\nStep: Divide the sum by 9\nStep: Add all the digits together\nStep: That is the remainder\nPage 36 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nSection 2 Quiz\nThe following quiz comprises 22 questions designed to assess your understanding of\nthe tricks taught in Section 2. Begin by identifying the appropriate trick to apply for each\nproblem, and then follow the steps to solve it. Answers are available at the end of the\nbook.\nQuiz\n1. 9 x 11 = 12. 9 x 25=\n2. 732/9; r= 13. 2981/9; r=\n3. 9 x 20 = 14. 56 x 99 =\n4. 39 x 99 = 15. 9 x 10 =\n5. 5702/9; r= 16. 64 x 99=\n6. 9 x 16 = 17. 9x8=\n7. 25 x 99 = 18. 9 x 15 =\n8. 684/9; r= 19. 9 x 2 =\n9. 9 x 6 = 20. 91 x 99 =\n10. 48 x 99 = 21. 9483/9; r=\n11. 4158/9; r= 22. 74 x 99 =\nTable 9: Section 2 Quiz\nPage 37 of 113"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nMath Wizards: Zerah Colburn\nZerah Colburn: Entertaining Calculations\nZerah Colburn, born in 1804 in Vermont, was a celebrated\nAmerican mathematical prodigy known for his exceptional lightning\ncalculation abilities. From an early age, Colburn displayed a\nremarkable aptitude for arithmetic, mastering multiplication tables\nup to 100 before he could even read or write. His extraordinary\ntalent was discovered by his father, who began showcasing young\nZerah's abilities in public performances. These demonstrations\nquickly garnered attention and financial support, enabling Colburn\nto study in Paris and London by the age of six, where he became\ninternationally renowned for his mental arithmetic skills.\nBy the time he was eight years old, Colburn was performing\nlightning calculations across Europe and was hailed as a \"singular\nphenomenon\" in the Annual Register. His mathematical feats\nincluded solving complex problems with incredible speed and\naccuracy. For example, in 1811, he calculated the number of days\nand hours since the beginning of the Christian Era and determined\nthe number of seconds in eleven years in just seconds. His\ntechniques involved breaking down large numbers into smaller"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "and hours since the beginning of the Christian Era and determined\nthe number of seconds in eleven years in just seconds. His\ntechniques involved breaking down large numbers into smaller\ncomponents, which he multiplied mentally to achieve accurate\nresults.\nColburn\u2019s performances were marked by his ability to solve\nchallenging mathematical problems on the spot, demonstrating his\nmastery of mental arithmetic. One of his notable techniques\ninvolved factoring numbers into smaller units to simplify\ncalculations. For instance, he successfully multiplied 21,734 by\n543 by first breaking 543 into 181 and 3, then performing the\nmultiplications in stages to reach the final answer of 11,801,562.\nHis methods, while highly impressive, required significant mental\neffort and skill.\nDespite his early fame, Colburn\u2019s public interest gradually\ndiminished as he grew older. By the age of twenty, he returned to\nthe United States and transitioned to a new career as a Methodist\npreacher. Colburn\u2019s life was cut short when he passed away at the\nage of thirty-five. Reflecting on his extraordinary abilities, Colburn\nnoted that his methods, while complex, allowed him to perform\ncalculations efficiently without the need for pen, ink, or paper. His"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "age of thirty-five. Reflecting on his extraordinary abilities, Colburn\nnoted that his methods, while complex, allowed him to perform\ncalculations efficiently without the need for pen, ink, or paper. His\nlegacy as one of the earliest lightning calculators remains a\ntestament to his exceptional mental arithmetic skills.\n\"Pure mathematics is, in its way, the poetry of logical ideas.\"\n\u2013 Albert Einstein\nPage 38 of 113"}


created_at: 2025-01-09 14:43:47
topic: persistent://default/default/155c55a6-dc21-4efd-bb5d-885bafc8bd13
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nSection 3\n3-M1: Two-Digit Multiplication (FOIL)\nWhen:\nMultiplying two-digit numbers\nHow:\n1. Multiply units\u2019 digits and remember the remainder.\n2. Cross-multiply the tens of one number and units of another; do both ways.\n3. Add both products and the previous remainder.\n4. Remember the tens digit of this number.\n5. Multiply tens digits and add the previous remainder.\n6. Write all digits.\nVisual Flow:\nStart\nProblem: 68 X 57 =?\nStep 1: Multiply units Digit\n8 x 7 = 56\nWrite down 6, carry over 5\nStep 2: Cross multiple units and tens\ndigits\n6 x 7 = 42 and 8 x 5 = 40\nStep 3: Sum of the results from Step\n2 plus carry over\n40 + 42 + 5 = 87\nWrite down 7 and carry over 8\nStep 4: Multiply tens digits\n6 x 5 = 30\nPage 39 of 113"}


created_at: 2025-01-09 14:43:47
topic: persistent://default/default/155c55a6-dc21-4efd-bb5d-885bafc8bd13
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nStep 5: Add results with carryover\n30 + 8 = 38\nWrite down 3\nStep 6: Combine results\n3876\nAnswer: 68 x 57 = 3876\nEnd\nFigure 8:3-M1: Two-Digit Multiplication (FOIL)\nExample and Practice Questions\nExample 1 Practice Questions\nProblem: 68 x 57 =?\nStep1: 8 x 7 = 56\nStep2: 6 x 7 = 42 & 8 x 5 = 40\nStep3: 42 + 40 + 5 = 87\n1. 74 x 43 =\nStep4: 6 x 5 = 30\nStep5: 30 + 8 = 38\n2. 36 x 42 =\nAnswer: 68 x 57 = 3876\n3. 58 x 84 =\nExample 2\nProblem: 14 x 82 =? 4. 24 x 92 =\nStep1: 4 x 2 = 8\nStep2: 1 x 2 = 2 & 4 x 8 = 32 5. 38 x 48 =\nStep3: 2+32=34\nStep4: 1 x 8 = 8 + 3 = 11\nAnswer: 14 x 82 = 1148\nTable 10: 3-M1: Two-Digit Multiplication (FOIL)\nPage 40 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\n3-M2: Multiply by 12\nWhen:\nMultiplying 12 and a number\nHow:\n1. Double the units digit.\n2. Double the tens digit.\n3. Add units digit plus any carryover.\n4. Double the hundreds.\n5. Add the tens digit plus carryover.\n6. The next digit is the hundreds digit plus any carryover.\nVisual Flow:\nStart\nProblem: 65 x 12 =?\nStep 1: Double Units Digit\n5 x 2 = 10\nWrite down 0, carry over 1\nStep 2: Double Tens digit\n6 x 2 = 12\nStep 3: Add Units digit plus carryover\n12 + 5 + 1 = 18\nWrite down 8 and carry over 1\nStep 5: Add tens digit with carryover\n6 + 1 = 7\nPage 41 of 113\n\nELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nStep 6: Combine results\n780\nAnswer: 65 x 12 = 780\nEnd\nFigure 9:3-M2: Multiply by 12\nExample and Practice Questions\nExample 1 Practice Questions\nProblem: 65 x 12 =?\nStep1: 5 x 2 = 10\nStep2: 6 x 2 = 12 & 5 x 1 = 5\nStep3: 12 + 5 + 1 = 18\n1. 32 x 12 =\nStep4: 1 x 6 = 6 + 1 = 7\nAnswer: 65 x 12 = 780\n2. 63 x 12 =\n3. 57 x 12 =\nExample 2\nProblem: 82 x 12 =? 4. 89 x 12 =\nStep1: 2 x 2 = 4\n5. 76 x 12 =\nStep2: 8 x 2 = 16 & 2 x 1 = 2\nStep3: 16 + 2 = 18\nStep4: 8 x 1 = 8 + 1 = 9\nAnswer: 82 x 12 = 984\nTable 11: 3-M2: Multiply by 12\nPage 42 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\n3-M3: Add to 10 Tens Digit Trick\nWhen:\nWhen the sum of both tens digits is 10 and the unit digits are the same\nHow:\n1. Multiply the units digits of each number. This is the last two digits of your answer (If\nthe answer is only one digit add a zero in the tens place)\n2. Multiply the tens digits and add the units digit to the product.\n3. The sum is the first two digits of your answer.\nVisual Flow:\nStart\nProblem: 47 x 67 =?\nStep 1: Multiply units digits\n7 x 7 = 49\nWrite down 49\nStep 2: Multiply tens digits\n4 x 6 = 24\nStep 3: Add results with units digit\n24 + 7 = 31\nStep 4: Combine results\n3149\nAnswer: 47 X 67 = 3149\nEnd\nFigure 10:3-M3: Add to 10 Tens Digit Trick\nPage 43 of 113\n\nELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nExample and Practice Questions\nExample 1 Practice Questions\nProblem: 47 x 67 =?\nStep1: 7 x 7 = 49\n1. 68 x 48 =\nStep2: 47 x 67 = 3149\nAnswer: 4 x 6 = 24 + 7 = 31\n2. 24 x 84 =\n3. 52 x 52 =\nExample 2\nProblem: 33 x 73 =? 4. 35 x 75 =\nStep1: 3 x 3 = 9\n5. 772 x 32 =\nStep2: 3 x 7 = 21 + 3 = 24\nAnswer: 33 x 73 = 2409\nTable 12: 3-M3: Add to 10 Tens Digit Trick\nPage 44 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\n3-M4: Add to 10 Units Digit Trick\nWhen:\nThe tens digits are the same and the units\u2019 digits add to 10.\nHow:\n1. Multiply the units digits. This is the last two digits of the answer.\n2. Add 1 to one of the ten digit.\n3. Multiply the new number with the other tens digit.\n4. This is the first one or two digits of the answer.\nVisual Flow:\nStart\nProblem: 86 x 84 =?\nStep 1: Multiply Units Digit\n6 x 4 = 24\nWrite down 24\nStep 2: Add 1 to the tens digit\n8 + 1 = 9\nStep 3: multiply the result of Step 2\nwith the second tens digit\n9 x 8 = 72\nStep 4: Combine results\n7224\nAnswer: 86 x 84 = 7224\nEnd\nFigure 11:3-M4: Add to 10 Units Digit Trick\nPage 45 of 113\n\nELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nExample and Practice Questions\nExample 1 Practice Questions\nProblem: 86 x 84 =?\nStep1: 6 x 4 = 24\nStep2: 8 x (8 + 1) = 72 1. 42 x 48 =\nAnswer: 86 x 84 = 7224\n2. 65 x 65 =\n3. 12 x 18 =\nExample 2\nProblem: 63 x 67 =? 4. 43 x 47 =\nStep1: 3 x 7 = 21\n5. 23 x 27 =\nStep2: 6 x (6 + 1) = 42\nAnswer: 4221\nTable 13: 3-M4: Add to 10 Units Digit Trick\nPage 46 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nSection 3 Order These Steps\nArrange the steps for the following tricks. Please write down the step number in the left\ncolumn to solve this math trick.\n1. Two-Digit Multiplication (FOIL):\nStep Numbers Actions\nStep: Write all digits\nStep: Multiply units\u2019 digits and remember the remainder\nStep: Cross-multiply the tens of one number and units of another; do\nboth ways\nStep: Remember the tens digit of this number\nStep: Multiply tens digits and add the previous remainder.\nStep: Add both products and the previous remainder.\n2. Multiply by 12:\nStep Numbers Actions\nStep: The next digit is double the tens digit plus the units digit plus\nany carrying\nStep: The first digit is double the units digit\nStep: The next digit is the hundreds digit plus any carrying\nStep: The next digit is double the hundreds plus the tens digit plus\ncarrying\n3. Add to 10 Tens Digit Trick:\nStep Numbers Actions\nStep: The sum is the first two digits of your answer\nStep: Multiply the last digit of each number. This is the last two digits\nof your answer (If the answer is only one digit add a zero in the\ntens place)\nStep: Multiply the tens digits and add the units digit to the product\n4. Add to 10 Units Digit Trick:"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "of your answer (If the answer is only one digit add a zero in the\ntens place)\nStep: Multiply the tens digits and add the units digit to the product\n4. Add to 10 Units Digit Trick:\nStep Numbers Actions\nStep: Multiply the new number with the other tens digit.\nStep: Add 1 to one of the ten digit.\nStep: Multiply the units digits. This is the last two digits of the answer.\nStep: This is the first one or two digits of the answer.\nPage 47 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nSection 3 Quiz\nThe following quiz comprises 28 questions designed to test your understanding of the\ntricks taught in Section 3. Begin by identifying the appropriate trick to apply for each\nproblem, and then follow the steps to solve it. Answers are available at the end of the\nbook.\nQuiz\n1. 24 x 92 = 15. 76 x 12 =\n2. 35 x 75 = 16. 76 x 74 =\n3. 58 x 84 = 17. 12 x 63 =\n4. 32 x 12 = 18. 25 x 25 =\n5. 74 x 43 = 19. 47 x 67 =\n6. 86 x 84 = 20. 65 x 23 =\n7. 25 x 99 = 21. 67 x 63 =\n8. 36 x 42 = 22. 92 x 12 =\n9. 72 x 32 = 23. 43 x 47 =\n10. 85 x 85 = 24. 83 x 88 =\n11. 38 x 48 = 25. 12 x 59 =\n12. 57 x 12 = 26. 89 x 12 =\n13. 24 x 84 = 27. 52 x 52 =\n14. 33 x 73 = 28. 95 x 95 =\nTable 14: Section 3 Quiz\nPage 48 of 113"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nMath Wizards: \u00c9variste Galois\nThe Mathematical Duel of \u00c9variste Galois\n\u00c9variste Galois, born in 1811, was a French mathematician whose\nwork laid the groundwork for modern abstract algebra. Galois was\na prodigious talent, displaying remarkable mathematical abilities\nfrom a young age. Despite his brief life, his contributions to\nmathematics, particularly in the area of group theory, have had a\nprofound and lasting impact. Galois's innovative approach to the\ntheory of equations and algebraic structures fundamentally\nchanged the landscape of mathematics.\nThe dramatic story of Galois is as well-known as his mathematical\nachievements. On May 30, 1832, at the age of twenty, Galois was\nkilled in a duel over a romantic entanglement. The legend has it\nthat the night before his death, he composed his most important\nmathematical theories, anticipating his demise and hoping to leave\na legacy for future mathematicians. In a poignant letter to his friend\nAuguste Chevalier, Galois described his recent discoveries and\nasked that they be reviewed by prominent mathematicians, such\nas Jacobi or Gauss.\nIn reality, the mathematical papers that Galois revised in his final"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "Auguste Chevalier, Galois described his recent discoveries and\nasked that they be reviewed by prominent mathematicians, such\nas Jacobi or Gauss.\nIn reality, the mathematical papers that Galois revised in his final\nhours were actually earlier submissions made when he was just\nseventeen. His contributions had already been recognized by the\nAcademy of Sciences before his death. Galois's work became\nentangled with political activism and personal disputes, leading to\nhis tragic death. His early involvement in radical politics and his\nimprisonment further complicated his life and career,\novershadowing his mathematical genius.\nGalois's assertion that his research would \"halt many savants in\ntheirs\" proved to be accurate. His development of group theory and\nexploration of polynomial equations introduced concepts that are\ncentral to modern algebra. Although his life was cut short, \u00c9variste\nGalois's pioneering work continues to influence and inspire\nmathematicians, affirming his place as one of the most innovative\nand influential figures in the history of mathematics.\n\"Mathematics is the music of reason.\" \u2013 James Joseph\nSylvester\nPage 49 of 113"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nSection 4\n4-M1: Multiplying by 11\nWhen:\nMultiplying 11 with a number\nHow:\n1. The last digit of the answer is the units digit of the non-11 number.\n2. The next digit of the answer is the sum of the tens and units digit of the non-11\nnumber.\n3. The next digit is the sum of the tens and hundreds of digits plus any carry.\n4. The first digit is the hundreds digit plus any carry.\nVisual Flow:\nStart\nProblem: 451 x 11 =?\nStep 1: Write down the units digit of\n451\nThat is 1\nStep 2: Add tens and units digit\n5 + 1 = 6\nWrite down 6 next to 1\nStep 3: Add the next two digits\n4 + 5 = 9\nWrite down 9 next to 61\nStep 4: Combine results\n961\nPage 50 of 113\n\nELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nAnswer: 451 x 11 = 961\nEnd\nFigure 12:4-M1: Multiplying by 11\nExample and Practice Questions\nExample 1 Practice Questions\nProblem: 451 x 11 =?\nStep 1: Units is 1\nStep 2: 5 + 1 = 6\nStep 3: 4 + 5 = 9\n1. 732 x 11=\nAnswer: 451 x 11 = 4961\n2. 486 x 11=\nExample 2 3. 215 x 11=\nProblem: 958 x 11 =?\n4. 349 x 11=\nStep 1: Units is 8.\nStep 2: 5 + 8 = 13 5. 154 x 11=\nStep 3: 9 + 5 = 14 + 1 = 15\nStep 4: 9 + 1 = 10\nAnswer: 958 x 11 = 10538\nTable 15: 4-M1: Multiplying by 11\nPage 51 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\n4-M2: Multiplying by 111\nWhen:\nMultiplying 11 with a number\nHow:\n1. How: The last digit is the units digit of the non-111 number.\n2. The next digit is the sum of the tens and the units digit plus any carry.\n3. The next digit is the sum of the units, tens, and hundreds digits, plus any carry.\n4. The next digit is the sum of the tens and hundreds of digits plus any carry.\n5. The next digit is the hundreds digit plus any carry.\nNOTE: you can use the two-finger method but with three fingers instead\nVisual Flow:\nStart\nProblem: 263 x 111 =?\nStep 1: Write down the units digit of\nnon-111\nWrite down 3\nStep 2: Add tens and units digit\n6 + 3 = 9\nWrite down 9 next to 3\nStep 3: Add hundreds, tens, and\nunits digits and any carryover\n2 + 6 + 3 = 11\nWrite down 1 and carry over 1\nStep 4: Add hundreds, and tens digits\nand any carryover\n2 + 6 = 8\nAdd carry over\n8 + 1 = 9\nPage 52 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nStep 5: Write down hundreds digit\nand any carryover\n2\nStep 6: Combine results\n29193\nAnswer: 263 x 11 = 29193\nEnd\nFigure 13:4-M2: Multiplying by 111\nExample and Practice Questions\nExample 1 Practice Questions\nProblem: 263 x 111 =?\nStep 1: Units is 3\nStep 2: 6 + 3 = 9\nStep 3: 2 + 6 + 3 = 11\nStep 4: 2 + 6 = 8 + 1 = 9 1. 384 x 111=\nAnswer: 263 x 111 = 29193\n2. 567 x 111=\nExample 2\n3. 149 x 111=\nProblem: 986 x 111 =?\n4. 278 x 111=\nStep 1: Units is 6\nStep 2: 8 + 6 = 14 5. 432 x 111=\nStep 3: 9 + 8 + 6 = 23 + 1 = 24\nStep 4: 9 + 8 = 17 + 2 = 19\nStep 5: 9 + 1 = 10\nAnswer: 986 x 111 = 109446\nTable 16: 4-M2: Multiplying by 111\nPage 53 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nSection 4 Order these Steps\nArrange the steps for the following tricks. Please write down the step number in the left\ncolumn to solve this math trick.\n1. Multiplying by 111:\nStep Numbers Actions\nStep: The next digit is the sum of the tens and units digit plus any\ncarry.\nStep: The last digit is the units digit of the non-111 number.\nStep: The next digit is the sum of the units, tens, and hundreds digits\nplus any carry.\nStep: The next digit is the hundreds digit plus any carry.\nStep: The next digit is the sum of the tens and hundreds of digits\nplus any carry.\nSection 4 Quiz\nThe following quiz comprises 16 questions designed to assess your understanding of\nthe tricks taught in Section 4. Begin by identifying the appropriate trick to apply for each\nproblem, and then follow the steps to solve it. Answers are available at the end of the\nbook.\nQuiz\n1. 482 x 111 = 9. 102 x 11 =\n2. 376 x 11 = 10. 482 x 111 =\n3. 697 x 11 = 11. 376 x 11 =\n4. 319 x 111 = 12. 528 x 111 =\n5. 241 x 11 = 13. 743 x 111 =\n6. 237 x 111 = 14. 509 x 11 =\n7. 854 x 11 = 15. 102 x 11 =\n8. 654 x 111 = 16. 468 x 111 =\nPage 54 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nTable 17: Section 4 Quiz\nMath Wizards: Srinivasa Ramanujan\nSrinivasa Ramanujan: Series of \u03c0\nSrinivasa Ramanujan, an Indian mathematician born in 1887 in\nErode, is celebrated for his extraordinary contributions to\nmathematics despite having minimal formal training. His\nmathematical prowess emerged in isolation, with Ramanujan\ndeveloping complex theories and results largely independently. His\nwork was characterized by a profound intuition and a unique\napproach to problems, which led him to formulate numerous\ngroundbreaking results in number theory, continued fractions, and\ninfinite series.\nRamanujan's mathematical genius attracted the attention of the\nBritish mathematician G.H. Hardy, who invited him to Cambridge\nUniversity in 1914. This collaboration proved highly fruitful, with\nRamanujan and Hardy producing significant research that\nenriched the field of mathematics. Ramanujan's work included the\ndiscovery of new results on the partition function, the famous\nRamanujan-Hardy number 1729, and the development of the\nhighly influential Ramanujan series for \u03c0.\nDespite his profound impact on mathematics, Ramanujan's career"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "discovery of new results on the partition function, the famous\nRamanujan-Hardy number 1729, and the development of the\nhighly influential Ramanujan series for \u03c0.\nDespite his profound impact on mathematics, Ramanujan's career\nwas tragically cut short by his early death at the age of 32 in 1920.\nHis work was often ahead of its time, and many of his results were\ninitially met with skepticism until later verified and appreciated.\nRamanujan's legacy endures through his contributions to\nmathematics, which continue to influence and inspire\ncontemporary mathematicians.\nRamanujan's life and work have been the subject of considerable\nstudy and admiration. His story exemplifies the power of innate\ntalent and dedication, and his mathematical results, often derived\nfrom intuition rather than formal proofs, remain a testament to his\nremarkable genius. His work continues to be explored and\nextended, ensuring that his mathematical contributions remain\nrelevant and influential in the field.\n\"In mathematics, the art of proposing a question must be\nheld of higher value than solving it.\" \u2013 Georg Cantor\nPage 55 of 113"}


created_at: 2025-01-09 14:43:47
topic: persistent://default/default/155c55a6-dc21-4efd-bb5d-885bafc8bd13
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nSection 5\n5-M1: Multiply by 25\nWhen\nMultiplying 25 with a number\nHow:\n1. Divide the non-25 number by 4.\n2. Add two zeroes at the end.\nVisual Flow:\nStart\nProblem: 25 x 64 =?\nStep 1: Divide non-25 number by 4\n64 / 4 = 16\nStep 2: Add two zero\nWrite down 1600\nAnswer: 25 x 64 = 1600\nEnd\nFigure 14: 5-M1: Multiply by 25\nPage 56 of 113\n\nELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nExample and Practice Questions\nExample 1 Practice Questions\nProblem: 25 x 64 =?\n1. 72 x 25 =\nStep1: 64 / 4 = 16\nAnswer: 25 x 64 = 1600 2. 16 x 25 =\n3. 25 x 108 =\nExample 2\n4. 44 x 25 =\nProblem: 25 x 0.24 =?\nStep1: 0.24 / 4 = 0.06 5. 2.04 x 25 =\nAnswer: 25 x 0.24 = 6\nTable 18: 5-M1: Multiply by 25\nPage 57 of 113\n\nELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\n5-M2: Multiply by 50\nWhen:\nMultiplying 50 with a number\nHow:\n1. Divide the non-50 number by 2.\n2. Add two zeroes at the end.\nVisual Flow:\nStart\nProblem: 50 x 48 =?\nStep 1: Divide non-50 number by 2\n48 / 2 = 24\nStep 2: Add two zero\nWrite down 2400\nAnswer: 50 x 48 = 2400\nEnd\nFigure 15: 5-M2: Multiply by 50\nPage 58 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nExample and Practice Questions\nExample 1 Practice Questions\nProblem: 50 x 48 =? 1. 50 x 42 =\nStep1: 48 / 2 = 24\n2. 85 x 50 =\nAnswer: 50 x 48 = 2400\nExample 2 3. 64 x 50 =\nProblem: 37 x 50 =? 4. 76 x 50 =\nStep1: 37 / 2 = 18.5\n5. 50 x 98 =\nAnswer: 37 x 50 = 1850\nTable 19: 5-M2: Multiply by 50\nPage 59 of 113\n\nELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\n5-M3: Multiply by 75\nWhen:\nMultiplying 75 with a number\nHow:\n1. Divide the non-75 number by 4.\n2. Multiply the answer by 3.\n3. Add two zeroes.\nVisual Flow:\nStart\nProblem: 48 x 75 =?\nStep 1: Divide non-75 number by 4\n48 / 4 = 12\nStep 2: Multiply the result by 3\n12 x 3 = 36\nStep 3: Add two zero\nWrite down 3600\nAnswer: 48 x 75 = 3600\nEnd\nFigure 16: 5-M3: Multiply by 75\nPage 60 of 113\n\nELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nExample and Practice Questions\nExample 1 Practice Questions\nProblem: 75 x 48 =?\nStep1: 48 / 4 = 12\n1. 75 x 64 =\nStep2: 12 x 3 = 36\nAnswer: 75 x 48 = 3600\n2. 75 x 18 =\nExample 2 3. 82 x 75 =\nProblem: 98 x 75 =?\n4. 34 x 75 =\nStep1: 98 / 4 = 24.5\nStep2: 24.5 x 3 = 72.5 5. 56 x 75 =\nAnswer: 98 x 75 = 7250\nTable 20: 5-M3: Multiply by 75\nPage 61 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\n5-M4: Multiply two numbers ending with 5\nWhen:\nFinding the product of two numbers ending with 5\nHow:\n1. Find the sum of the tens digits.\n2. If the sum is even, then the last two digits of the answer are 25.\n3. If the sum is odd, then the last two digits of the answer are 75.\n4. Find half of the sum (do not worry about any remainder)\n5. Multiply the tens digits.\n6. Add the answers from steps 4 and 5. This is the first two digits of the answer.\nVisual Flow:\nStart\nProblem: 45 x 35 =?\nStep 1: Add tens digit\n4 + 3 = 7\nStep 2: since the result is odd\nWrite down 75\nStep 3: Divide the result of Step 1 by\n2\n7 / 2 = 3\nStep 4: Multiply tens digits\n4 x 3 = 12\nPage 62 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nStep 5: Add the answer from Step 3\nand Step 4\n3 + 12 = 15\nStep 6: Combine results\n1575\nAnswer: 45 x 35 = 1575\nEnd\nFigure 17: 5-M4: Multiply two numbers ending with 5\nExample and Practice Questions\nExample 1 Practice Questions\nProblem: 45 x 35 =?\nStep1: 4 + 3 = 7 (odd)\nStep2: 7 / 2 = 3\nStep3: 4 x 3 = 12\n1. 95 x 45 =\nStep4: 12 + 3 = 15\nAnswer: 45 x 35 = 1575 2. 85 x 65 =\n3. 65 x 25 =\nExample 2\nProblem: 95 x 55 =? 4. 35 x 65 =\nStep1: 9 + 5 = 14 (even)\n5. 95 x 15 =\nStep2: 14 / 2 = 7\nStep3: 9 x 5 = 45\nStep4: 7 + 45 = 52\nAnswer: 95 x 55 = 5225\nTable 21: 5-M4: Multiply two numbers ending with 5\nPage 63 of 113\n\nELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\n5-M5: Multiplying by 15\nWhen:\nFinding the product of a number and 15\nHow:\n1. Divide the non-15 number by 2.\n2. Add the Answer to the non-15 number.\n3. Add a zero at the end.\nVisual Flow\nStart\nProblem: 22 x 15 =?\nStep 1: divide non-15 number by 2\n22 / 2 = 11\nStep 2: Add the result of Step 1 to the\nnon-15 number\n22 + 11 = 33\nStep 3: Add a zero at the end\n330\nAnswer: 22 x 15 = 330\nEnd\nFigure 18: 5-M5: Multiplying by 15\nPage 64 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nExample and Practice Questions\nExample 1 Practice Questions\nProblem: 22 x 15 =?\nStep1: 22 / 2 = 11\nStep2: 11 + 22 = 33\n1. 88 x 15 =\nStep3: 330\nAnswer: 22 x 15 = 330\n2. 96 x 15 =\n3. 12 x 15 =\nExample 2\nProblem: 42 x 15 =? 4. 24 x 15 =\nStep1: 42 / 2 = 21\n5. 76 x 15 =\nStep2: 21 + 42 = 63\nStep3: 630\nAnswer: 42 x 15 = 630\nTable 22:M4: Multiplying by 15\nPage 65 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nSection 5 Order These Steps\nArrange the steps for the following tricks. Please write down the step number in the left\ncolumn to solve this math trick.\n1. Multiply by 25:\nStep Numbers Actions\nStep: Add two zeroes at the end\nStep: Divide the non-25 number by 4\n2. Multiply by 50:\nStep Numbers Actions\nStep: Add two zeroes at the end.\nStep: Divide the non-50 number by 2.\n3. Multiply by 75:\nStep Numbers Actions\nStep: Multiply the answer by 3\nStep: Divide the non-75 number by 4\nStep: Add two zeroes.\n4. Multiply two numbers ending with 5:\nStep Numbers Actions\nStep: If the sum is even, then the last two digits of the\nanswer are 25.\nStep: Find the sum of the tens digits\nStep: Find half of the sum (do not worry about any\nremainder)\nStep: If the sum is odd, then the last two digits of the\nanswer are 75.\nStep: Add the answers from steps 2 and 3. This is the first\ntwo digits of the answer.\nStep: Multiply the tens digits.\n5. Multiply by 15:\nStep Numbers Actions\nStep: Add a zero at the end\nStep: Add the Answer to the non-15 number\nStep: Divide the non-15 number by 2\nPage 66 of 113"}


created_at: 2025-01-09 14:43:47
topic: persistent://default/default/155c55a6-dc21-4efd-bb5d-885bafc8bd13
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nSection 5 Quiz\nThe following quiz comprises 28 questions designed to assess your understanding of\nthe tricks taught in Section 5. Begin by identifying the appropriate trick to apply for each\nproblem, and then follow the steps to solve it. Answers are available at the end of the\nbook.\nQuiz\n1. 25 x 72 = 15. 75 x 34 =\n2. 50 x 42 = 16. 65 x 25 =\n3. 75 x 64 = 17. 25 x 2.04 =\n4. 45 x 35 = 18. 50 x 98 =\n5. 15 x 16 = 19. 15 x 56 =\n6. 50 x 85 = 20. 25 x 81 =\n7. 75 x 18 = 21. 35 x 65 =\n8. 95 x 55 = 22. 50 x 57 =\n9. 25 x 108 = 23. 75 x 20 =\n10. 50 x 64 = 24. 95 x 15 =\n11. 75 x 82 = 25. 25 x 0.56 =\n12. 85 x 65 = 26. 50 x 0.8 =\n13. 25 x 44 = 27. 75 x 92 =\n14. 15 x 76 = 28. 55 x 45 =\nTable 23: 5-M4: Multiply two numbers ending with 5\nPage 67 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nMath Wizards: Alexander Craig Aitken\nA Piece of Pi for Alexander Craig Aitken\nAlexander Craig Aitken (1895\u20131967), a distinguished professor of\nmathematics at the University of Edinburgh, is celebrated for his\nexceptional memory and mental calculation skills. One of his most\nimpressive feats was memorizing the value of \u03c0 (pi) to 1,000\ndecimal places. During a lecture, Aitken demonstrated his\nremarkable ability by reciting the first 250 digits of \u03c0 and then\ncontinuing from the 551st digit for an additional 150 digits, all\nwithout making a single error. His proficiency in handling such a\nvast array of digits highlighted his extraordinary cognitive skills.\nAitken's approach to memorization was unique and differed from\nconventional methods. He employed a technique that he described\nas the antithesis of intense concentration. Instead of focusing\nnarrowly, he broke the digits into manageable chunks of fifty and\nmemorized them using rhythmic patterns. This auditory method\nallowed him to recall extensive sequences of numbers with ease,\nshowcasing his innovative and relaxed approach to mental\ncalculation.\nIn addition to his prowess in memorizing \u03c0, Aitken demonstrated"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "allowed him to recall extensive sequences of numbers with ease,\nshowcasing his innovative and relaxed approach to mental\ncalculation.\nIn addition to his prowess in memorizing \u03c0, Aitken demonstrated\nimpressive mental arithmetic abilities. He was known for quickly\nand accurately performing complex calculations, such as\nmultiplying five-digit numbers. During a demonstration of desk\ncalculators, Aitken astounded onlookers by instantly providing the\ncorrect product of two large numbers, proving his mental\ncalculation skills were as remarkable as his memory.\nIronically, Aitken noticed a decline in his mental calculation abilities\nafter he began using a desk calculator, which he believed might\nsignal the potential extinction of mental calculation skills. Despite\nhis concerns, his legacy remains significant. Aitken\u2019s contributions\nto mathematics and his extraordinary mental feats continue to\ninspire and intrigue mathematicians and enthusiasts, affirming his\nplace as a remarkable figure in the history of mental arithmetic.\n\"Mathematics is not about numbers, equations,\ncomputations, or algorithms: it is about\nunderstanding.\"\n\u2013 William Paul Thurston\nPage 68 of 113"}


created_at: 2025-01-09 14:43:47
topic: persistent://default/default/155c55a6-dc21-4efd-bb5d-885bafc8bd13
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nSection 6\n6-M1: Multiplying two numbers both over 100\nWhen:\nFinding the product of two numbers that are greater than 100.\nHow:\n1. Determine how much each number is over 100.\n2. Multiply the differences; this is the last two digits of the answer.\n3. Pick the larger number and add the difference of the other number. This is the\nfirst three digits of the answer.\nVisual Flow:\nStart\nProblem: 103 x 105 =?\nStep 1: Multiply units digits\n3 x 5 = 15\nStep 2: Add the First number and the\nunits digit of the second number\n105 + 3 = 108\nStep 3: Combine results\n10815\nAnswer: 103 x 105 = 10815\nEnd\nFigure 19: 6-M1: Multiplying two numbers both over 100\nPage 69 of 113\n\nELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nExample and Practice Questions\nExample 1 Practice Questions\nProblem: 103 x 105 =?\nStep1: 3 x 5 = 15\n1. 102 x 104 =\nStep2: 103 + 5 = 108\nAnswer: 103 x 105 = 10815\n2. 105 x 106 =\n3. 103 x 109 =\nExample 2\nProblem: 109 x 112 =? 4. 111 x 104 =\nStep1: 9 x 12 = 108\n5. 112 x 107 =\nStep2: 109 + 12 = 121\nAnswer: 109 x 112 = 12208\nTable 24: 6-M1: Multiplying two numbers both over 100\nPage 70 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\n6-M2: Multiplying two numbers both under 100\nWhen:\nFinding the product of two numbers that are less than 100.\nHow:\n1. Determine how much each number is over 100.\n2. Multiply the differences; this is the last two digits of the answer.\n3. Pick the larger number and subtract the difference of the other number. This is\nthe first three digits of the answer.\nVisual Flow:\nStart\nProblem: 97 x 96 =?\nStep 1: Subtract both numbers from\n100\n100 \u2013 97 = 3\n100 \u2013 96 = 4\nStep 2: Multiply both results of step 1\n3 x 4 = 12\nStep 3: Pick the larger number and\nsubtract the difference from the\nother number\n97 \u2013 4 = 93\nStep 4: Combine results\n9312\nPage 71 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nAnswer: 97 x 96 = 9312\nEnd\nFigure 20: 6-M2: Multiplying two numbers both under 100\nExample and Practice Questions\nExample 1 Practice Questions\nProblem: 97 x 96 =?\nStep1: 100 -97 = 3 & 100 \u2013 96 = 4\nStep2: 3 x 4 = 12\n1. 96 x 93 =\nStep3: 96 \u2013 3 = 93\nAnswer: 97 x 96 = 9312 2. 90 x 94 =\nExample 2 3. 97 x 92 =\nProblem: 94 x 98 =? 4. 95 x 91 =\nStep1: 100 \u2013 94 = 6 & 100 \u2013 98 = 2\n5. 90 x 91 =\nStep2: 6 x 2 = 12\nStep3: 94 \u2013 2 = 92\nAnswer: 94 x 98 = 9212\nTable 25: 6-M2: Multiplying two numbers both under 100\nPage 72 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\n6-M3: Multiplying two numbers over/under 100\nWhen:\nFinding the product of a number greater than 100 and a number less than 100.\nHow:\n1. Determine how much each number is either over or under 100. The number less\nthan 100 should be considered negative.\n2. Multiply the two numbers.\n3. Add the product to 100 (you should be subtracting). This is the last two digits of\nthe answer.\n4. Add the two numbers from step 1.\n5. Add or subtract that number from 99.\nVisual Flow:\nStart\nProblem: 103 x 98 =?\nStep 1: subtract 100 from both\nnumbers\n103 \u2013 100 = 3\n98 - 100 = -2\nStep 2: multiply the results of the last\nstep\n3 x -2 = -6\nStep 3: Add the result of step 2 to\n100\n100 + (-6) = 94\nStep 4: Add the results of the Step 1\n3 + (-2) = 1\nPage 73 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nStep 5: Add the result of step 4 from\n99\n99 + 1 = 100\nStep 5: Combine results\n10094\nAnswer: 103 x 98 = 10094\nEnd\nFigure 21: 6-M3: Multiplying two numbers over/under 100.\nExample and Practice Questions\nExample 1 Practice Questions\nProblem: 103 x 98 =?\nStep1: 3x (-2) = -6\nStep2: 100 + (-6) = 94\nStep3: 3 + (-2) = 1\n1. 106 x 94 =\nStep4: 99 + 1 = 100\n2. 104 x 90 =\nAnswer: 103 x 98 = 10094\n3. 108 x 91 =\nExample 2\n4. 92 x 109 =\nProblem: 108 x 92 =?\nStep1: 8x (-8) = -64 5. 111 x 93 =\nStep2: 100 + (-64) = 32\nStep3: 8 + (-8) = 0\nStep4: 99 + 0 = 99\nAnswer: 108 x 92 = 9932\nTable 26: 6-M3: Multiplying two numbers over/under 100\nPage 74 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nSection 6 Order these Steps\nArrange the steps for the following tricks. Please write down the step number in the left\ncolumn to solve this math trick.\n1. Multiplying two numbers both over 100:\nStep Numbers Actions\nStep: Pick the larger number and add the difference of the\nother number. This is the first three digits of the\nanswer.\nStep: Determine how much each number is over 100\nStep: Multiply the differences; this is the last two digits of\nthe answer.\n2. Multiplying two numbers both under 100:\nStep Numbers Actions\nStep: Multiply the differences; this is the last two digits of\nthe answer.\nStep: Determine how much each number is over 100\nStep: Pick the larger number and add the difference of the\nother number. This is the first three digits of the\nanswer.\n3. Multiplying two numbers over/under 100:\nStep Numbers Actions\nStep: Add or subtract that number from 99\nStep: Add the two numbers from step 1.\nStep: Add the product to 100 (you should be subtracting).\nThis is the last two digits of the answer.\nStep: Multiply the two numbers\nStep: Determine how much each number is either over or\nunder 100. The number less than 100 should be\nconsidered negative\nPage 75 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nSection 6 Quiz\nThe following quiz comprises 24 questions designed to assess your understanding of\nthe tricks taught in Section 6. Begin by identifying the appropriate trick to apply for each\nproblem, and then follow the steps to solve it. Answers are available at the end of the\nbook.\nQuiz\n1. 102 x 107= 13. 103 x 113 =\n2. 85 x 92 = 14. 112 x 119 =\n3. 105 x 98 = 15. 89 x 93 =\n4. 104 x 110 = 16. 76 x 84 =\n5. 98 x 87 = 17. 97 x 95 =\n6. 110 x 87 = 18. 92 x 88 =\n7. 115 x 120 = 19. 108 x 92 =\n8. 106 x 108 = 20. 115 x 89 =\n9. 94 x 96 = 21. 103 x 94 =\n10. 101 x 95 = 22. 112 x 91 =\n11. 111 x 115 = 23. 109 x 87 =\n12. 120 x 125 = 24. 94 x 96 =\nTable 27: Section 6 Quiz\nPage 76 of 113"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nMath Wizards: Thomas Fuller\nThomas Fuller: Learned Men and Great Fools\nThomas Fuller, born in Africa in 1710, stands out as a remarkable\nfigure in the history of mental arithmetic despite the severe\nlimitations imposed on him by slavery and illiteracy. As an enslaved\nindividual in Virginia, Fuller had no access to formal education. Yet,\nhe demonstrated an extraordinary ability for numerical\ncalculations, teaching himself to count and developing a unique\naptitude for arithmetic through practical experiences. He would\ncount everyday items such as grains of wheat, seeds, and even\nhairs in a cow\u2019s tail, which honed his skills in mental computation.\nFuller\u2019s mathematical talents were not confined to simple counting.\nHe applied his skills to more complex problems, such as\ncalculating the number of shingles needed for roofing or the posts\nrequired for fencing. His proficiency in handling these calculations\ncaught the attention of others, leading to challenges from two\nPennsylvanians who tested his abilities with sophisticated\nproblems. For instance, he correctly solved a complex exponential\ngrowth problem involving sows and pigs, arriving at the accurate"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "Pennsylvanians who tested his abilities with sophisticated\nproblems. For instance, he correctly solved a complex exponential\ngrowth problem involving sows and pigs, arriving at the accurate\ntotal of 34,588,806 within ten minutes.\nDespite his lack of formal education, Fuller\u2019s arithmetic prowess\nwas widely recognized. Upon his death in 1790, the Columbian\nCentinel reported that he could perform calculations with\nremarkable speed and accuracy, often surpassing the capabilities\nof many educated individuals using writing tools. Fuller\u2019s ability to\nquickly provide answers to intricate problems demonstrated not\nonly his natural talent but also his remarkable mental agility.\nFuller\u2019s response to the lack of formal education was both insightful\nand philosophical. When asked if he regretted not receiving a\ntraditional education, he expressed contentment with his self-\ntaught skills, remarking that \u201cmany learned men be great fools.\u201d\nHis story reflects his remarkable resilience and intellectual\nprowess, illustrating that exceptional mental abilities can emerge\neven in the face of significant social and educational\ndisadvantages. Thomas Fuller\u2019s legacy endures as a testament to\nhis extraordinary mental arithmetic skills and his ability to"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "prowess, illustrating that exceptional mental abilities can emerge\neven in the face of significant social and educational\ndisadvantages. Thomas Fuller\u2019s legacy endures as a testament to\nhis extraordinary mental arithmetic skills and his ability to\novercome formidable obstacles.\n\"Do not worry too much about your difficulties in\nmathematics, I can assure you mine are still greater.\u201d\n\u2013 Albert Einstein\nPage 77 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nSection 7\n7-M1: Multiplying Two digits number by 101\nWhen:\nFinding the product of a number and 101\nHow:\n1. Recognize that multiplying by 101.\n2. Write the number twice.\nVisual Flow :\nStart\nProblem: 22 x 101 =?\nStep 1: Write non 101 twice\n2222\nAnswer: 22 x 101 = 2222\nEnd\nFigure 22: 7-M1: Multiplying Two digits number by 101\nExample and Practice Questions\nExample 1 Practice Questions\n1. 23 x 101=\nProblem: 22 x 101 =?\nAnswer: 22 x 101 = 2222\n2. 12 x 101 =\nExample 2 3. 56 x 101 =\n4. 35 x 101 =\nProblem: 43 x 101 =?\nAnswer: 43 x 101 = 4343\n5. 46 x 101 =\nTable 28: 7-M1: Multiplying Two digits number by 101\nPage 78 of 113\n\nELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\n7-M2: Multiplying Three digits number by 101\nWhen:\nFinding the product of a number and 101\nHow:\n1. Write the tens and units digit.\n2. Add the hundredth digit to the whole number.\n3. Write the answer of the Step 2 in front of Step 1\nVisual Flow :\nStart\nProblem: 234 x 101 =?\nStep 1: write tens and units digit\n34\nStep 2: Add the hundredth digit to the\nwhole number\n234 + 2 = 236\nStep 3: Combine results\n23634\nAnswer: 263 x 11 = 29193\nEnd\nFigure 23:7-M2: Multiplying Three digits number by 101\nPage 79 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nExample and Practice Questions\nExample 1 Practice Questions\nProblem: 234 x 101 =?\nStep1: 34\nStep2: 234 + 2 = 236\n1. 123 x 101=\nStep3: 23624\nAnswer: 234 x 101 = 23634\n2. 225 x 101 =\n3. 560 x 101 =\nExample 2\nProblem: 346 x 101 =? 4. 356 x 101 =\nStep1: 46\n5. 416 x 101 =\nStep2: 346 + 3 = 349\nStep3: 34946\nAnswer: 346 x 101 = 34946\nTable 29: 7-M2: Multiplying Three digits number by 101\nPage 80 of 113\n\nELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\n7-M3: Multiplying Two digits number by 1001\nWhen:\nFinding the product of a two-digit number and 1001\nHow:\n1. Write the number one time.\n2. Add 0 at the end of the number.\n3. Write the number one more time.\nVisual Flow :\nStart\nProblem: 35 x 1001 =?\nStep 1: write down the non-1001\nnumber once\nWrite down 35\nStep 2: Add zero at the end of step 1\n350\nStep 3: write the number again\n35035\nAnswer: 263 x 11 = 29193\nEnd\nFigure 24:7-M3: Multiplying Two digits number by 1001\nPage 81 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nExample and Practice Questions\nExample 1 Practice Questions\nProblem: 35 x 1001 =?\nStep1: 35\nStep2: 350\n1. 12 x 1001=\nStep3: 35035\nAnswer: 35 x 1001 = 35035\n2. 26 x 1001 =\nExample 2 3. 52 x 1001 =\nProblem: 41 x 1001 =? 4. 37 x 1001 =\nStep1: 41\n5. 89 x 1001 =\nStep2: 410\nStep3: 41041\nAnswer: 41 x 1001 = 41041\nTable 30: 7-M3: Multiplying Two digits number by 1001\nPage 82 of 113\n\nELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\n7-M4: Multiplying Three digits number by 1001\nWhen:\nFinding the product of a three-digit number and 1001\nHow:\n1. Recognize that multiplying by 1001.\n2. Write the number twice.\nVisual Flow:\nStart\nProblem: 465 x 1001 = ?\nStep 2: Add the last two digit\n6 + 3 = write the number twice\nAnswer: 465 x 1001 = 465465\nEnd\nFigure 25:7-M4: Multiplying Three digits number by 1001.\nExample and Practice Questions\nExample 1 Practice Questions\nProblem: 123 x 1001 =? 1. 789 x 1001=\nAnswer: 123 x 1001 = 123123\n2. 822 x 1001 =\nExample 2 3. 567 x 1001 =\nProblem: 465 x 1001 =? 4. 379 x 1001 =\nAnswer: 465 x 1001 = 465465\n5. 987 x 1001 =\nTable 31: 7-M4: Multiplying Three digits number by 1001\nPage 83 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\n7-M5: Multiplying four digits number by 1001\nWhen:\nFinding the product of a four-digit number and 1001\nHow:\n1. Write the hundredth, tens, and units digit.\n2. Add the first digit to the whole number\n3. Write thousand, hundredth, and tens digit.\nVisual Flow :\nStart\nProblem: 3125 x 1001 = ?\nStep 1: Write the hundredth, tens,\nand units digit\nWrite down 125\nStep 2:Add first digit to the whole\nnumber\n3125 + 3 = 3128\nStep 3: Combine results\n3128125\nAnswer: 3125 x 1001 = 3128125\nEnd\nFigure 26:7-M5: Multiplying four digits number by 1001\nPage 84 of 113\n\nELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nExample and Practice Questions\nExample 1 Practice Questions\nProblem: 3125 x 1001 =?\nStep1: 125\n1. 1278 x 1001=\nStep2: 3125 + 3 = 3128\nAnswer: 3125 x 1001 = 3128125\n2. 3626 x 1001 =\n3. 5322 x 1001 =\nExample 2\n4. 3766 x 1001 =\nProblem: 4651 x 1001 =?\nStep1: 651\n5. 1805 x 1001 =\nStep2: 4651 + 4 = 4655\nAnswer: 4651 x 1001 = 4655651\nTable 32: 7-M5: Multiplying four digits number by 1001\nPage 85 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nSection 7 Order these Steps\nArrange the steps for the following tricks. Please write down the step number in the left\ncolumn to solve this math trick.\n1. Multiplying Two digits number by 101:\nStep Numbers Actions\nStep: Write the number twice\nStep: Recognize that multiplying by 101\n2. Multiplying Three digits number by 101:\nStep Numbers Actions\nStep: Write the answer of Step 2 in front of Step 1\nStep: Write the tens and units digit\nStep: Add the hundredth digit to the whole number\n3. Multiplying Two digits number by 1001:\nStep Numbers Actions\nStep: Write the number one more time\nStep: Write the number one time\nStep: Add 0 at the end of the number\n4. Multiplying Three digits number by 1001:\nStep Numbers Actions\nStep: Write the number twice\nStep: Recognize that multiplying by 1001\n5. Multiplying four digits number by 1001:\nStep Numbers Actions\nStep: Write thousand, hundredth and tens digit\nStep: Write thousand, hundredth and tens digit\nStep: Add the first digit to the whole number\nPage 86 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nSection 7 Quiz\nThe following quiz comprises 20 questions designed to assess your understanding of\nthe tricks taught in Section 7. Begin by identifying the appropriate trick to apply for each\nproblem, and then follow the steps to solve it. Answers are available at the end of the\nbook.\nQuiz\n1. 231 x 1001 11. 52 x 101\n2. 345 x 101 12. 2355 x 1001\n3. 64 x 101 13. 34 x 101\n4. 54 x 1001 14. 212 x 101\n5. 124 x 101 15. 580 x 1001\n6. 2122 x 1001 16. 453 x 101\n7. 23 x 1001 17. 341 x 1001\n8. 467 x 101 18. 78 x 1001\n9. 870 x 101 19. 232 x 101\n10. 97 x 101 20. 124 x 101\nTable 33: Section 7 Quiz\nPage 87 of 113"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nMath Wizards: Carl Friedrich Gauss\nCarl Friedrich Gauss: Mathematical Prodigy\nPrince of Mathematicians,\" Gauss's early talent became evident in\nhis childhood. By the age of seven, he had already demonstrated\nexceptional mathematical abilities, famously calculating the sum of\nthe first 100 integers quickly by recognizing the arithmetic series\npattern. This early display of genius foreshadowed his future\nachievements and established him as a prodigy in mathematics.\nGauss's contributions span numerous areas, including number\ntheory, statistics, and astronomy. One of his seminal works, the\n\"Disquisitiones Arithmeticae\" (1801), laid the foundation for\nmodern number theory. In this work, Gauss introduced concepts\nsuch as modular arithmetic and the theory of quadratic residues,\nwhich have had a lasting impact on the field. Additionally, his work\nin statistics and the method of least squares revolutionized data\nanalysis and error minimization, providing tools that are\nfundamental in modern statistical practice.\nIn addition to his theoretical achievements, Gauss made significant\ncontributions to practical mathematics and astronomy. His work on"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "analysis and error minimization, providing tools that are\nfundamental in modern statistical practice.\nIn addition to his theoretical achievements, Gauss made significant\ncontributions to practical mathematics and astronomy. His work on\nthe orbit of the asteroid Ceres, which he successfully predicted\nusing his own method of least squares, demonstrated his\nexceptional skill in both theoretical and applied mathematics.\nGauss also made important advancements in differential\ngeometry, where his work on curvature and the \"Theorema\nEgregium\" established foundational principles that continue to\ninfluence geometric theory.\nGauss's legacy is marked by his profound impact on many\nbranches of mathematics and science. His insights and\nmethodologies have shaped modern mathematical thought and\npractice. Despite his considerable achievements, Gauss remained\nrelatively modest and reclusive, preferring to let his work speak for\nitself. Today, his contributions are celebrated for their depth and\nenduring significance, cementing Carl Friedrich Gauss as a\ntowering figure in the history of mathematics.\n\"The essence of mathematics lies in its freedom.\"\n\u2013 Georg Cantor\nPage 88 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nSection 8\n8-M1: Multiply A number with an Equal Number of 9's\nWhen:\nMultiply two-digit numbers by 99 or a three-digit number by 999\nHow:\n1. Subtract 1 from the number.\n2. Write the Answer on the left-hand side.\n3. Subtract each digit from 99.\n4. Write the answer next to the First answer.\nVisual Flow :\nStart\nProblem: 65 x 99 =?\nStep 1: Subtract 1 from 65\n65 - 1 = 64\nStep 2: The first part of the answer\n64\nStep 3: Subtract step 1 answer from\n99\n99 \u2013 64 = 35\nStep 4: Combine results\n6435\nAnswer: 65 x 99 = 6435\nEnd\nPage 89 of 113\n\nELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nFigure 27:8-M1: Multiply A number with an Equal Number of 9's\nExample and Practice Questions\nExample 1 Practice Questions\nProblem: 65 x 99 =?\nStep1: 65 -1 = 64\nStep2: 64\n1. 78 x 99 =\nStep3: 99 - 64 = 35\nAnswer: 65 x 99 = 6435 2. 29 x 99 =\nExample 2 3. 37 x 99 =\nProblem: 24 x 99 =?\n4. 33 x 99 =\nStep1: 24 \u2013 1 = 23\nStep2: 23 5. 49 x 99 =\nStep3: 99 \u2013 23 = 76\nAnswer: 24 x 99 = 2376\nTable 34: 8-M1: Multiply A number with an Equal Number of 9's\nPage 90 of 113"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\n8-M2: Multiply the number of digits less than the\nnumber of 9s\nWhen:\nMultiple the number of digits is less than the number of 9s (22 x 999 or 234 x 9999)\nHow:\n1. Add a zero in front of the number.\n2. First subtract 1 from the number\n3. Subtract each digit of the result from 9.\nVisual Flow :\nStart\nProblem: 23 x 999 =?\nStep 1: add a zero in front of the\nnumber\nWrite down 023\nStep 2: Subtract 1 from the number\n23 - 1 = 22\nStep 3: Subtract each digit from 9\n9 \u2013 0 = 9\n9 \u2013 2 = 7\n9 \u2013 2 = 7\nStep 4: Combine results\n22977\nPage 91 of 113\n\nELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nAnswer: 23 x 999 = 22977\nEnd\nFigure 28:8-M2: Multiply the number of digits less than the number of 9s\nExample and Practice Questions\nExample 1 Practice Questions\nProblem: 23 x 999 =?\nStep1: 023\nStep2: 23 \u2013 1 = 22\nStep3: 9 \u2013 0 = 9\n1. 42 x 999 =\nStep4: 9 \u2013 2 = 7\nStep5: 9 \u2013 2 = 7 2. 67 x 999 =\nAnswer: 23 x 999 = 22977\n3. 34 x 999 =\nExample 2\n4. 52 x 999 =\nProblem: 36 x 999 =?\nStep1: 036 5. 27 x 999 =\nStep2: 36 \u2013 1 = 35\nStep3: 9 \u2013 0 = 9\nStep4: 9 \u2013 3 = 6\nStep5: 9 \u2013 5 = 4\nAnswer: 36 x 999 = 35964\nTable 35: 8-M2: Multiply the number of digits less than the number of 9s\nPage 92 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nSection 8 Order these Steps\nArrange the steps for the following tricks. Please write down the step number in the left\ncolumn to solve this math trick.\n1. Multiply A number with an Equal Number of 9's:\nStep Numbers Actions\nStep: Write the Answer on the left-hand side\nStep: Subtract 1 from the number\nStep: Write the answer next to the First answer\nStep: Subtract each digit from 99\n2. Multiply the number of digits less than the number of 9s:\nStep Numbers Actions\nStep: Subtract each digit of the result from 9\nStep: Add a zero in front of the number\nStep: First subtract 1 from the number\nPage 93 of 113"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nSection 8 Quiz\nThe following quiz comprises 20 questions designed to assess your understanding of\nthe tricks taught in Section 8 Begin by identifying the appropriate trick to apply for each\nproblem, and then follow the steps to solve it. Answers are available at the end of the\nbook.\nQuiz\n1. 45 x 99 11. 76 x 99\n2. 78 x 999 12. 43 X 999\n3. 98 x 99 13. 73 x 99\n4. 76 x 999 14. 28 x 99\n5. 34 X 999 15. 62 X 999\n6. 22 x 99 16. 84 x 99\n7. 56 X 999 17. 37 x 99\n8. 88 x 99 18. 84 X 999\n9. 26 x 99 19. 64 X 999\n10. 54 X 999 20. 48 x 99\nTable 36: Section 8 Quiz\nPage 94 of 113"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nMath Wizards: George Parker Bidder\nGeorge Parker Bidder: The Calculating Engineer\nGeorge Parker Bidder (1806\u20131878) was an extraordinary English\nmathematician and mental calculator who captivated audiences\nwith his remarkable mental arithmetic abilities. Born in a small\nvillage in Devon, England, Bidder\u2019s talents became apparent at a\nyoung age. Despite his lack of formal education, he displayed an\nexceptional aptitude for mental calculations, which he honed\nthrough self-study and practical applications. His abilities attracted\nconsiderable attention, leading to his performances at scientific\nand mathematical societies across Britain.\nBidder\u2019s skills in mental arithmetic were showcased through\nnumerous public demonstrations. He was known for his ability to\nperform complex calculations quickly and accurately, such as\nmultiplying large numbers or extracting square roots. One of his\nmost famous feats involved calculating the cube root of a large\nnumber with astonishing speed, a skill that earned him widespread\nrecognition and acclaim. His performances were not merely\ndemonstrations of raw talent but were underpinned by his deep\nunderstanding of mathematical principles and techniques."}


created_at: 2025-01-09 14:43:47
topic: persistent://default/default/155c55a6-dc21-4efd-bb5d-885bafc8bd13
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "number with astonishing speed, a skill that earned him widespread\nrecognition and acclaim. His performances were not merely\ndemonstrations of raw talent but were underpinned by his deep\nunderstanding of mathematical principles and techniques.\nIn addition to his public demonstrations, Bidder\u2019s mathematical\nabilities earned him a position as a consulting engineer and\nsurveyor. He applied his skills to practical problems, including\ncalculating measurements for engineering projects and providing\nsolutions for various technical challenges. His ability to perform\nmental calculations with precision proved invaluable in his\nprofessional work, further solidifying his reputation as a remarkable\nmathematical mind.\nBidder\u2019s contributions to mathematics and mental calculation were\nsignificant, though his work is less well-known today compared to\nother historical figures. His extraordinary mental feats and practical\napplications of arithmetic demonstrated the potential of the human\nmind to achieve remarkable feats of calculation. George Parker\nBidder remains a celebrated figure in the history of mathematics\nfor his impressive mental arithmetic skills and his ability to apply\nthese skills to real-world problems."}


created_at: 2025-01-09 14:43:47
topic: persistent://default/default/155c55a6-dc21-4efd-bb5d-885bafc8bd13
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "mind to achieve remarkable feats of calculation. George Parker\nBidder remains a celebrated figure in the history of mathematics\nfor his impressive mental arithmetic skills and his ability to apply\nthese skills to real-world problems.\n\"The study of mathematics, like the Nile, begins in\nminuteness but ends in magnificence.\" \u2013 Charles\nCaleb Colton\nPage 95 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nSection 9\n9-A1: Sum of even numbers between 1 and 100\nWhen:\nAdd all even numbers starting with 2 to 100.\nHow:\n1. Divide the last even number by 2.\n2. Multiply the result (n) by n + 1.\nVisual Flow:\nStart\nProblem: Sum of even numbers\nbetween 1 and 20 =?\nStep 1: divide the last even number\nby 2\n20 / 2 = 10\nStep 2: Multiple the result of step 1\nby same number plus 1\n10 x (10 +1) = 110\nAnswer: Sum of even numbers\nbetween 1 and 20 = 110\nEnd\nFigure 29:9-A1: Sum of even numbers between 1 and 100\nPage 96 of 113\n\nELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nExample and Practice Questions\nExample 1 Practice Questions\nProblem: Sum of even numbers\nbetween 1 and 20 =?\nStep1: 20 / 2 = 10\nStep2: 10 x (10 + 1) 1. Sum of even numbers between 1 and\nStep3: 10 x 11 10 =\n2. Sum of even numbers between 1 and\nAnswer: 2 + 4 + 6\u2026+20 = 110\n40 =?\n3. Sum of even numbers between 1 and\nExample 2\n30 =\nProblem: Sum of even numbers 4. Sum of even numbers between 1 and\nbetween 1 and 100 =? 60 =\n5. Sum of even numbers between 1 and\nStep1: 100 / 2 = 50\n80 =\nStep2: 50 x (50 + 1)\nStep3: 50 x 51\nAnswer: 2 + 4 + 6\u2026+100 = 2550\nTable 37: 9-A1: Sum of even numbers between 1 and 100\nPage 97 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\n9-A2: Sum of odd numbers between 1 and 100\nWhen:\nAdd all odd numbers starting 1.\nHow:\n1. Divide the last number by 2.\n2. Multiply the result (n) by n.\nVisual Flow :\nStart\nProblem: Sum of odd numbers\nbetween 1 and 100 =?\nStep 1: Step 1: divide the last even\nnumber by 2\n100 / 2 = 50\nStep 2: Multiple the result of step 1\nby the same number\n50 x 50 = 2500\nAnswer: Sum of odd numbers\nbetween 1 and 100 = 2500\nEnd\nFigure 30:9-A2: Sum of odd numbers between 1 and 100\nPage 98 of 113\n\nELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nExample and Practice Questions\nExample 1 Practice Questions\nProblem: Sum of odd numbers\nbetween 1 and 100 =?\nStep1: 100 / 2 = 50 1. Sum of odd numbers between 1 and\nStep2: 50 x 50 10 =\n2. Sum of odd numbers between 1 and\nAnswer: 3 + 5 + 7\u2026+100 = 2500\n30 =\n3. Sum of odd numbers between 1 and\nExample 2 24 =\n4. Sum of odd numbers between 1 and\nProblem: Sum of odd numbers\n50 =\nbetween 1 and 20 =?\n5. Sum of odd numbers between 1 and\nStep1: 20 / 2 = 10\n80 =\nStep2: 10 x 10\nAnswer: 3 + 5 + 7\u2026+20 = 100\nTable 38: A2: Sum of odd numbers between 1 and 100\nPage 99 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nSection 9 Order these Steps\nArrange the steps for the following tricks. Please write down the step number in the left\ncolumn to solve this math trick.\n1. The sum of even numbers between 1 and 100:\nStep Numbers Actions\nStep: Multiply the result (n) by n + 1\nStep: Divide the last even number by 2\n2. Sum of odd numbers between 1 and 100:\nStep Numbers Actions\nStep: Multiply the result (n) by n\nStep: Divide the last number by 2\nSection 9 Quiz\nThe following quiz comprises 10 questions designed to assess your understanding of\nthe tricks taught in Section 9. Begin by identifying the appropriate trick to apply for each\nproblem, and then follow the steps to solve it. Answers are available at the end of the\nbook.\nQuiz\nSum of even numbers Sum of odd numbers between\n1. 6.\nbetween 1 and 30 1 and 50\nSum of odd numbers between Sum of even numbers\n2. 7.\n1 and 20 between 1 and 80\nSum of even numbers Sum of even numbers\n3. 8.\nbetween 1 and 60 between 1 and 90\nSum of odd numbers between Sum of odd numbers between\n4. 9.\n1 and 30 1 and 90\nThe sum of even numbers Sum of odd numbers between\n5. 10.\nbetween 1 and 100 1 and 60\nPage 100 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nMath Wizards: Aryabhata\nAryabhata: The Value System and ZERO\nAryabhata, an ancient Indian mathematician and astronomer born\nin 476 CE, is one of the most influential figures in early Indian\nmathematics. His seminal work, the Aryabhatiya, written in 499\nCE, is a comprehensive text that covers a wide range of\nmathematical and astronomical topics. Aryabhata introduced and\ndeveloped concepts such as the place value system and zero,\nwhich were foundational for the evolution of mathematics in India\nand beyond.\nIn his Aryabhatiya, Aryabhata made significant contributions to\ntrigonometry, providing sine tables, and developing formulas for\nsolving spherical triangles. He also tackled problems related to\narithmetic and algebra, presenting solutions for linear and\nquadratic equations. Aryabhata's work on the approximation of \u03c0\n(pi) was remarkably accurate for its time, reflecting his advanced\nunderstanding of geometry and numerical methods.\nAryabhata's contributions were not limited to mathematics; he also\nmade important advancements in astronomy. He proposed a\nheliocentric model where the Earth rotates on its axis, explaining\nthe apparent daily motion of celestial bodies. His methods for"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "made important advancements in astronomy. He proposed a\nheliocentric model where the Earth rotates on its axis, explaining\nthe apparent daily motion of celestial bodies. His methods for\ncalculating eclipses and the positions of planets were highly\nadvanced and demonstrated a deep understanding of celestial\nmechanics.\nDespite the passage of centuries, Aryabhata's influence remains\nprofound. His mathematical and astronomical concepts were\nfurther developed by later scholars and have had a lasting impact\non both fields. Aryabhata's work laid the groundwork for future\nmathematical innovations in India and contributed to the global\nmathematical heritage. His legacy is celebrated as a pioneer who\nbridged the ancient and modern worlds of science and\nmathematics.\n\"Without mathematics, there\u2019s nothing you can do.\nEverything around you is mathematics. Everything around you\nis numbers.\" \u2013 Shakuntala Devi\nPage 101 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nAnswer Sheet\nSection 1 Practice\n1-A1 1-S1 1-S2 1-P1\n689 215 551 28.8\n1,411 605 171 48\n1,194 5,907 665 53.2\n1,190 1,763 409 80.1\n1,680 167 546 30.72\nSection 1 Quiz\n1. 4211 (1-S1) 15. 126 (1-P1)\n2. 1109 (1-A1) 16. 1877 (1-S1)\n3. 63.75 (1-P1) 17. 514 (1-S2)\n4. 541 (1-S1) 18. 379 (1-S2)\n5. 67.6 (1-P1) 19. 960 (1-A1)\n6. 622 (1-S1) 20. 564 (1-S2)\n7. 1,237 (1-A1) 21. 331 (1-S2)\n8. 115.5 (1-P1) 22. 44 (1-P1)\n9. 1231 (1-A1) 23. 5110 (1-S1)\n10. 7535 (1-S1) 24. 48 (1-P1)\n11. 376 (1-S2) 25. 1528 (1-A1)\n12. 40.8 (1-P1) 26. 75 (1-S2)\n13. 823 (1-A1) 27. 366 (1-S1)\n14. 143 (1-S2) 28. 1197 (1-A1)\nPage 102 of 113\n\nELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nSection 2 Practice\n2-T1 2-M1 2-D1\n27 4,653 r=6\n45 5,247 r=1\n63 7,524 r=0\n117 8,118 r=7\n126 9,306 r=1\nSection 2 Quiz\n1. 99 (2-T1) 12. 225 (2-T1)\n2. r=3 (2-D1) 13. r=2 (2-D1)\n3. 180 (2-T1) 14. 5544 (2-M1)\n4. 3861 (2-M1) 15. 90 (2-T1)\n5. r=2 (2-D1) 16. 6336 (2-M1)\n6. 144 (2-T1) 17. 72 (2-T1)\n7. 2475 (2-M1) 18. 135 (2-T1)\n8. r=6 (2-D1) 19. 18 (2-T1)\n9. 54 (2-T1) 20. 9009 (2-M1)\n10. 4752 (2-M1) 21. r=6 (2-D1)\n11. r=0 (2-D1) 22. 7326 (2-M1)\nPage 103 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nSection 3 Practice\n3-M1 3-M2 3-M3 3-M4\n3,182 384 3,264 2,016\n1,512 756 2,016 4,225\n4,872 684 2,704 216\n2,208 1,068 2,625 2,021\n1,824 912 2,304 621\nSection 3 Quiz\n1. 2,208 (3-M1) 15. 912 (3-M2)\n2. 2,625 (3-M3) 16. 5,624 (3-M4)\n3. 4,872 (3-M1) 17. 756 (3-M2)\n4. 384 (3-M2) 18. 625 (3-M4)\n5. 3,182 (3-M1) 19. 3,149 (3-M3)\n6. 7,224 (3-M4) 20. 1,495 (3-M1)\n7. 3,264 (3-M3) 21. 4,221 (3-M4)\n8. 1,512 (3-M1) 22. 1,104 (3-M2)\n9. 2,304 (3-M3) 23. 2,021 (3-M4)\n10. 7,225 (3-M4) 24. 7,221 (3-M1)\n11. 1,824 (3-M1) 25. 708 (3-M2)\n12. 684 (3-M2) 26. 1,068 (3-M2)\n13. 2,016 (3-M3) 27. 2,704 (3-M3)\n14. 2,409 (3-M3) 28. 9,025 (3-M4)\nPage 104 of 113\n\nELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nSection 4 Practice\n4-M1 4-M2\n8,052 42,624\n5,346 62,937\n2,365 16,539\n3,839 30,858\n1,694 47,952\nSection 4 Quiz\n1. 53,502 (4-M2) 9. 1,122 (4-M1)\n2. 4,136 (4-M1) 10. 53,502 (4-M2)\n3. 7,667 (4-M1) 11. 4,136 (4-M1)\n4. 35,409 (4-M2) 12. 58,608 (4-M2)\n5. 2,651 (4-M1) 13. 82,473 (4-M2)\n6. 26,307 (4-M2) 14. 5,599 (4-M1)\n7. 9,394 (4-M1) 15. 1,122 (4-M1)\n8. 72,594 (4-M2) 16. 51,948 (4-M2)\nPage 105 of 113"}


created_at: 2025-01-09 14:43:47
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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nSection 5 Practice\n5-M1 5-M2 5-M3 5-M4 5-M5\n1,800 2,100 4,800 4,275 1,320\n400 4,250 1,350 5,525 1,440\n2,700 3,200 6,150 1,625 180\n1,100 3,800 2,550 2,275 360\n51 4,900 4,250 1,425 1,140\nSection 5 Quiz\n1. 1,800 (5-M1) 15. 2,550 (5-M3)\n2. 2,100 (5-M2) 16. 1,625 (5-M4)\n3. 4,800 (5-M3) 17. 51.0 (5-M1)\n4. 1,575 (5-M4) 18. 4,900 (5-M2)\n5. 240 (5-M5) 19. 8,40 (5-M5)\n6. 4,250 (5-M2) 20. 2,025 (5-M1)\n7. 1,350 (5-M3) 21. 2,275 (5-M4)\n8. 5,225 (5-M4) 22. 2,850 (5-M2)\n9. 2,700 (5-M1) 23. 1,500 (5-M3)\n10. 3,200 (5-M2) 24. 1,425 (5-M4)\n11. 6,150 (5-M3) 25. 14.0 (5-M1)\n12. 5,525 (5-M4) 26. 40.0 (5-M2)\n13. 1,100 (5-M1) 27. 6,900 (5-M3)\n14. 1,140 (5-M5) 28. 2,475 (5-M4)\nPage 106 of 113"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nSection 6 Practice\n6-M1 6-M2 6-M3\n10,608 8,928 9,964\n11,130 8,460 9,360\n11,227 8,924 9,828\n11,544 8,645 10,028\n11,984 8,190 10,323\nSection 6 Quiz\n1. 10,914 (6-M1) 13. 11,639 (6-M1)\n2. 7,820 (6-M2) 14. 13,328 (6-M1)\n3. 10,290 (6-M3) 15. 8,277 (6-M2)\n4. 11,440 (6-M1) 16. 6,384 (6-M2)\n5. 8,526 (6-M2) 17. 9,215 (6-M2)\n6. 9,570 (6-M3) 18. 8,096 (6-M2)\n7. 13,800 (6-M1) 19. 9,936 (6-M3)\n8. 11,448 (6-M1) 20. 10,235 (6-M3)\n9. 9,024 (6-M2) 21. 9,682 (6-M3)\n10. 9,595 (6-M3) 22. 10,192 (6-M3)\n11. 12,765 (6-M1) 23. 9,483 (6-M3)\n12. 15,000 (6-M1) 24. 9,024 (6-M2)\nPage 107 of 113"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nSection 7 Practice\n7-M1 7-M2 7-M3 7-M4 7-M5\n2,323 12,423 12,012 789,789 1,279,278\n1,212 22,725 26,026 822,822 3,629,626\n5,656 56,560 52,052 567,567 5,327,322\n3,535 35,956 37,037 379,379 3,769,766\n4,646 42,016 89,089 987,987 1,806,805\nSection 7 Quiz\n1. 231,231 (7-M4) 11. 5,252 (7-M1)\n2. 34,845 (7-M2) 12. 2,357,355 (7-M5)\n3. 6,464 (7-M1) 13. 3,434 (7-M1)\n4. 54,054 (7-M3) 14. 21,412 (7-M2)\n5. 124 x 101 (7-M2) 15. 580,580 (7-M4)\n6. 12,524 (7-M5) 16. 45,753 (7-M2)\n7. 23,023 (7-M3) 17. 341,341 (7-M4)\n8. 47,167 (7-M2) 18. 78,078 (7-M3)\n9. 87,870 (7-M42) 19. 23,432 (7-M2)\n10. 9,797 (7-M1) 20. 12,524 (7-M2)\nPage 108 of 113\n\nELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nSection 8 Practice\n8-M1 8-M2\n7,722 41,958\n2,871 66,933\n3,663 33,966\n3,267 51,948\n4,851 26,973\nSection 8 Quiz\n1. 4,455 (8 \u2013 M1) 11. 7, 524 (8 \u2013 M1)\n2. 77,922 (8 \u2013 M2) 12. 42, 957 (8 \u2013 M2)\n3. 9,702 (8 \u2013 M1) 13. 7, 227 (8 \u2013 M1)\n4. 75,924 (8 \u2013 M2) 14. 2, 772 (8 \u2013 M1)\n5. 33, 966 (8 \u2013 M2) 15. 61, 938 (8 \u2013 M2)\n6. 2,178 (8 \u2013 M1) 16. 8, 316 (8 \u2013 M1)\n7. 55, 944 (8 \u2013 M2) 17. 3, 663 (8 \u2013 M1)\n8. 8, 712 (8 \u2013 M1) 18. 83, 916 (8 \u2013 M2)\n9. 2, 574 (8 \u2013 M1) 19. 63, 936 (8 \u2013 M2)\n10. 53, 946 (8 \u2013 M2) 20. 4, 752 (8 \u2013 M1)\nPage 109 of 113"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nSection 9 Practice\n9-A1 9-A2\n30 25\n420 225\n240 144\n930 625\n1,640 1,600\nSection 9 Quiz\n1. 240 (9 \u2013 A1) 6. 625 (9 \u2013 A2)\n2. 100 (9 \u2013 A2) 7. 1,640 (9 \u2013 A1)\n3. 930 (9 \u2013 A1) 8. 2,070 (9 \u2013 A1)\n4. 225 (9 \u2013 A2) 9. 2,050 (9 \u2013 A2)\n5. 2,550 (9 \u2013 A1) 10. 900 (9 \u2013 A2)\nPage 110 of 113"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nBibliography\nRAPID CALCULATION\nCutler, Ann, and Rudolph McShane. The Trachtenberg Speed System of Basic\nMathematics. New York: Doubleday, 1960.\nDevi, Shakuntala. Figuring: The Joys of Numbers. New York: Basic Books, 1964.\nTirtha, Jagadguru Swami Bharati Krishna, Shankaracharya of Govard-\nhana Pitha. Vedic Mathematics or \u201cSixteen Simple Mathematical For- mulae from the\nVedas.\u201d Banaras, India: Hindu University Press, 1965.\nMenninger, K. Calculator\u2019s Cunning. New York: Basic Books, 1964.\nMEMORY\nLorayne, Harry, and Jerry Lucas. The Memory Book. New York: Ballan- tine Books,\n1974.\nRECREATIONAL MATHEMATICS\nGardner, Martin. Magic and Mystery. New York: Random House, 1956. \u2014\u2014\u2014.\nMathematical Carnival. Washington, D.C.: Mathematical Associa-\ntion of America, 1965.\nStewart, Ian. Game, Set, and Math: Enigmas and Conundrums. New York:\nPenguin Books, 1989.\nADVANCED MATHEMATICS (BY ARTHUR BENJAMIN)\nBenjamin, Arthur T., and Jennifer J. Quinn. Proofs That Really Count: The Art of\nCombinatorial Proof. Washington: Mathematical Association of America, 2003.\nBenjamin, Arthur T., and Kan Yasuda. \u201cMagic \u2018Squares\u2019 Indeed!,\u201d The American\nMathematical Monthly 106, no. 2 (February 1999): 152\u201356.\nUseful Links"}


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{"total_pages": 113, "Title": "Elementary Mathematics: Core Skills", "Author": "Ice Venkatesh", "chroma:document": "ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)\nAbout The Author\nIce Venkatesh, a recognized state medalist, has\nexcelled in her own academic and professional\npursuits. She also mentored numerous students to\nachieve top honors in prestigious mathematics\ncompetitions and examinations. Her unwavering\ndedication, passion, and expertise make her an\ninvaluable asset to the educational community, where\nshe strives to ensure every student receives the\nhighest quality instruction and support needed to\nsucceed.\nIce has combined her profound knowledge of\nSingapore, Vedic, and Mental Math into this\ncomprehensive book series. This initiative is designed\nto provide students with a strong foundation in\nessential mathematical concepts that are crucial for\ntheir academic journey. The first book in the series focuses on the fundamental principles\nof math, aligning with the curriculum requirements and preparing students for competitive\nmath events such as the AMC (American Mathematics Competitions), TMSCA (Texas\nMath and Science Coaches Association), and the Texas Math League.\nIce is helping The Village School as a Teacher\u2019s Assistant (TA) for the mathematics\ndepartment. She loves teaching math, cultivating problem-solving skills, and building the"}


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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Elementary Mathematics:
Core Skills (Book 1)
Mathematics core concepts that enhance computational fluency
for TMSCA and Math Olympiad
By: Ice Venkatesh
Author and Founder of Vibba LLC
Page 1 of 113

ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Published in 2024 by Vibba LLC
https://Vibba.com
Copyright © 2024 by Ice Venkatesh
@Vibba LLC All rights reserved. No part of this publication may be reproduced, stored in a retrieval
system, or transmitted in any form or by any means, electronic, mechanical, photocopying, or
otherwise, without the publisher's prior permission.
Editor: Arti Venkatesh
Page 2 of 113

ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
I am dedicating my first book to my Mathematics Teacher,
Mr. Doug Bell
Mathematics Teacher
THE VILLAGE SCHOOL
A NORD ANGLIA EDUCATION SCHOOL
Houston TX USA
Page 3 of 113


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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
I am dedicating my first book to my Mathematics Teacher,
Mr. Doug Bell
Mathematics Teacher
THE VILLAGE SCHOOL
A NORD ANGLIA EDUCATION SCHOOL
Houston TX USA
Page 3 of 113

ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Acknowledgments
This book would not have been possible without the incredible support
from Village School, whose belief in my vision fueled my journey. I am
deeply grateful to my mom, Arti Venkatesh, for her pivotal role in
typesetting the initial draft and for always encouraging me to chase my
dreams.
A special acknowledgment to those who have inspired and empowered
me to embrace the wonders of mathematics and explore the limitless
possibilities it holds. You sparked my passion for mathematical
adventures, and for that, I am forever thankful.
To my friends at Village School—you are my constant source of
inspiration, reminding me every day of the power of curiosity,
perseverance, and friendship. Together, we are changing the way the
world sees math!
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Table of content
Foreword by Bill Delbrugge .......................................................................................... 8
Foreword by Cindy Proske ........................................................................................... 9
Introduction ................................................................................................................. 10
Importance of this Book .............................................................................................. 14
How to use this Book series ....................................................................................... 16
Path to Expert ............................................................................................................. 17
Section 1 .................................................................................................................... 18
1-A1: Addition using Mental MATH ........................................................................ 18
1-S1: Subtraction Method 1 ................................................................................... 20


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1-A1: Addition using Mental MATH ........................................................................ 18
1-S1: Subtraction Method 1 ................................................................................... 20
1-S2: Subtraction Method 2 ................................................................................... 22
1-P1: Percentage Trick .......................................................................................... 25
Section 1 Order these Steps ................................................................................... 27
Practice Quiz: .......................................................................................................... 28
Math Wizards: Shakuntala Devi .............................................................................. 29
Section 2 .................................................................................................................... 30
2-T1: Table of 9 Trick ............................................................................................. 30
2-M1: Multiply by 99 ............................................................................................... 32


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2-T1: Table of 9 Trick ............................................................................................. 30
2-M1: Multiply by 99 ............................................................................................... 32
2-D1: Dividing by 9 ................................................................................................. 34
Section 2 Order these Steps ................................................................................... 36
Section 2 Quiz ......................................................................................................... 37
Math Wizards: Zerah Colburn ................................................................................. 38
Section 3 .................................................................................................................... 39
3-M1: Two-Digit Multiplication (FOIL) ..................................................................... 39
3-M2: Multiply by 12 ............................................................................................... 41
3-M3: Add to 10 Tens Digit Trick ............................................................................ 43


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3-M2: Multiply by 12 ............................................................................................... 41
3-M3: Add to 10 Tens Digit Trick ............................................................................ 43
3-M4: Add to 10 Units Digit Trick ............................................................................ 45
Section 3 Order These Steps .................................................................................. 47
Section 3 Quiz ......................................................................................................... 48
Math Wizards: Évariste Galois ................................................................................ 49
Section 4 .................................................................................................................... 50
4-M1: Multiplying by 11 ........................................................................................... 50
4-M2: Multiplying by 111 ......................................................................................... 52
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Section 4 Order these Steps ................................................................................... 54
Section 4 Quiz ......................................................................................................... 54
Math Wizards: Srinivasa Ramanujan....................................................................... 55
Section 5 ..................................................................................................................... 56
5-M1: Multiply by 25 ................................................................................................ 56
5-M2: Multiply by 50 ................................................................................................ 58
5-M3: Multiply by 75 ................................................................................................ 60
5-M4: Multiply two numbers ending with 5 ............................................................... 62
5-M5: Multiplying by 15 ............................................................................................ 64
Section 5 Order these Steps ................................................................................... 66


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5-M5: Multiplying by 15 ............................................................................................ 64
Section 5 Order these Steps ................................................................................... 66
Math Wizards: Alexander Craig Aitken .................................................................... 68
Section 6 ..................................................................................................................... 69
6-M1: Multiplying two numbers both over 100 ......................................................... 69
6-M2: Multiplying two numbers both under 100 ....................................................... 71
6-M3: Multiplying two numbers over/under 100 ....................................................... 73
Section 6 Order these Steps ................................................................................... 75
Section 6 Quiz ......................................................................................................... 76
Math Wizards: Thomas Fuller .................................................................................. 77


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Section 6 Quiz ......................................................................................................... 76
Math Wizards: Thomas Fuller .................................................................................. 77
Section 7 ..................................................................................................................... 78
7-M1: Multiplying Two digits number by 101 ........................................................... 78
7-M2: Multiplying Three digits number by 101 ......................................................... 79
7-M3: Multiplying Two digits number by 1001 ......................................................... 81
7-M4: Multiplying Three digits number by 1001 ....................................................... 83
7-M5: Multiplying four digits number by 1001 .......................................................... 84
Section 7 Order these Steps ................................................................................... 86
Section 7 Quiz ......................................................................................................... 87
Math Wizards: Carl Friedrich Gauss ........................................................................ 88


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Section 7 Quiz ......................................................................................................... 87
Math Wizards: Carl Friedrich Gauss ........................................................................ 88
Section 8 ..................................................................................................................... 89
8-M1: Multiply A number with an Equal Number of 9's ............................................ 89
8-M2: Multiply the number of digits less than the number of 9s ............................... 91
Section 8 Order these Steps ................................................................................... 93
Section 8 Quiz ......................................................................................................... 94
Math Wizards: George Parker Bidder ...................................................................... 95
Section 9 ..................................................................................................................... 96
9-A1: Sum of even numbers between 1 and 100 .................................................... 96
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
9-A2: Sum of odd numbers between 1 and 100 ...................................................... 98
Section 9 Order these Steps ................................................................................. 100
Section 9 Quiz ....................................................................................................... 100
Math Wizards: Aryabhata ...................................................................................... 101
Answer Sheet ........................................................................................................... 102
Bibliography .............................................................................................................. 111
Useful Links .............................................................................................................. 111
About The Author ..................................................................................................... 113
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Foreword by Bill Delbrugge
Ice Venkatesh has truly inspired us all with her fantastic new math book. Ice has
masterfully created a book that breaks down difficult concepts into small steps,
helping scholars find success.
Ice has a passion for mathematics and is dedicated to making it accessible to all,
which is truly remarkable. As Benjamin Franklin once said, "An investment in
knowledge pays the best interest." Through this book, Ice has made a powerful
investment, one that will surely inspire countless minds to explore, understand,
and appreciate the beauty of math.
Ice, your ability to simplify complex concepts and present them in a way that
sparks curiosity and understanding is a gift. This book is more than just a
resource; it is a guide that helps learners at all levels unlock the true potential of
mathematics. You have transformed math from something that may seem
intimidating into a tool that empowers people, encouraging them to think critically
and solve problems in new and innovative ways.
Your work will undoubtedly have a lasting impact, providing both students and
educators with the knowledge and confidence they need to succeed. Thank you


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and solve problems in new and innovative ways.
Your work will undoubtedly have a lasting impact, providing both students and
educators with the knowledge and confidence they need to succeed. Thank you
for sharing your love for math with the world and for making this investment in the
minds of future generations.
Educationally Yours,
Bill Delbrugge
Head of School, The Village School
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Foreword by Cindy Proske
Having started the Village Middle School Math Team in 1996, it warms my heart
to see this team not only thriving but producing students like Ice Venkatesh who
is willing to share her tips and tricks with other students wanting to participate in
math competitions.
The hard work and perseverance, it takes to be successful in math competitions
is not an easy task. With Ice’s help, other students will be able to have a roadmap
to guide them to their success. The math and science team not only helps
students solve complex problems, but it gives them a place to form friendships as
well as learn about teamwork, perseverance, and the thrill of competition. These
skills last a lifetime for these competitors and increase their understanding and
skills in the classroom.
I am so proud that Ice Venkatesh will help others embrace the beauty of numbers
and the excitement of the challenge. What YOU Do Matters, Ice!
Best Wishes,
Cindy Proske
Director of Middle School, The Village School
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Introduction
This is a great book for Middle School students who want to excel and enjoy mathematics.
This book will help to improve your problem-solving and critical thinking skills.
To make learning math more engaging and accessible, various innovative methods have
been developed, including Vedic Mathematics, Singapore Mathematics, and Mental
Mathematics. These approaches are taught in numerous countries around the world, each
with unique features and benefits. Core Skills is an extremely easy-to-learn book, filled with
such well-known concepts. Let us understand a little bit more about these concepts.
Page 10 of 113


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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Vedic Mathematics
Vedic Mathematics is an ancient system of Indian mathematics that consists of a set of
techniques or "sutras" (aphorisms) to solve mathematical problems in a more straightforward
and faster way. These techniques are derived from ancient Indian texts known as the Vedas,
specifically the "Atharvaveda". The system was popularized in the 20th century by Jagadguru
Swami Bharati Krishna Tirthaji Maharaj, who authored a book titled "Vedic Mathematics" in
1965, which compiled 16 sutras (formulas) and 13 sub-sutras (sub-formulas) that cover a
wide range of mathematical concepts.
Key Features of Vedic Mathematics
Simplicity and Speed: The methods in Vedic Mathematics are designed to be simple and
enable quick calculations. The concepts are very efficient compared to conventional
methods, especially for algebra, geometry, and higher levels.
Versatility: The techniques can be applied to a variety of mathematical problems, including
addition, subtraction, multiplication, division, square roots, cube roots, algebraic equations,
and complex calculations.
Mental Calculation: Vedic Mathematics emphasizes mental math techniques, encouraging


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addition, subtraction, multiplication, division, square roots, cube roots, algebraic equations,
and complex calculations.
Mental Calculation: Vedic Mathematics emphasizes mental math techniques, encouraging
the user to perform calculations in their head without the need for paper or calculators.
Flexibility: There are often multiple ways to solve a problem using Vedic methods, providing
flexibility, and encouraging creative approaches.
Pattern Recognition: Vedic Mathematics often involves recognizing patterns and using
them to simplify complex calculations. For instance, certain multiplication techniques involve
specific patterns that reduce the complexity of the calculation.
Vedic Math Example:
One Vedic Mathematics technique is called “Ekadhikena Purvena,” which means “one more
than the previous one.” This is particularly useful for finding squares of numbers ending in 5.
For example, to find the square of 15:
Take the number before 5, which is 1.
Add 1 to it, so 1 + 1 = 2.
Multiply this sum by the original number before 5, so 2 x 1 = 2.
Write 25 at the end. The result is 225.
So, 152=225.
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Singapore Mathematics
Singapore Mathematics, similar to Vedic Mathematics, focuses on efficiency and
understanding and is a teaching method that originates from the national mathematics
curriculum of Singapore. It is recognized for its clear and structured approach which aims to
develop student’s mathematical abilities and problem-solving skills. Here is a brief overview:
Conceptual Understanding: Singapore Math emphasizes a deep understanding of core
mathematical concepts rather than rote memorization of procedures.
CPA Approach: It implies the Concrete-Pictorial-Abstract (CPA) approach, where students
first use tangible objects to understand a concept (Concrete), then represent these concepts
with pictures or diagrams (Pictorial), and finally use mathematical symbols to express the
concepts (Abstract).
Problem Solving: Students are taught to solve problems in several ways, using model
drawing, number bonds, and mental calculations.
Mastery: The curriculum is designed for students to achieve mastery of a topic before
moving on to more complex ideas.
Both Vedic and Singapore Mathematics encourage mental calculation and a deeper


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drawing, number bonds, and mental calculations.
Mastery: The curriculum is designed for students to achieve mastery of a topic before
moving on to more complex ideas.
Both Vedic and Singapore Mathematics encourage mental calculation and a deeper
understanding of mathematical principles, but they differ in their origins, cultural context, and
specific techniques used for teaching and learning mathematics. Singapore Math is adopted
in various countries for its effectiveness in teaching mathematics to young learners.
Singapore Math Example:
Singapore Mathematics often uses model drawing to solve word problems. For instance, if a
problem states that John has 20 apples and gives away 8, how many does he have left? A
Singapore Math approach would be to draw a bar model:
Draw a bar and split it into two parts.
Label one part with 20 (total apples) and the other part with 8 (apples given away).
Subtract the second part from the first to find the answer.
John has 20−8=12
apples left.
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Mental Mathematics:
Mental Mathematics, also known as mental math or mental arithmetic, is the ability to perform
mathematical calculations in your head without the use of paper, pencil, or a calculator. It
involves using various techniques and strategies to solve math problems quickly and
accurately. Here are some key aspects of Mental Mathematics:
Number Sense: Developing a strong understanding of numbers and their relationships.
Mental Strategies: Using techniques like breaking down problems, rounding numbers, and
using known facts to simplify calculations.
Practice: Regular practice to improve speed and accuracy in mental calculations.
Applications: Useful in everyday situations like calculating tips, making quick estimates,
and solving problems on the spot.
Mental Mathematics is similar to Vedic and Singapore Mathematics and it emphasizes
mental agility and efficient problem-solving. It is a valuable skill that can enhance overall
mathematical understanding and confidence.
Mental Math Example:
A mental math strategy for multiplication is to break numbers into easier chunks. For
example, to multiply 14 by 5:
Multiply 10 by 5 to get 50.
Multiply 4 by 5 to get 20.


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mathematical understanding and confidence.
Mental Math Example:
A mental math strategy for multiplication is to break numbers into easier chunks. For
example, to multiply 14 by 5:
Multiply 10 by 5 to get 50.
Multiply 4 by 5 to get 20.
Add the two results together: 50 + 20 = 70.
So, 14 × 5 = 70
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Importance of this Book
Math is indeed a foundational subject with a wide array of benefits. It helps develop critical
thinking and problem-solving skills, which are essential for academic success but also for
navigating everyday life. Here are some ways math skills are applied in daily life:
1. Critical Thinking: Math teaches logical reasoning and the ability to approach problems
methodically.
2. Problem-Solving: Mathematical problems often require identifying patterns, making
connections, and applying known methods to new situations.
3. Attention to Detail: Solving math problems accurately requires careful attention to detail
and precision.
4. Perseverance: Working through challenging problems builds resilience and
persistence, valuable traits in life.
5. Practical value: By incorporating math into everyday activities and other academic
subjects, kids can see its practical value and develop a stronger appreciation for the
subject.
Who will benefit from this book?
This book is designed to benefit a wide range of students, from those who are just beginning
to learn foundational math concepts to those who have developed a passion for mathematics


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subject.
Who will benefit from this book?
This book is designed to benefit a wide range of students, from those who are just beginning
to learn foundational math concepts to those who have developed a passion for mathematics
and are looking to prepare for math competitions. It provides a comprehensive and structured
approach to learning that caters to the diverse needs and abilities of young learners.
For Beginners
For children at the initial stages of their mathematical journey, this book offers an engaging
and accessible introduction to essential math concepts such as Vedic Mathematics,
Singapore Math, and Mental Math.
For Advanced Learners
For students who have already developed a strong interest in mathematics, this book serves
as a valuable resource to deepen their understanding and enhance their skills. Advanced
learners will find challenging problems and sophisticated techniques that push the
boundaries of their knowledge and abilities.
Preparing for Math Competitions
This book is an indispensable tool for students aspiring to excel in math competitions. It
includes a variety of problem-solving strategies, tips, and tricks that are commonly used in


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Preparing for Math Competitions
This book is an indispensable tool for students aspiring to excel in math competitions. It
includes a variety of problem-solving strategies, tips, and tricks that are commonly used in
competitive math. The exercises and problems are designed to simulate the types of
questions encountered in competitions, providing students with ample practice and
exposure.
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
How to Use Elementary Mathematics: Core Skills Book I
Elementary Mathematics: Core Skills Book I serves as an essential guide, introducing
fundamental tricks that lay the groundwork for advanced mathematical understanding. The
book is meticulously structured to ensure that children can grasp the underlying concepts
and also learn how to effectively solve mathematical problems.
The book is divided into sections, with each section dedicated to explaining concepts that
are either similar or help solve similar types of problems. This organization allows for a
coherent and logical progression through various mathematical topics, making it easier for
students to build on their knowledge as they advance.
Each section encompasses multiple tricks, each designed to simplify and elucidate complex
mathematical problems.
To reinforce learning, every trick is accompanied by two examples. These examples provide
a visual and practical understanding of the trick, demonstrating its application and
effectiveness.
To ensure that students can apply the concepts independently, each trick is followed by five
practice problems. These problems are carefully crafted to reinforce the trick and provide


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effectiveness.
To ensure that students can apply the concepts independently, each trick is followed by five
practice problems. These problems are carefully crafted to reinforce the trick and provide
ample practice.
At the end of each section, there is a comprehensive quiz covering all the tricks taught within
that section. This quiz serves as an assessment tool, allowing students to review and
consolidate their understanding of the material.
Additional Resources
Answer sheets for all practice problems and section quizzes are provided at the end of the
book. For further convenience and support, these answer sheets are also available on the
Vibba website (Vibba.com). This dual availability ensures that students and educators have
easy access to the answers, facilitating a smooth learning and teaching experience.
In conclusion, Elementary Mathematics: Core Skills Book I is an invaluable resource for
young learners. Its structured approach, clear explanations, and ample practice opportunities
ensure that students develop a solid foundation in mathematics. By mastering the tricks and
concepts presented in this book, students will be well-prepared for more advanced
mathematical studies and problem-solving challenges.
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
How to use this Book series
This Elementary Mathematics book focuses on Core skills and tricks required to get started.
This book is our first book to get started on the mathematics tricks series. Total of 5 books
series.
Intermediate Competent Advanced
Core Skills Expert Skills
Tricks Focus area Skills Skills Skills
(4th Grade) (8th Grade)
(5th Grade) (6th Grade) (7th Grade)
Addition
Subtraction
Multiplication
Division
Power
Square Root
Table
Fraction
Integers
Complement
Average
Factors
Union
Set Theory
Quadratic
Midpoint
Triangular
Base
Sequence
Conversion
Logical Reasoning
Diagonals
Each book will give an easy step-by-step approach so that you can understand and
memorize how to solve these Mathematics tricks. On average each book will have 25 ~ 35
tricks covering various Mathematics topics with many different and easy-to-master tricks from
Vedic Math, Singapore Math, and Mental Math areas.
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Path to Expert
This Elementary Mathematics Book: Core Skills is the starting point for a kid to excel in
Math, one of the key components for STEM (Science, Technology, Engineering, and
Mathematics) education. Besides this physical book, you can access our Cloud-based
Website (https://Vibba.com) which provides in-depth online classes for kids who want to
excel in math with all these tricks. These tricks will help kids get a head start in TMSCA,
AMC, Math Counts, and Texas Math League and excel in their educational journey.
Page 17 of 113

ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Section 1
1-A1: Addition using Mental MATH
When:
Adding complex two or three-digit numbers
How:
1. Add units digits.
2. Add tens digits plus the remainder if any
3. Add hundreds digits plus any remainder if any
Visual Flow:
Start
Problem: 576 + 687 =?
Step 1: Add units digits
6 + 7 = 13
Write down 3, carry over 1
Step 2: Add ten digits
7 + 8 = 15
Step 3: Add carry 1
15 + 1 = 16
Write down 6, carry over 1
Step 4: Add Hundreds Digits
5 + 6 = 11
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Step 5: Add carry 1
11 + 1 = 12
Write down 12
Answer: 576 + 687 = 1263
End
Figure 1: 1-A1: Addition using Mental MATH Visual Flow
Example and Practice Questions
Example 1 Practice Questions
Problem: 576 + 687 =?
Step 1: 6 + 7 = 13
Step 2: 7 + 8 = 15 + 1 = 16
1. 457 + 232 =
Step 3: 5 + 6 = 11 + 1 = 12
Answer: 576 + 687 = 1263
2. 869 + 542 =
3. 385 + 809 =
Example 2
Problem: 576 + 687 =? 4. 215 + 975 =
Step 1: 6 + 7 = 13
5. 748 + 932 =
Step 2: 7 + 8 = 15 + 1 = 16
Step 3: 5 + 6 = 11 + 1 = 12
Answer: 576 + 687 = 1263
Table 1-A1: Addition using Mental Math
Page 19 of 113

ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
1-S1: Subtraction Method 1
When:
Subtracting from a number that is a multiple of 10
How:
1. Subtract the last digit from 10.
2. Subtract all other digits from 9.
Visual Flow:
Start
Problem: 1000 - 374
Step 1: Subtract Units Digit
10 - 4 = 6
Write down 6
Step 2: Subtract Tens Digit
9 - 7 = 2
Write down 2
Step 3: Subtract Hundreds Digit
9 - 3 = 6
Write down 6
Answer: 1000 - 374 = 626
End
Figure 2: 1-S1: Subtraction Method 1 Visual Flow
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Example and Practice Questions
Example 1 Practice Questions
Problem: 1000 - 374
Step 1: 10 – 4 = 6
Step 2: 9 – 7 = 2
Step 3: 9 – 3 = 6 1. 1000 - 785 =
Answer: 1000 – 374 = 626
2. 1000 - 395 =
Example 2 3. 10000 - 4093 =
Problem: 10000 – 5748 =?
4. 10000 - 8237 =
Step 1: 10 – 8 = 2
Step 2: 9 – 4 = 5
5. 10000 - 9833 =
Step 3: 9 – 7 = 2
Step 4: 9 – 5 = 4
Answer: 10000 – 5748 = 4252
Table 2:1-S1: Subtraction Method 1
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
1-S2: Subtraction Method 2
When:
Need to find the difference between the two numbers.
How:
1. Take the difference of two units digits.
2. Subtract that from ten.
3. Take the difference between the tens digits.
4. Subtract that from 9.
5. Take the difference between the hundreds digits.
6. Subtract that from 9.
7. Keep continuing till the second to last digit
8. On the last digit, find the difference between the two numbers
9. Subtract one from that difference
Visual Flow:
Start
Problem: 914 – 325 =?
Step 1: Find the difference between
units digits
5 - 4 = 1
Step 2: Subtract the result of step 1
from 10
10 - 1 = 9
Step 3: Find the difference between
tens digits
2– 1 = 1
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Step 4: Subtract the result of step 3
from 9
9 – 1 = 8
Step 5: Find the difference between
hundreds digits
9 – 3 = 6
Step 6: Subtract the result of step 5
from 9
6 – 1 = 5
Answer: 914 – 325 = 589
End
Figure 3: 1-S2: Subtraction Method 2 Visual Flow
Page 23 of 113

ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Example and Practice Questions
Example 1 Practice Questions
Problem: 743 - 469 =
Step 1: 9 – 3 = 6
Step 2: 10 - 6 = 4
Step 3: 6 – 4 = 2
Step 4: 9 – 2 = 7
Step 5: 7 – 4 = 3 (this is a special case 1. 785 – 234 =
if the top digit is greater than the
bottom) 2. 395 – 224 =
Step 6: 3 – 1 = 2
3. 893 - 228 =
Answer: 743 – 469 = 274
Example 2 4. 658 - 249 =
Problem: 934 – 325 =
5. 833 – 287 =
Step 1: 5 – 4 = 1
Step 2: 10 – 1 = 9
Step 3: 3 – 2 = 1
Step 4: 1 – 1 = 0
Step 5: 9 – 3 = 6
Answer: 934 – 325 = 609
Table 3:1-S2: Subtraction Method 2
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
1-P1: Percentage Trick
When:
Trying to find the percentage of a number
How:
1. Put the percent on a simpler number (ending in 0 or 5 preferably)
2. Round up or down if units digit is not 0.
3. Then find the percentage using a new number.
4. Subtract the percentage of difference when rounding.
5. 45% of 90
6. 90% of 45
7. (100-10) % of 45
8. 100% of 45=45
9. 10% of 45=4.5
10. 45-4.5=40.5
11. 45% of 90 = 40.5
Visual Flow:
Start
Problem: 15% of 95
Step 1: Find 95% of 15
Step 2: (100 – 5)% = 15
Step 3: Find 100% of 15
15
Step 4: 5% of 15
.75
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Step 5: Subtract Step 5 from Step 3
15 - 0.75 = 14.25
Answer: 15% of 95 = 14.25
End
Figure 4: 1-P1: Percentage Trick Visual Flow
Example and Practice Questions
Example 1 Practice Questions
Problem: 64% of 50=?
Step 1: 50% of 64
Step 2: (100 - 50) % of 64
Step 3: 100% of 64 = 64
Step 4: 50% of 64 = 32
1. 72% of 40 =
Step 5: 64 – 32 = 32
Answer: 64% of 50 = 32
2. 60% of 80 =
3. 56% of 95 =
Example 2
Problem: 15% of 95=? 4. 89% of 90 =
Step 1: 95% of 15
5. 48% of 64 =
Step 2: (100-5) % of 15
Step 3: 100% of 15 = 15
Step 4: 5% of 15 = 0.75
Step 5: 15 - 0.75 = 14.25
Answer: 15% of 95 = 14.25
Table 4:1-P1: Percentage Trick
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Section 1 Order these Steps
Arrange the steps for the following tricks. Please write down the step number in the left
column to solve this math trick.
1. Adding two complex two or three-digit numbers:
Step Numbers Actions
Step: Add tens digits and put the remainder under the
hundreds answer
Step: Add the first hundred digits
Step: Add the unit digits but put any remainder under the
tens digit answer
2. The minuend is a multiple of ten:
Step Numbers Actions
Step: Subtract the last digit from 9
Step: Subtract the last digit from 10
3. Need to find the difference of two numbers:
Step Numbers Actions
Step: Take the difference between the hundred’s digits
Step: Subtract that from 9
Step: Take the difference of two units’ digits
Step: Subtract that from ten
Step: Take the difference between the ten’s digits
Step: Subtract that from 9
Step: Keep continuing…
4. Trying to find the percentage of a number:
Step Numbers Actions
Step: Subtract the last digit from 9
Step: Subtract the last digit from 10
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Practice Quiz:
The following quiz comprises 28 questions designed to assess your understanding of the
tricks taught in Section 1. Begin by identifying the appropriate trick to apply for each
problem, and then follow the steps to solve it. Answers are available at the end of the book.
Quiz
1. 10000 – 5789 = 15. 63% of 200=
2. 691 + 418 = 16. 10000 – 8123 =
3. 85% of 75 = 17. 902 – 388 =
4. 1000 – 459 = 18. 1658 – 279 =
5. 52% of 130 = 19. 374 + 586 =
6. 1000 – 378 = 20. 942 – 378 =
7. 274 + 963 = 21. 746 – 415 =
8. 77% of 150 22. 55% of 80 =
9. 582 + 649 = 23. 10000 – 4890 =
10. 10000 – 2465 = 24. 48% of 100 =
11. 849 – 473 = 25. 857 + 671 =
12. 48% of 85 = 26. 531 – 456 =
13. 537 + 286 = 27. 1000 – 634 =
14. 382 – 239 = 28. 391 + 806 =
Table 5: Section 1 Quiz
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Math Wizards: Shakuntala Devi
Shakuntala Devi: That is Incalculable!
Shakuntala Devi, born in 1939, is celebrated for her incredible aptitude in mental
arithmetic, earning her the title of "Human Calculator." Her mathematical
prowess was first showcased at the age of five, and she quickly became
renowned for her ability to perform complex calculations with astonishing speed.
Devi's talents brought her international recognition, and she performed at various
prestigious events, impressing audiences with her seemingly supernatural
calculation skills.
One of Devi's most notable achievements occurred in 1976, when The New York
Times reported her remarkable ability to add and multiply large numbers in
record time. She successfully added the numbers 25,842, 111,201,721,
370,247,830, and 55,511,315, and then multiplied the sum by 9,878, arriving at
a correct total of 5,559,369,456,432 in under twenty seconds. This feat
highlighted her extraordinary mental agility and earned her a place among the
world's top mathematicians.
Despite her impressive feats, some of Devi’s records have been met with
skepticism due to inadequate documentation and the challenging nature of


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world's top mathematicians.
Despite her impressive feats, some of Devi’s records have been met with
skepticism due to inadequate documentation and the challenging nature of
verifying such rapid calculations. Her most famous claim involved multiplying two
thirteen-digit numbers in just twenty seconds during a 1980 demonstration at
Imperial College, London. Although her answer of
18,947,668,177,995,426,773,730 was correct, the conditions of the test have
been debated, with some experts questioning the feasibility of replicating the feat
under strict conditions.
Regardless of the controversies, Shakuntala Devi's legacy remains significant.
Her ability to perform rapid calculations and her contributions to the field of
mathematics, including her authorship of a book on mental arithmetic,
underscore her exceptional talent. Devi's work continues to inspire and captivate
mathematicians and enthusiasts around the world, affirming her place as one of
history’s greatest mathematical prodigies.
"Mathematics is the most beautiful and most powerful creation
of the human spirit." – Stefan Banach
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Section 2
2-T1: Table of 9 Trick
When:
Finding the product of a nine and a number
How:
1. Take the non-nine number and subtract one.
2. This is your tens digit for the answer.
3. Subtract the new number from nine.
4. This is your units digit for the answer.
Visual Flow:
Start
Problem: 9 x 4 =?
Step 1: Subtract 1 from the second
number
4 – 1 = 3
Step 2: Subtract Result from 9
9 – 3 = 6
Write down 6 (units digit)
Step 3: Combine Results
36
Answer: 9 x 4 =36
End
Figure 5: 2-T1: Table of 9 Trick
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Example and Practice Questions
Example 1 Practice Questions
Problem: 9 x 4 =?
Step1: 4 – 1 = 3 1. 9 x 3 =
Step2: 9 – 3 = 6
Answer: 9 x 4 = 36 (30 + 6) 2. 9 x 5 =
3. 9 x 7 =
Example 2
Problem: 9 x 12 =? 4. 9 x 13 =
Step1: 12 – 1 = 11
Step2: 9 – 11 = -2 5. 9 x 14 =
Answer: 9 x 12 = 108 (110 - 2)
Table 6:2-T1: Table of 9 Trick
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
2-M1: Multiply by 99
When:
Multiplying any two-digit number with 99
How:
1. Subtract 1 from the non-99 number; this is the first two digits of the answer.
2. Subtract the tens digit of the non-99 number from 9; this is the tens digit of the
answer.
3. Subtract the unit digit of the difference from step 1 from 9; this is the unit digit of the
answer.
Visual Flow:
Start
Problem: 68 x 99 =?
Step 1: Subtract 1 from the non-99
number
68 – 1 = 67
Write down 67 (part of the final
answer)
Step 2: Subtract each digit from 9
9 – 6 = 3
9 – 7 =2
Write down the remaining part of the
answer
Step 3: Combine Results
6732
Answer: 68 x 99 = 6732
End
Figure 6:2-M1: Multiply by 99
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Example and Practice Questions
Example 1 Practice Questions
Problem: 68 x 99 =?
Step1: 68 – 1 = 67
1. 47 x 99 =
Step2: 9 – 6 = 3
Step3: 9 – 7 = 2
2. 53 x 99 =
Answer: 68 x 99 = 6732
3. 76 x 99 =
Example 2
Problem: 87 x 99 =? 4. 82 x 99 =
Step1: 87 – 1 = 86
5. 94 x 99 =
Step2: 9 – 8 = 1
Step3: 9 – 6 = 3
Answer: 87 x 99 = 8613
Table 7:2-M1: Multiply by 99
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
2-D1: Dividing by 9
When:
Finding the remainder of a number when the divisor is 9.
How:
1. Add all the digits together.
2. Divide the sum by 9.
3. That is the remainder.
Visual Flow:
Start
Problem: 865 / 9 =?
Step 1: Add digits of the first number
8 + 6 + 5 = 19
Step 2: Divide the Sum by 9
19 / 9 = 2
With remainder 1
Answer: 865 / 9 = 96 Remainder = 1
End
Figure 7:2-D1: Dividing by 9
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Example and Practice Questions
Example 1 Practice Questions
Problem: 865 / 9 =?
Step1: 8 + 6 + 5 = 19 1. 573 / 9; r=
Step2: 19 / 9 = 2 r1
Answer: Remainder=1 2. 829 / 9; r=
3. 999 / 9; r=
Example 2
Problem: 2468 / 9 =? 4. 1546 / 9; r=
Step1: 2 + 4 + 6 + 8 = 20
Step2: 20 / 9 = 2 r2 5. 4312 / 9; r=
Answer: Remainder = 2
Table 8:2-D1: Dividing by 9
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Section 2 Order these Steps
Arrange the steps for the following tricks. Please write down the step number in the left
column to solve this math trick.
1. Table of 9 Trick:
Step Numbers Actions
Step: This is your tens digit for the answer
Step: Subtract the new number from nine
Step: Take the non-nine number and subtract one
Step: This is your units digit for the answer
2. Multiply by 99:
Step Numbers Actions
Step: Subtract the units digit of the difference from step 1 from 9; this
is the units digit of the answer.
Step: Subtract the tens digit of the non-99 number from 9; this is the
tens digit of the answer
Step: Subtract 1 from the non 99 number; this is the first two digits of
the answer
3. Dividing by 9:
Step Numbers Actions
Step: Divide the sum by 9
Step: Add all the digits together
Step: That is the remainder
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Section 2 Quiz
The following quiz comprises 22 questions designed to assess your understanding of
the tricks taught in Section 2. Begin by identifying the appropriate trick to apply for each
problem, and then follow the steps to solve it. Answers are available at the end of the
book.
Quiz
1. 9 x 11 = 12. 9 x 25=
2. 732/9; r= 13. 2981/9; r=
3. 9 x 20 = 14. 56 x 99 =
4. 39 x 99 = 15. 9 x 10 =
5. 5702/9; r= 16. 64 x 99=
6. 9 x 16 = 17. 9x8=
7. 25 x 99 = 18. 9 x 15 =
8. 684/9; r= 19. 9 x 2 =
9. 9 x 6 = 20. 91 x 99 =
10. 48 x 99 = 21. 9483/9; r=
11. 4158/9; r= 22. 74 x 99 =
Table 9: Section 2 Quiz
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Math Wizards: Zerah Colburn
Zerah Colburn: Entertaining Calculations
Zerah Colburn, born in 1804 in Vermont, was a celebrated
American mathematical prodigy known for his exceptional lightning
calculation abilities. From an early age, Colburn displayed a
remarkable aptitude for arithmetic, mastering multiplication tables
up to 100 before he could even read or write. His extraordinary
talent was discovered by his father, who began showcasing young
Zerah's abilities in public performances. These demonstrations
quickly garnered attention and financial support, enabling Colburn
to study in Paris and London by the age of six, where he became
internationally renowned for his mental arithmetic skills.
By the time he was eight years old, Colburn was performing
lightning calculations across Europe and was hailed as a "singular
phenomenon" in the Annual Register. His mathematical feats
included solving complex problems with incredible speed and
accuracy. For example, in 1811, he calculated the number of days
and hours since the beginning of the Christian Era and determined
the number of seconds in eleven years in just seconds. His
techniques involved breaking down large numbers into smaller


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and hours since the beginning of the Christian Era and determined
the number of seconds in eleven years in just seconds. His
techniques involved breaking down large numbers into smaller
components, which he multiplied mentally to achieve accurate
results.
Colburn’s performances were marked by his ability to solve
challenging mathematical problems on the spot, demonstrating his
mastery of mental arithmetic. One of his notable techniques
involved factoring numbers into smaller units to simplify
calculations. For instance, he successfully multiplied 21,734 by
543 by first breaking 543 into 181 and 3, then performing the
multiplications in stages to reach the final answer of 11,801,562.
His methods, while highly impressive, required significant mental
effort and skill.
Despite his early fame, Colburn’s public interest gradually
diminished as he grew older. By the age of twenty, he returned to
the United States and transitioned to a new career as a Methodist
preacher. Colburn’s life was cut short when he passed away at the
age of thirty-five. Reflecting on his extraordinary abilities, Colburn
noted that his methods, while complex, allowed him to perform
calculations efficiently without the need for pen, ink, or paper. His


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age of thirty-five. Reflecting on his extraordinary abilities, Colburn
noted that his methods, while complex, allowed him to perform
calculations efficiently without the need for pen, ink, or paper. His
legacy as one of the earliest lightning calculators remains a
testament to his exceptional mental arithmetic skills.
"Pure mathematics is, in its way, the poetry of logical ideas."
– Albert Einstein
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Section 3
3-M1: Two-Digit Multiplication (FOIL)
When:
Multiplying two-digit numbers
How:
1. Multiply units’ digits and remember the remainder.
2. Cross-multiply the tens of one number and units of another; do both ways.
3. Add both products and the previous remainder.
4. Remember the tens digit of this number.
5. Multiply tens digits and add the previous remainder.
6. Write all digits.
Visual Flow:
Start
Problem: 68 X 57 =?
Step 1: Multiply units Digit
8 x 7 = 56
Write down 6, carry over 5
Step 2: Cross multiple units and tens
digits
6 x 7 = 42 and 8 x 5 = 40
Step 3: Sum of the results from Step
2 plus carry over
40 + 42 + 5 = 87
Write down 7 and carry over 8
Step 4: Multiply tens digits
6 x 5 = 30
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Step 5: Add results with carryover
30 + 8 = 38
Write down 3
Step 6: Combine results
3876
Answer: 68 x 57 = 3876
End
Figure 8:3-M1: Two-Digit Multiplication (FOIL)
Example and Practice Questions
Example 1 Practice Questions
Problem: 68 x 57 =?
Step1: 8 x 7 = 56
Step2: 6 x 7 = 42 & 8 x 5 = 40
Step3: 42 + 40 + 5 = 87
1. 74 x 43 =
Step4: 6 x 5 = 30
Step5: 30 + 8 = 38
2. 36 x 42 =
Answer: 68 x 57 = 3876
3. 58 x 84 =
Example 2
Problem: 14 x 82 =? 4. 24 x 92 =
Step1: 4 x 2 = 8
Step2: 1 x 2 = 2 & 4 x 8 = 32 5. 38 x 48 =
Step3: 2+32=34
Step4: 1 x 8 = 8 + 3 = 11
Answer: 14 x 82 = 1148
Table 10: 3-M1: Two-Digit Multiplication (FOIL)
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
3-M2: Multiply by 12
When:
Multiplying 12 and a number
How:
1. Double the units digit.
2. Double the tens digit.
3. Add units digit plus any carryover.
4. Double the hundreds.
5. Add the tens digit plus carryover.
6. The next digit is the hundreds digit plus any carryover.
Visual Flow:
Start
Problem: 65 x 12 =?
Step 1: Double Units Digit
5 x 2 = 10
Write down 0, carry over 1
Step 2: Double Tens digit
6 x 2 = 12
Step 3: Add Units digit plus carryover
12 + 5 + 1 = 18
Write down 8 and carry over 1
Step 5: Add tens digit with carryover
6 + 1 = 7
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Step 6: Combine results
780
Answer: 65 x 12 = 780
End
Figure 9:3-M2: Multiply by 12
Example and Practice Questions
Example 1 Practice Questions
Problem: 65 x 12 =?
Step1: 5 x 2 = 10
Step2: 6 x 2 = 12 & 5 x 1 = 5
Step3: 12 + 5 + 1 = 18
1. 32 x 12 =
Step4: 1 x 6 = 6 + 1 = 7
Answer: 65 x 12 = 780
2. 63 x 12 =
3. 57 x 12 =
Example 2
Problem: 82 x 12 =? 4. 89 x 12 =
Step1: 2 x 2 = 4
5. 76 x 12 =
Step2: 8 x 2 = 16 & 2 x 1 = 2
Step3: 16 + 2 = 18
Step4: 8 x 1 = 8 + 1 = 9
Answer: 82 x 12 = 984
Table 11: 3-M2: Multiply by 12
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
3-M3: Add to 10 Tens Digit Trick
When:
When the sum of both tens digits is 10 and the unit digits are the same
How:
1. Multiply the units digits of each number. This is the last two digits of your answer (If
the answer is only one digit add a zero in the tens place)
2. Multiply the tens digits and add the units digit to the product.
3. The sum is the first two digits of your answer.
Visual Flow:
Start
Problem: 47 x 67 =?
Step 1: Multiply units digits
7 x 7 = 49
Write down 49
Step 2: Multiply tens digits
4 x 6 = 24
Step 3: Add results with units digit
24 + 7 = 31
Step 4: Combine results
3149
Answer: 47 X 67 = 3149
End
Figure 10:3-M3: Add to 10 Tens Digit Trick
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Example and Practice Questions
Example 1 Practice Questions
Problem: 47 x 67 =?
Step1: 7 x 7 = 49
1. 68 x 48 =
Step2: 47 x 67 = 3149
Answer: 4 x 6 = 24 + 7 = 31
2. 24 x 84 =
3. 52 x 52 =
Example 2
Problem: 33 x 73 =? 4. 35 x 75 =
Step1: 3 x 3 = 9
5. 772 x 32 =
Step2: 3 x 7 = 21 + 3 = 24
Answer: 33 x 73 = 2409
Table 12: 3-M3: Add to 10 Tens Digit Trick
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
3-M4: Add to 10 Units Digit Trick
When:
The tens digits are the same and the units’ digits add to 10.
How:
1. Multiply the units digits. This is the last two digits of the answer.
2. Add 1 to one of the ten digit.
3. Multiply the new number with the other tens digit.
4. This is the first one or two digits of the answer.
Visual Flow:
Start
Problem: 86 x 84 =?
Step 1: Multiply Units Digit
6 x 4 = 24
Write down 24
Step 2: Add 1 to the tens digit
8 + 1 = 9
Step 3: multiply the result of Step 2
with the second tens digit
9 x 8 = 72
Step 4: Combine results
7224
Answer: 86 x 84 = 7224
End
Figure 11:3-M4: Add to 10 Units Digit Trick
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Example and Practice Questions
Example 1 Practice Questions
Problem: 86 x 84 =?
Step1: 6 x 4 = 24
Step2: 8 x (8 + 1) = 72 1. 42 x 48 =
Answer: 86 x 84 = 7224
2. 65 x 65 =
3. 12 x 18 =
Example 2
Problem: 63 x 67 =? 4. 43 x 47 =
Step1: 3 x 7 = 21
5. 23 x 27 =
Step2: 6 x (6 + 1) = 42
Answer: 4221
Table 13: 3-M4: Add to 10 Units Digit Trick
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Section 3 Order These Steps
Arrange the steps for the following tricks. Please write down the step number in the left
column to solve this math trick.
1. Two-Digit Multiplication (FOIL):
Step Numbers Actions
Step: Write all digits
Step: Multiply units’ digits and remember the remainder
Step: Cross-multiply the tens of one number and units of another; do
both ways
Step: Remember the tens digit of this number
Step: Multiply tens digits and add the previous remainder.
Step: Add both products and the previous remainder.
2. Multiply by 12:
Step Numbers Actions
Step: The next digit is double the tens digit plus the units digit plus
any carrying
Step: The first digit is double the units digit
Step: The next digit is the hundreds digit plus any carrying
Step: The next digit is double the hundreds plus the tens digit plus
carrying
3. Add to 10 Tens Digit Trick:
Step Numbers Actions
Step: The sum is the first two digits of your answer
Step: Multiply the last digit of each number. This is the last two digits
of your answer (If the answer is only one digit add a zero in the
tens place)
Step: Multiply the tens digits and add the units digit to the product
4. Add to 10 Units Digit Trick:


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of your answer (If the answer is only one digit add a zero in the
tens place)
Step: Multiply the tens digits and add the units digit to the product
4. Add to 10 Units Digit Trick:
Step Numbers Actions
Step: Multiply the new number with the other tens digit.
Step: Add 1 to one of the ten digit.
Step: Multiply the units digits. This is the last two digits of the answer.
Step: This is the first one or two digits of the answer.
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Section 3 Quiz
The following quiz comprises 28 questions designed to test your understanding of the
tricks taught in Section 3. Begin by identifying the appropriate trick to apply for each
problem, and then follow the steps to solve it. Answers are available at the end of the
book.
Quiz
1. 24 x 92 = 15. 76 x 12 =
2. 35 x 75 = 16. 76 x 74 =
3. 58 x 84 = 17. 12 x 63 =
4. 32 x 12 = 18. 25 x 25 =
5. 74 x 43 = 19. 47 x 67 =
6. 86 x 84 = 20. 65 x 23 =
7. 25 x 99 = 21. 67 x 63 =
8. 36 x 42 = 22. 92 x 12 =
9. 72 x 32 = 23. 43 x 47 =
10. 85 x 85 = 24. 83 x 88 =
11. 38 x 48 = 25. 12 x 59 =
12. 57 x 12 = 26. 89 x 12 =
13. 24 x 84 = 27. 52 x 52 =
14. 33 x 73 = 28. 95 x 95 =
Table 14: Section 3 Quiz
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Math Wizards: Évariste Galois
The Mathematical Duel of Évariste Galois
Évariste Galois, born in 1811, was a French mathematician whose
work laid the groundwork for modern abstract algebra. Galois was
a prodigious talent, displaying remarkable mathematical abilities
from a young age. Despite his brief life, his contributions to
mathematics, particularly in the area of group theory, have had a
profound and lasting impact. Galois's innovative approach to the
theory of equations and algebraic structures fundamentally
changed the landscape of mathematics.
The dramatic story of Galois is as well-known as his mathematical
achievements. On May 30, 1832, at the age of twenty, Galois was
killed in a duel over a romantic entanglement. The legend has it
that the night before his death, he composed his most important
mathematical theories, anticipating his demise and hoping to leave
a legacy for future mathematicians. In a poignant letter to his friend
Auguste Chevalier, Galois described his recent discoveries and
asked that they be reviewed by prominent mathematicians, such
as Jacobi or Gauss.
In reality, the mathematical papers that Galois revised in his final


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Auguste Chevalier, Galois described his recent discoveries and
asked that they be reviewed by prominent mathematicians, such
as Jacobi or Gauss.
In reality, the mathematical papers that Galois revised in his final
hours were actually earlier submissions made when he was just
seventeen. His contributions had already been recognized by the
Academy of Sciences before his death. Galois's work became
entangled with political activism and personal disputes, leading to
his tragic death. His early involvement in radical politics and his
imprisonment further complicated his life and career,
overshadowing his mathematical genius.
Galois's assertion that his research would "halt many savants in
theirs" proved to be accurate. His development of group theory and
exploration of polynomial equations introduced concepts that are
central to modern algebra. Although his life was cut short, Évariste
Galois's pioneering work continues to influence and inspire
mathematicians, affirming his place as one of the most innovative
and influential figures in the history of mathematics.
"Mathematics is the music of reason." – James Joseph
Sylvester
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Section 4
4-M1: Multiplying by 11
When:
Multiplying 11 with a number
How:
1. The last digit of the answer is the units digit of the non-11 number.
2. The next digit of the answer is the sum of the tens and units digit of the non-11
number.
3. The next digit is the sum of the tens and hundreds of digits plus any carry.
4. The first digit is the hundreds digit plus any carry.
Visual Flow:
Start
Problem: 451 x 11 =?
Step 1: Write down the units digit of
451
That is 1
Step 2: Add tens and units digit
5 + 1 = 6
Write down 6 next to 1
Step 3: Add the next two digits
4 + 5 = 9
Write down 9 next to 61
Step 4: Combine results
961
Page 50 of 113

ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Answer: 451 x 11 = 961
End
Figure 12:4-M1: Multiplying by 11
Example and Practice Questions
Example 1 Practice Questions
Problem: 451 x 11 =?
Step 1: Units is 1
Step 2: 5 + 1 = 6
Step 3: 4 + 5 = 9
1. 732 x 11=
Answer: 451 x 11 = 4961
2. 486 x 11=
Example 2 3. 215 x 11=
Problem: 958 x 11 =?
4. 349 x 11=
Step 1: Units is 8.
Step 2: 5 + 8 = 13 5. 154 x 11=
Step 3: 9 + 5 = 14 + 1 = 15
Step 4: 9 + 1 = 10
Answer: 958 x 11 = 10538
Table 15: 4-M1: Multiplying by 11
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
4-M2: Multiplying by 111
When:
Multiplying 11 with a number
How:
1. How: The last digit is the units digit of the non-111 number.
2. The next digit is the sum of the tens and the units digit plus any carry.
3. The next digit is the sum of the units, tens, and hundreds digits, plus any carry.
4. The next digit is the sum of the tens and hundreds of digits plus any carry.
5. The next digit is the hundreds digit plus any carry.
NOTE: you can use the two-finger method but with three fingers instead
Visual Flow:
Start
Problem: 263 x 111 =?
Step 1: Write down the units digit of
non-111
Write down 3
Step 2: Add tens and units digit
6 + 3 = 9
Write down 9 next to 3
Step 3: Add hundreds, tens, and
units digits and any carryover
2 + 6 + 3 = 11
Write down 1 and carry over 1
Step 4: Add hundreds, and tens digits
and any carryover
2 + 6 = 8
Add carry over
8 + 1 = 9
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Step 5: Write down hundreds digit
and any carryover
2
Step 6: Combine results
29193
Answer: 263 x 11 = 29193
End
Figure 13:4-M2: Multiplying by 111
Example and Practice Questions
Example 1 Practice Questions
Problem: 263 x 111 =?
Step 1: Units is 3
Step 2: 6 + 3 = 9
Step 3: 2 + 6 + 3 = 11
Step 4: 2 + 6 = 8 + 1 = 9 1. 384 x 111=
Answer: 263 x 111 = 29193
2. 567 x 111=
Example 2
3. 149 x 111=
Problem: 986 x 111 =?
4. 278 x 111=
Step 1: Units is 6
Step 2: 8 + 6 = 14 5. 432 x 111=
Step 3: 9 + 8 + 6 = 23 + 1 = 24
Step 4: 9 + 8 = 17 + 2 = 19
Step 5: 9 + 1 = 10
Answer: 986 x 111 = 109446
Table 16: 4-M2: Multiplying by 111
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Section 4 Order these Steps
Arrange the steps for the following tricks. Please write down the step number in the left
column to solve this math trick.
1. Multiplying by 111:
Step Numbers Actions
Step: The next digit is the sum of the tens and units digit plus any
carry.
Step: The last digit is the units digit of the non-111 number.
Step: The next digit is the sum of the units, tens, and hundreds digits
plus any carry.
Step: The next digit is the hundreds digit plus any carry.
Step: The next digit is the sum of the tens and hundreds of digits
plus any carry.
Section 4 Quiz
The following quiz comprises 16 questions designed to assess your understanding of
the tricks taught in Section 4. Begin by identifying the appropriate trick to apply for each
problem, and then follow the steps to solve it. Answers are available at the end of the
book.
Quiz
1. 482 x 111 = 9. 102 x 11 =
2. 376 x 11 = 10. 482 x 111 =
3. 697 x 11 = 11. 376 x 11 =
4. 319 x 111 = 12. 528 x 111 =
5. 241 x 11 = 13. 743 x 111 =
6. 237 x 111 = 14. 509 x 11 =
7. 854 x 11 = 15. 102 x 11 =
8. 654 x 111 = 16. 468 x 111 =
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Table 17: Section 4 Quiz
Math Wizards: Srinivasa Ramanujan
Srinivasa Ramanujan: Series of π
Srinivasa Ramanujan, an Indian mathematician born in 1887 in
Erode, is celebrated for his extraordinary contributions to
mathematics despite having minimal formal training. His
mathematical prowess emerged in isolation, with Ramanujan
developing complex theories and results largely independently. His
work was characterized by a profound intuition and a unique
approach to problems, which led him to formulate numerous
groundbreaking results in number theory, continued fractions, and
infinite series.
Ramanujan's mathematical genius attracted the attention of the
British mathematician G.H. Hardy, who invited him to Cambridge
University in 1914. This collaboration proved highly fruitful, with
Ramanujan and Hardy producing significant research that
enriched the field of mathematics. Ramanujan's work included the
discovery of new results on the partition function, the famous
Ramanujan-Hardy number 1729, and the development of the
highly influential Ramanujan series for π.
Despite his profound impact on mathematics, Ramanujan's career


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discovery of new results on the partition function, the famous
Ramanujan-Hardy number 1729, and the development of the
highly influential Ramanujan series for π.
Despite his profound impact on mathematics, Ramanujan's career
was tragically cut short by his early death at the age of 32 in 1920.
His work was often ahead of its time, and many of his results were
initially met with skepticism until later verified and appreciated.
Ramanujan's legacy endures through his contributions to
mathematics, which continue to influence and inspire
contemporary mathematicians.
Ramanujan's life and work have been the subject of considerable
study and admiration. His story exemplifies the power of innate
talent and dedication, and his mathematical results, often derived
from intuition rather than formal proofs, remain a testament to his
remarkable genius. His work continues to be explored and
extended, ensuring that his mathematical contributions remain
relevant and influential in the field.
"In mathematics, the art of proposing a question must be
held of higher value than solving it." – Georg Cantor
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Section 5
5-M1: Multiply by 25
When
Multiplying 25 with a number
How:
1. Divide the non-25 number by 4.
2. Add two zeroes at the end.
Visual Flow:
Start
Problem: 25 x 64 =?
Step 1: Divide non-25 number by 4
64 / 4 = 16
Step 2: Add two zero
Write down 1600
Answer: 25 x 64 = 1600
End
Figure 14: 5-M1: Multiply by 25
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Example and Practice Questions
Example 1 Practice Questions
Problem: 25 x 64 =?
1. 72 x 25 =
Step1: 64 / 4 = 16
Answer: 25 x 64 = 1600 2. 16 x 25 =
3. 25 x 108 =
Example 2
4. 44 x 25 =
Problem: 25 x 0.24 =?
Step1: 0.24 / 4 = 0.06 5. 2.04 x 25 =
Answer: 25 x 0.24 = 6
Table 18: 5-M1: Multiply by 25
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
5-M2: Multiply by 50
When:
Multiplying 50 with a number
How:
1. Divide the non-50 number by 2.
2. Add two zeroes at the end.
Visual Flow:
Start
Problem: 50 x 48 =?
Step 1: Divide non-50 number by 2
48 / 2 = 24
Step 2: Add two zero
Write down 2400
Answer: 50 x 48 = 2400
End
Figure 15: 5-M2: Multiply by 50
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Example and Practice Questions
Example 1 Practice Questions
Problem: 50 x 48 =? 1. 50 x 42 =
Step1: 48 / 2 = 24
2. 85 x 50 =
Answer: 50 x 48 = 2400
Example 2 3. 64 x 50 =
Problem: 37 x 50 =? 4. 76 x 50 =
Step1: 37 / 2 = 18.5
5. 50 x 98 =
Answer: 37 x 50 = 1850
Table 19: 5-M2: Multiply by 50
Page 59 of 113

ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
5-M3: Multiply by 75
When:
Multiplying 75 with a number
How:
1. Divide the non-75 number by 4.
2. Multiply the answer by 3.
3. Add two zeroes.
Visual Flow:
Start
Problem: 48 x 75 =?
Step 1: Divide non-75 number by 4
48 / 4 = 12
Step 2: Multiply the result by 3
12 x 3 = 36
Step 3: Add two zero
Write down 3600
Answer: 48 x 75 = 3600
End
Figure 16: 5-M3: Multiply by 75
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Example and Practice Questions
Example 1 Practice Questions
Problem: 75 x 48 =?
Step1: 48 / 4 = 12
1. 75 x 64 =
Step2: 12 x 3 = 36
Answer: 75 x 48 = 3600
2. 75 x 18 =
Example 2 3. 82 x 75 =
Problem: 98 x 75 =?
4. 34 x 75 =
Step1: 98 / 4 = 24.5
Step2: 24.5 x 3 = 72.5 5. 56 x 75 =
Answer: 98 x 75 = 7250
Table 20: 5-M3: Multiply by 75
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
5-M4: Multiply two numbers ending with 5
When:
Finding the product of two numbers ending with 5
How:
1. Find the sum of the tens digits.
2. If the sum is even, then the last two digits of the answer are 25.
3. If the sum is odd, then the last two digits of the answer are 75.
4. Find half of the sum (do not worry about any remainder)
5. Multiply the tens digits.
6. Add the answers from steps 4 and 5. This is the first two digits of the answer.
Visual Flow:
Start
Problem: 45 x 35 =?
Step 1: Add tens digit
4 + 3 = 7
Step 2: since the result is odd
Write down 75
Step 3: Divide the result of Step 1 by
2
7 / 2 = 3
Step 4: Multiply tens digits
4 x 3 = 12
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Step 5: Add the answer from Step 3
and Step 4
3 + 12 = 15
Step 6: Combine results
1575
Answer: 45 x 35 = 1575
End
Figure 17: 5-M4: Multiply two numbers ending with 5
Example and Practice Questions
Example 1 Practice Questions
Problem: 45 x 35 =?
Step1: 4 + 3 = 7 (odd)
Step2: 7 / 2 = 3
Step3: 4 x 3 = 12
1. 95 x 45 =
Step4: 12 + 3 = 15
Answer: 45 x 35 = 1575 2. 85 x 65 =
3. 65 x 25 =
Example 2
Problem: 95 x 55 =? 4. 35 x 65 =
Step1: 9 + 5 = 14 (even)
5. 95 x 15 =
Step2: 14 / 2 = 7
Step3: 9 x 5 = 45
Step4: 7 + 45 = 52
Answer: 95 x 55 = 5225
Table 21: 5-M4: Multiply two numbers ending with 5
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
5-M5: Multiplying by 15
When:
Finding the product of a number and 15
How:
1. Divide the non-15 number by 2.
2. Add the Answer to the non-15 number.
3. Add a zero at the end.
Visual Flow
Start
Problem: 22 x 15 =?
Step 1: divide non-15 number by 2
22 / 2 = 11
Step 2: Add the result of Step 1 to the
non-15 number
22 + 11 = 33
Step 3: Add a zero at the end
330
Answer: 22 x 15 = 330
End
Figure 18: 5-M5: Multiplying by 15
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Example and Practice Questions
Example 1 Practice Questions
Problem: 22 x 15 =?
Step1: 22 / 2 = 11
Step2: 11 + 22 = 33
1. 88 x 15 =
Step3: 330
Answer: 22 x 15 = 330
2. 96 x 15 =
3. 12 x 15 =
Example 2
Problem: 42 x 15 =? 4. 24 x 15 =
Step1: 42 / 2 = 21
5. 76 x 15 =
Step2: 21 + 42 = 63
Step3: 630
Answer: 42 x 15 = 630
Table 22:M4: Multiplying by 15
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Section 5 Order These Steps
Arrange the steps for the following tricks. Please write down the step number in the left
column to solve this math trick.
1. Multiply by 25:
Step Numbers Actions
Step: Add two zeroes at the end
Step: Divide the non-25 number by 4
2. Multiply by 50:
Step Numbers Actions
Step: Add two zeroes at the end.
Step: Divide the non-50 number by 2.
3. Multiply by 75:
Step Numbers Actions
Step: Multiply the answer by 3
Step: Divide the non-75 number by 4
Step: Add two zeroes.
4. Multiply two numbers ending with 5:
Step Numbers Actions
Step: If the sum is even, then the last two digits of the
answer are 25.
Step: Find the sum of the tens digits
Step: Find half of the sum (do not worry about any
remainder)
Step: If the sum is odd, then the last two digits of the
answer are 75.
Step: Add the answers from steps 2 and 3. This is the first
two digits of the answer.
Step: Multiply the tens digits.
5. Multiply by 15:
Step Numbers Actions
Step: Add a zero at the end
Step: Add the Answer to the non-15 number
Step: Divide the non-15 number by 2
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Section 5 Quiz
The following quiz comprises 28 questions designed to assess your understanding of
the tricks taught in Section 5. Begin by identifying the appropriate trick to apply for each
problem, and then follow the steps to solve it. Answers are available at the end of the
book.
Quiz
1. 25 x 72 = 15. 75 x 34 =
2. 50 x 42 = 16. 65 x 25 =
3. 75 x 64 = 17. 25 x 2.04 =
4. 45 x 35 = 18. 50 x 98 =
5. 15 x 16 = 19. 15 x 56 =
6. 50 x 85 = 20. 25 x 81 =
7. 75 x 18 = 21. 35 x 65 =
8. 95 x 55 = 22. 50 x 57 =
9. 25 x 108 = 23. 75 x 20 =
10. 50 x 64 = 24. 95 x 15 =
11. 75 x 82 = 25. 25 x 0.56 =
12. 85 x 65 = 26. 50 x 0.8 =
13. 25 x 44 = 27. 75 x 92 =
14. 15 x 76 = 28. 55 x 45 =
Table 23: 5-M4: Multiply two numbers ending with 5
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Math Wizards: Alexander Craig Aitken
A Piece of Pi for Alexander Craig Aitken
Alexander Craig Aitken (1895–1967), a distinguished professor of
mathematics at the University of Edinburgh, is celebrated for his
exceptional memory and mental calculation skills. One of his most
impressive feats was memorizing the value of π (pi) to 1,000
decimal places. During a lecture, Aitken demonstrated his
remarkable ability by reciting the first 250 digits of π and then
continuing from the 551st digit for an additional 150 digits, all
without making a single error. His proficiency in handling such a
vast array of digits highlighted his extraordinary cognitive skills.
Aitken's approach to memorization was unique and differed from
conventional methods. He employed a technique that he described
as the antithesis of intense concentration. Instead of focusing
narrowly, he broke the digits into manageable chunks of fifty and
memorized them using rhythmic patterns. This auditory method
allowed him to recall extensive sequences of numbers with ease,
showcasing his innovative and relaxed approach to mental
calculation.
In addition to his prowess in memorizing π, Aitken demonstrated


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allowed him to recall extensive sequences of numbers with ease,
showcasing his innovative and relaxed approach to mental
calculation.
In addition to his prowess in memorizing π, Aitken demonstrated
impressive mental arithmetic abilities. He was known for quickly
and accurately performing complex calculations, such as
multiplying five-digit numbers. During a demonstration of desk
calculators, Aitken astounded onlookers by instantly providing the
correct product of two large numbers, proving his mental
calculation skills were as remarkable as his memory.
Ironically, Aitken noticed a decline in his mental calculation abilities
after he began using a desk calculator, which he believed might
signal the potential extinction of mental calculation skills. Despite
his concerns, his legacy remains significant. Aitken’s contributions
to mathematics and his extraordinary mental feats continue to
inspire and intrigue mathematicians and enthusiasts, affirming his
place as a remarkable figure in the history of mental arithmetic.
"Mathematics is not about numbers, equations,
computations, or algorithms: it is about
understanding."
– William Paul Thurston
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Section 6
6-M1: Multiplying two numbers both over 100
When:
Finding the product of two numbers that are greater than 100.
How:
1. Determine how much each number is over 100.
2. Multiply the differences; this is the last two digits of the answer.
3. Pick the larger number and add the difference of the other number. This is the
first three digits of the answer.
Visual Flow:
Start
Problem: 103 x 105 =?
Step 1: Multiply units digits
3 x 5 = 15
Step 2: Add the First number and the
units digit of the second number
105 + 3 = 108
Step 3: Combine results
10815
Answer: 103 x 105 = 10815
End
Figure 19: 6-M1: Multiplying two numbers both over 100
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Example and Practice Questions
Example 1 Practice Questions
Problem: 103 x 105 =?
Step1: 3 x 5 = 15
1. 102 x 104 =
Step2: 103 + 5 = 108
Answer: 103 x 105 = 10815
2. 105 x 106 =
3. 103 x 109 =
Example 2
Problem: 109 x 112 =? 4. 111 x 104 =
Step1: 9 x 12 = 108
5. 112 x 107 =
Step2: 109 + 12 = 121
Answer: 109 x 112 = 12208
Table 24: 6-M1: Multiplying two numbers both over 100
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
6-M2: Multiplying two numbers both under 100
When:
Finding the product of two numbers that are less than 100.
How:
1. Determine how much each number is over 100.
2. Multiply the differences; this is the last two digits of the answer.
3. Pick the larger number and subtract the difference of the other number. This is
the first three digits of the answer.
Visual Flow:
Start
Problem: 97 x 96 =?
Step 1: Subtract both numbers from
100
100 – 97 = 3
100 – 96 = 4
Step 2: Multiply both results of step 1
3 x 4 = 12
Step 3: Pick the larger number and
subtract the difference from the
other number
97 – 4 = 93
Step 4: Combine results
9312
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Answer: 97 x 96 = 9312
End
Figure 20: 6-M2: Multiplying two numbers both under 100
Example and Practice Questions
Example 1 Practice Questions
Problem: 97 x 96 =?
Step1: 100 -97 = 3 & 100 – 96 = 4
Step2: 3 x 4 = 12
1. 96 x 93 =
Step3: 96 – 3 = 93
Answer: 97 x 96 = 9312 2. 90 x 94 =
Example 2 3. 97 x 92 =
Problem: 94 x 98 =? 4. 95 x 91 =
Step1: 100 – 94 = 6 & 100 – 98 = 2
5. 90 x 91 =
Step2: 6 x 2 = 12
Step3: 94 – 2 = 92
Answer: 94 x 98 = 9212
Table 25: 6-M2: Multiplying two numbers both under 100
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
6-M3: Multiplying two numbers over/under 100
When:
Finding the product of a number greater than 100 and a number less than 100.
How:
1. Determine how much each number is either over or under 100. The number less
than 100 should be considered negative.
2. Multiply the two numbers.
3. Add the product to 100 (you should be subtracting). This is the last two digits of
the answer.
4. Add the two numbers from step 1.
5. Add or subtract that number from 99.
Visual Flow:
Start
Problem: 103 x 98 =?
Step 1: subtract 100 from both
numbers
103 – 100 = 3
98 - 100 = -2
Step 2: multiply the results of the last
step
3 x -2 = -6
Step 3: Add the result of step 2 to
100
100 + (-6) = 94
Step 4: Add the results of the Step 1
3 + (-2) = 1
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Step 5: Add the result of step 4 from
99
99 + 1 = 100
Step 5: Combine results
10094
Answer: 103 x 98 = 10094
End
Figure 21: 6-M3: Multiplying two numbers over/under 100.
Example and Practice Questions
Example 1 Practice Questions
Problem: 103 x 98 =?
Step1: 3x (-2) = -6
Step2: 100 + (-6) = 94
Step3: 3 + (-2) = 1
1. 106 x 94 =
Step4: 99 + 1 = 100
2. 104 x 90 =
Answer: 103 x 98 = 10094
3. 108 x 91 =
Example 2
4. 92 x 109 =
Problem: 108 x 92 =?
Step1: 8x (-8) = -64 5. 111 x 93 =
Step2: 100 + (-64) = 32
Step3: 8 + (-8) = 0
Step4: 99 + 0 = 99
Answer: 108 x 92 = 9932
Table 26: 6-M3: Multiplying two numbers over/under 100
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Section 6 Order these Steps
Arrange the steps for the following tricks. Please write down the step number in the left
column to solve this math trick.
1. Multiplying two numbers both over 100:
Step Numbers Actions
Step: Pick the larger number and add the difference of the
other number. This is the first three digits of the
answer.
Step: Determine how much each number is over 100
Step: Multiply the differences; this is the last two digits of
the answer.
2. Multiplying two numbers both under 100:
Step Numbers Actions
Step: Multiply the differences; this is the last two digits of
the answer.
Step: Determine how much each number is over 100
Step: Pick the larger number and add the difference of the
other number. This is the first three digits of the
answer.
3. Multiplying two numbers over/under 100:
Step Numbers Actions
Step: Add or subtract that number from 99
Step: Add the two numbers from step 1.
Step: Add the product to 100 (you should be subtracting).
This is the last two digits of the answer.
Step: Multiply the two numbers
Step: Determine how much each number is either over or
under 100. The number less than 100 should be
considered negative
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Section 6 Quiz
The following quiz comprises 24 questions designed to assess your understanding of
the tricks taught in Section 6. Begin by identifying the appropriate trick to apply for each
problem, and then follow the steps to solve it. Answers are available at the end of the
book.
Quiz
1. 102 x 107= 13. 103 x 113 =
2. 85 x 92 = 14. 112 x 119 =
3. 105 x 98 = 15. 89 x 93 =
4. 104 x 110 = 16. 76 x 84 =
5. 98 x 87 = 17. 97 x 95 =
6. 110 x 87 = 18. 92 x 88 =
7. 115 x 120 = 19. 108 x 92 =
8. 106 x 108 = 20. 115 x 89 =
9. 94 x 96 = 21. 103 x 94 =
10. 101 x 95 = 22. 112 x 91 =
11. 111 x 115 = 23. 109 x 87 =
12. 120 x 125 = 24. 94 x 96 =
Table 27: Section 6 Quiz
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Math Wizards: Thomas Fuller
Thomas Fuller: Learned Men and Great Fools
Thomas Fuller, born in Africa in 1710, stands out as a remarkable
figure in the history of mental arithmetic despite the severe
limitations imposed on him by slavery and illiteracy. As an enslaved
individual in Virginia, Fuller had no access to formal education. Yet,
he demonstrated an extraordinary ability for numerical
calculations, teaching himself to count and developing a unique
aptitude for arithmetic through practical experiences. He would
count everyday items such as grains of wheat, seeds, and even
hairs in a cow’s tail, which honed his skills in mental computation.
Fuller’s mathematical talents were not confined to simple counting.
He applied his skills to more complex problems, such as
calculating the number of shingles needed for roofing or the posts
required for fencing. His proficiency in handling these calculations
caught the attention of others, leading to challenges from two
Pennsylvanians who tested his abilities with sophisticated
problems. For instance, he correctly solved a complex exponential
growth problem involving sows and pigs, arriving at the accurate


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Pennsylvanians who tested his abilities with sophisticated
problems. For instance, he correctly solved a complex exponential
growth problem involving sows and pigs, arriving at the accurate
total of 34,588,806 within ten minutes.
Despite his lack of formal education, Fuller’s arithmetic prowess
was widely recognized. Upon his death in 1790, the Columbian
Centinel reported that he could perform calculations with
remarkable speed and accuracy, often surpassing the capabilities
of many educated individuals using writing tools. Fuller’s ability to
quickly provide answers to intricate problems demonstrated not
only his natural talent but also his remarkable mental agility.
Fuller’s response to the lack of formal education was both insightful
and philosophical. When asked if he regretted not receiving a
traditional education, he expressed contentment with his self-
taught skills, remarking that “many learned men be great fools.”
His story reflects his remarkable resilience and intellectual
prowess, illustrating that exceptional mental abilities can emerge
even in the face of significant social and educational
disadvantages. Thomas Fuller’s legacy endures as a testament to
his extraordinary mental arithmetic skills and his ability to


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prowess, illustrating that exceptional mental abilities can emerge
even in the face of significant social and educational
disadvantages. Thomas Fuller’s legacy endures as a testament to
his extraordinary mental arithmetic skills and his ability to
overcome formidable obstacles.
"Do not worry too much about your difficulties in
mathematics, I can assure you mine are still greater.”
– Albert Einstein
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Section 7
7-M1: Multiplying Two digits number by 101
When:
Finding the product of a number and 101
How:
1. Recognize that multiplying by 101.
2. Write the number twice.
Visual Flow :
Start
Problem: 22 x 101 =?
Step 1: Write non 101 twice
2222
Answer: 22 x 101 = 2222
End
Figure 22: 7-M1: Multiplying Two digits number by 101
Example and Practice Questions
Example 1 Practice Questions
1. 23 x 101=
Problem: 22 x 101 =?
Answer: 22 x 101 = 2222
2. 12 x 101 =
Example 2 3. 56 x 101 =
4. 35 x 101 =
Problem: 43 x 101 =?
Answer: 43 x 101 = 4343
5. 46 x 101 =
Table 28: 7-M1: Multiplying Two digits number by 101
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
7-M2: Multiplying Three digits number by 101
When:
Finding the product of a number and 101
How:
1. Write the tens and units digit.
2. Add the hundredth digit to the whole number.
3. Write the answer of the Step 2 in front of Step 1
Visual Flow :
Start
Problem: 234 x 101 =?
Step 1: write tens and units digit
34
Step 2: Add the hundredth digit to the
whole number
234 + 2 = 236
Step 3: Combine results
23634
Answer: 263 x 11 = 29193
End
Figure 23:7-M2: Multiplying Three digits number by 101
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Example and Practice Questions
Example 1 Practice Questions
Problem: 234 x 101 =?
Step1: 34
Step2: 234 + 2 = 236
1. 123 x 101=
Step3: 23624
Answer: 234 x 101 = 23634
2. 225 x 101 =
3. 560 x 101 =
Example 2
Problem: 346 x 101 =? 4. 356 x 101 =
Step1: 46
5. 416 x 101 =
Step2: 346 + 3 = 349
Step3: 34946
Answer: 346 x 101 = 34946
Table 29: 7-M2: Multiplying Three digits number by 101
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
7-M3: Multiplying Two digits number by 1001
When:
Finding the product of a two-digit number and 1001
How:
1. Write the number one time.
2. Add 0 at the end of the number.
3. Write the number one more time.
Visual Flow :
Start
Problem: 35 x 1001 =?
Step 1: write down the non-1001
number once
Write down 35
Step 2: Add zero at the end of step 1
350
Step 3: write the number again
35035
Answer: 263 x 11 = 29193
End
Figure 24:7-M3: Multiplying Two digits number by 1001
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Example and Practice Questions
Example 1 Practice Questions
Problem: 35 x 1001 =?
Step1: 35
Step2: 350
1. 12 x 1001=
Step3: 35035
Answer: 35 x 1001 = 35035
2. 26 x 1001 =
Example 2 3. 52 x 1001 =
Problem: 41 x 1001 =? 4. 37 x 1001 =
Step1: 41
5. 89 x 1001 =
Step2: 410
Step3: 41041
Answer: 41 x 1001 = 41041
Table 30: 7-M3: Multiplying Two digits number by 1001
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
7-M4: Multiplying Three digits number by 1001
When:
Finding the product of a three-digit number and 1001
How:
1. Recognize that multiplying by 1001.
2. Write the number twice.
Visual Flow:
Start
Problem: 465 x 1001 = ?
Step 2: Add the last two digit
6 + 3 = write the number twice
Answer: 465 x 1001 = 465465
End
Figure 25:7-M4: Multiplying Three digits number by 1001.
Example and Practice Questions
Example 1 Practice Questions
Problem: 123 x 1001 =? 1. 789 x 1001=
Answer: 123 x 1001 = 123123
2. 822 x 1001 =
Example 2 3. 567 x 1001 =
Problem: 465 x 1001 =? 4. 379 x 1001 =
Answer: 465 x 1001 = 465465
5. 987 x 1001 =
Table 31: 7-M4: Multiplying Three digits number by 1001
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
7-M5: Multiplying four digits number by 1001
When:
Finding the product of a four-digit number and 1001
How:
1. Write the hundredth, tens, and units digit.
2. Add the first digit to the whole number
3. Write thousand, hundredth, and tens digit.
Visual Flow :
Start
Problem: 3125 x 1001 = ?
Step 1: Write the hundredth, tens,
and units digit
Write down 125
Step 2:Add first digit to the whole
number
3125 + 3 = 3128
Step 3: Combine results
3128125
Answer: 3125 x 1001 = 3128125
End
Figure 26:7-M5: Multiplying four digits number by 1001
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Example and Practice Questions
Example 1 Practice Questions
Problem: 3125 x 1001 =?
Step1: 125
1. 1278 x 1001=
Step2: 3125 + 3 = 3128
Answer: 3125 x 1001 = 3128125
2. 3626 x 1001 =
3. 5322 x 1001 =
Example 2
4. 3766 x 1001 =
Problem: 4651 x 1001 =?
Step1: 651
5. 1805 x 1001 =
Step2: 4651 + 4 = 4655
Answer: 4651 x 1001 = 4655651
Table 32: 7-M5: Multiplying four digits number by 1001
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Section 7 Order these Steps
Arrange the steps for the following tricks. Please write down the step number in the left
column to solve this math trick.
1. Multiplying Two digits number by 101:
Step Numbers Actions
Step: Write the number twice
Step: Recognize that multiplying by 101
2. Multiplying Three digits number by 101:
Step Numbers Actions
Step: Write the answer of Step 2 in front of Step 1
Step: Write the tens and units digit
Step: Add the hundredth digit to the whole number
3. Multiplying Two digits number by 1001:
Step Numbers Actions
Step: Write the number one more time
Step: Write the number one time
Step: Add 0 at the end of the number
4. Multiplying Three digits number by 1001:
Step Numbers Actions
Step: Write the number twice
Step: Recognize that multiplying by 1001
5. Multiplying four digits number by 1001:
Step Numbers Actions
Step: Write thousand, hundredth and tens digit
Step: Write thousand, hundredth and tens digit
Step: Add the first digit to the whole number
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Section 7 Quiz
The following quiz comprises 20 questions designed to assess your understanding of
the tricks taught in Section 7. Begin by identifying the appropriate trick to apply for each
problem, and then follow the steps to solve it. Answers are available at the end of the
book.
Quiz
1. 231 x 1001 11. 52 x 101
2. 345 x 101 12. 2355 x 1001
3. 64 x 101 13. 34 x 101
4. 54 x 1001 14. 212 x 101
5. 124 x 101 15. 580 x 1001
6. 2122 x 1001 16. 453 x 101
7. 23 x 1001 17. 341 x 1001
8. 467 x 101 18. 78 x 1001
9. 870 x 101 19. 232 x 101
10. 97 x 101 20. 124 x 101
Table 33: Section 7 Quiz
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Math Wizards: Carl Friedrich Gauss
Carl Friedrich Gauss: Mathematical Prodigy
Prince of Mathematicians," Gauss's early talent became evident in
his childhood. By the age of seven, he had already demonstrated
exceptional mathematical abilities, famously calculating the sum of
the first 100 integers quickly by recognizing the arithmetic series
pattern. This early display of genius foreshadowed his future
achievements and established him as a prodigy in mathematics.
Gauss's contributions span numerous areas, including number
theory, statistics, and astronomy. One of his seminal works, the
"Disquisitiones Arithmeticae" (1801), laid the foundation for
modern number theory. In this work, Gauss introduced concepts
such as modular arithmetic and the theory of quadratic residues,
which have had a lasting impact on the field. Additionally, his work
in statistics and the method of least squares revolutionized data
analysis and error minimization, providing tools that are
fundamental in modern statistical practice.
In addition to his theoretical achievements, Gauss made significant
contributions to practical mathematics and astronomy. His work on


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analysis and error minimization, providing tools that are
fundamental in modern statistical practice.
In addition to his theoretical achievements, Gauss made significant
contributions to practical mathematics and astronomy. His work on
the orbit of the asteroid Ceres, which he successfully predicted
using his own method of least squares, demonstrated his
exceptional skill in both theoretical and applied mathematics.
Gauss also made important advancements in differential
geometry, where his work on curvature and the "Theorema
Egregium" established foundational principles that continue to
influence geometric theory.
Gauss's legacy is marked by his profound impact on many
branches of mathematics and science. His insights and
methodologies have shaped modern mathematical thought and
practice. Despite his considerable achievements, Gauss remained
relatively modest and reclusive, preferring to let his work speak for
itself. Today, his contributions are celebrated for their depth and
enduring significance, cementing Carl Friedrich Gauss as a
towering figure in the history of mathematics.
"The essence of mathematics lies in its freedom."
– Georg Cantor
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Section 8
8-M1: Multiply A number with an Equal Number of 9's
When:
Multiply two-digit numbers by 99 or a three-digit number by 999
How:
1. Subtract 1 from the number.
2. Write the Answer on the left-hand side.
3. Subtract each digit from 99.
4. Write the answer next to the First answer.
Visual Flow :
Start
Problem: 65 x 99 =?
Step 1: Subtract 1 from 65
65 - 1 = 64
Step 2: The first part of the answer
64
Step 3: Subtract step 1 answer from
99
99 – 64 = 35
Step 4: Combine results
6435
Answer: 65 x 99 = 6435
End
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Figure 27:8-M1: Multiply A number with an Equal Number of 9's
Example and Practice Questions
Example 1 Practice Questions
Problem: 65 x 99 =?
Step1: 65 -1 = 64
Step2: 64
1. 78 x 99 =
Step3: 99 - 64 = 35
Answer: 65 x 99 = 6435 2. 29 x 99 =
Example 2 3. 37 x 99 =
Problem: 24 x 99 =?
4. 33 x 99 =
Step1: 24 – 1 = 23
Step2: 23 5. 49 x 99 =
Step3: 99 – 23 = 76
Answer: 24 x 99 = 2376
Table 34: 8-M1: Multiply A number with an Equal Number of 9's
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
8-M2: Multiply the number of digits less than the
number of 9s
When:
Multiple the number of digits is less than the number of 9s (22 x 999 or 234 x 9999)
How:
1. Add a zero in front of the number.
2. First subtract 1 from the number
3. Subtract each digit of the result from 9.
Visual Flow :
Start
Problem: 23 x 999 =?
Step 1: add a zero in front of the
number
Write down 023
Step 2: Subtract 1 from the number
23 - 1 = 22
Step 3: Subtract each digit from 9
9 – 0 = 9
9 – 2 = 7
9 – 2 = 7
Step 4: Combine results
22977
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Answer: 23 x 999 = 22977
End
Figure 28:8-M2: Multiply the number of digits less than the number of 9s
Example and Practice Questions
Example 1 Practice Questions
Problem: 23 x 999 =?
Step1: 023
Step2: 23 – 1 = 22
Step3: 9 – 0 = 9
1. 42 x 999 =
Step4: 9 – 2 = 7
Step5: 9 – 2 = 7 2. 67 x 999 =
Answer: 23 x 999 = 22977
3. 34 x 999 =
Example 2
4. 52 x 999 =
Problem: 36 x 999 =?
Step1: 036 5. 27 x 999 =
Step2: 36 – 1 = 35
Step3: 9 – 0 = 9
Step4: 9 – 3 = 6
Step5: 9 – 5 = 4
Answer: 36 x 999 = 35964
Table 35: 8-M2: Multiply the number of digits less than the number of 9s
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Section 8 Order these Steps
Arrange the steps for the following tricks. Please write down the step number in the left
column to solve this math trick.
1. Multiply A number with an Equal Number of 9's:
Step Numbers Actions
Step: Write the Answer on the left-hand side
Step: Subtract 1 from the number
Step: Write the answer next to the First answer
Step: Subtract each digit from 99
2. Multiply the number of digits less than the number of 9s:
Step Numbers Actions
Step: Subtract each digit of the result from 9
Step: Add a zero in front of the number
Step: First subtract 1 from the number
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Section 8 Quiz
The following quiz comprises 20 questions designed to assess your understanding of
the tricks taught in Section 8 Begin by identifying the appropriate trick to apply for each
problem, and then follow the steps to solve it. Answers are available at the end of the
book.
Quiz
1. 45 x 99 11. 76 x 99
2. 78 x 999 12. 43 X 999
3. 98 x 99 13. 73 x 99
4. 76 x 999 14. 28 x 99
5. 34 X 999 15. 62 X 999
6. 22 x 99 16. 84 x 99
7. 56 X 999 17. 37 x 99
8. 88 x 99 18. 84 X 999
9. 26 x 99 19. 64 X 999
10. 54 X 999 20. 48 x 99
Table 36: Section 8 Quiz
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Math Wizards: George Parker Bidder
George Parker Bidder: The Calculating Engineer
George Parker Bidder (1806–1878) was an extraordinary English
mathematician and mental calculator who captivated audiences
with his remarkable mental arithmetic abilities. Born in a small
village in Devon, England, Bidder’s talents became apparent at a
young age. Despite his lack of formal education, he displayed an
exceptional aptitude for mental calculations, which he honed
through self-study and practical applications. His abilities attracted
considerable attention, leading to his performances at scientific
and mathematical societies across Britain.
Bidder’s skills in mental arithmetic were showcased through
numerous public demonstrations. He was known for his ability to
perform complex calculations quickly and accurately, such as
multiplying large numbers or extracting square roots. One of his
most famous feats involved calculating the cube root of a large
number with astonishing speed, a skill that earned him widespread
recognition and acclaim. His performances were not merely
demonstrations of raw talent but were underpinned by his deep
understanding of mathematical principles and techniques.


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number with astonishing speed, a skill that earned him widespread
recognition and acclaim. His performances were not merely
demonstrations of raw talent but were underpinned by his deep
understanding of mathematical principles and techniques.
In addition to his public demonstrations, Bidder’s mathematical
abilities earned him a position as a consulting engineer and
surveyor. He applied his skills to practical problems, including
calculating measurements for engineering projects and providing
solutions for various technical challenges. His ability to perform
mental calculations with precision proved invaluable in his
professional work, further solidifying his reputation as a remarkable
mathematical mind.
Bidder’s contributions to mathematics and mental calculation were
significant, though his work is less well-known today compared to
other historical figures. His extraordinary mental feats and practical
applications of arithmetic demonstrated the potential of the human
mind to achieve remarkable feats of calculation. George Parker
Bidder remains a celebrated figure in the history of mathematics
for his impressive mental arithmetic skills and his ability to apply
these skills to real-world problems.


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mind to achieve remarkable feats of calculation. George Parker
Bidder remains a celebrated figure in the history of mathematics
for his impressive mental arithmetic skills and his ability to apply
these skills to real-world problems.
"The study of mathematics, like the Nile, begins in
minuteness but ends in magnificence." – Charles
Caleb Colton
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Section 9
9-A1: Sum of even numbers between 1 and 100
When:
Add all even numbers starting with 2 to 100.
How:
1. Divide the last even number by 2.
2. Multiply the result (n) by n + 1.
Visual Flow:
Start
Problem: Sum of even numbers
between 1 and 20 =?
Step 1: divide the last even number
by 2
20 / 2 = 10
Step 2: Multiple the result of step 1
by same number plus 1
10 x (10 +1) = 110
Answer: Sum of even numbers
between 1 and 20 = 110
End
Figure 29:9-A1: Sum of even numbers between 1 and 100
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Example and Practice Questions
Example 1 Practice Questions
Problem: Sum of even numbers
between 1 and 20 =?
Step1: 20 / 2 = 10
Step2: 10 x (10 + 1) 1. Sum of even numbers between 1 and
Step3: 10 x 11 10 =
2. Sum of even numbers between 1 and
Answer: 2 + 4 + 6…+20 = 110
40 =?
3. Sum of even numbers between 1 and
Example 2
30 =
Problem: Sum of even numbers 4. Sum of even numbers between 1 and
between 1 and 100 =? 60 =
5. Sum of even numbers between 1 and
Step1: 100 / 2 = 50
80 =
Step2: 50 x (50 + 1)
Step3: 50 x 51
Answer: 2 + 4 + 6…+100 = 2550
Table 37: 9-A1: Sum of even numbers between 1 and 100
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
9-A2: Sum of odd numbers between 1 and 100
When:
Add all odd numbers starting 1.
How:
1. Divide the last number by 2.
2. Multiply the result (n) by n.
Visual Flow :
Start
Problem: Sum of odd numbers
between 1 and 100 =?
Step 1: Step 1: divide the last even
number by 2
100 / 2 = 50
Step 2: Multiple the result of step 1
by the same number
50 x 50 = 2500
Answer: Sum of odd numbers
between 1 and 100 = 2500
End
Figure 30:9-A2: Sum of odd numbers between 1 and 100
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Example and Practice Questions
Example 1 Practice Questions
Problem: Sum of odd numbers
between 1 and 100 =?
Step1: 100 / 2 = 50 1. Sum of odd numbers between 1 and
Step2: 50 x 50 10 =
2. Sum of odd numbers between 1 and
Answer: 3 + 5 + 7…+100 = 2500
30 =
3. Sum of odd numbers between 1 and
Example 2 24 =
4. Sum of odd numbers between 1 and
Problem: Sum of odd numbers
50 =
between 1 and 20 =?
5. Sum of odd numbers between 1 and
Step1: 20 / 2 = 10
80 =
Step2: 10 x 10
Answer: 3 + 5 + 7…+20 = 100
Table 38: A2: Sum of odd numbers between 1 and 100
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Section 9 Order these Steps
Arrange the steps for the following tricks. Please write down the step number in the left
column to solve this math trick.
1. The sum of even numbers between 1 and 100:
Step Numbers Actions
Step: Multiply the result (n) by n + 1
Step: Divide the last even number by 2
2. Sum of odd numbers between 1 and 100:
Step Numbers Actions
Step: Multiply the result (n) by n
Step: Divide the last number by 2
Section 9 Quiz
The following quiz comprises 10 questions designed to assess your understanding of
the tricks taught in Section 9. Begin by identifying the appropriate trick to apply for each
problem, and then follow the steps to solve it. Answers are available at the end of the
book.
Quiz
Sum of even numbers Sum of odd numbers between
1. 6.
between 1 and 30 1 and 50
Sum of odd numbers between Sum of even numbers
2. 7.
1 and 20 between 1 and 80
Sum of even numbers Sum of even numbers
3. 8.
between 1 and 60 between 1 and 90
Sum of odd numbers between Sum of odd numbers between
4. 9.
1 and 30 1 and 90
The sum of even numbers Sum of odd numbers between
5. 10.
between 1 and 100 1 and 60
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Math Wizards: Aryabhata
Aryabhata: The Value System and ZERO
Aryabhata, an ancient Indian mathematician and astronomer born
in 476 CE, is one of the most influential figures in early Indian
mathematics. His seminal work, the Aryabhatiya, written in 499
CE, is a comprehensive text that covers a wide range of
mathematical and astronomical topics. Aryabhata introduced and
developed concepts such as the place value system and zero,
which were foundational for the evolution of mathematics in India
and beyond.
In his Aryabhatiya, Aryabhata made significant contributions to
trigonometry, providing sine tables, and developing formulas for
solving spherical triangles. He also tackled problems related to
arithmetic and algebra, presenting solutions for linear and
quadratic equations. Aryabhata's work on the approximation of π
(pi) was remarkably accurate for its time, reflecting his advanced
understanding of geometry and numerical methods.
Aryabhata's contributions were not limited to mathematics; he also
made important advancements in astronomy. He proposed a
heliocentric model where the Earth rotates on its axis, explaining
the apparent daily motion of celestial bodies. His methods for


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made important advancements in astronomy. He proposed a
heliocentric model where the Earth rotates on its axis, explaining
the apparent daily motion of celestial bodies. His methods for
calculating eclipses and the positions of planets were highly
advanced and demonstrated a deep understanding of celestial
mechanics.
Despite the passage of centuries, Aryabhata's influence remains
profound. His mathematical and astronomical concepts were
further developed by later scholars and have had a lasting impact
on both fields. Aryabhata's work laid the groundwork for future
mathematical innovations in India and contributed to the global
mathematical heritage. His legacy is celebrated as a pioneer who
bridged the ancient and modern worlds of science and
mathematics.
"Without mathematics, there’s nothing you can do.
Everything around you is mathematics. Everything around you
is numbers." – Shakuntala Devi
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Answer Sheet
Section 1 Practice
1-A1 1-S1 1-S2 1-P1
689 215 551 28.8
1,411 605 171 48
1,194 5,907 665 53.2
1,190 1,763 409 80.1
1,680 167 546 30.72
Section 1 Quiz
1. 4211 (1-S1) 15. 126 (1-P1)
2. 1109 (1-A1) 16. 1877 (1-S1)
3. 63.75 (1-P1) 17. 514 (1-S2)
4. 541 (1-S1) 18. 379 (1-S2)
5. 67.6 (1-P1) 19. 960 (1-A1)
6. 622 (1-S1) 20. 564 (1-S2)
7. 1,237 (1-A1) 21. 331 (1-S2)
8. 115.5 (1-P1) 22. 44 (1-P1)
9. 1231 (1-A1) 23. 5110 (1-S1)
10. 7535 (1-S1) 24. 48 (1-P1)
11. 376 (1-S2) 25. 1528 (1-A1)
12. 40.8 (1-P1) 26. 75 (1-S2)
13. 823 (1-A1) 27. 366 (1-S1)
14. 143 (1-S2) 28. 1197 (1-A1)
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Section 2 Practice
2-T1 2-M1 2-D1
27 4,653 r=6
45 5,247 r=1
63 7,524 r=0
117 8,118 r=7
126 9,306 r=1
Section 2 Quiz
1. 99 (2-T1) 12. 225 (2-T1)
2. r=3 (2-D1) 13. r=2 (2-D1)
3. 180 (2-T1) 14. 5544 (2-M1)
4. 3861 (2-M1) 15. 90 (2-T1)
5. r=2 (2-D1) 16. 6336 (2-M1)
6. 144 (2-T1) 17. 72 (2-T1)
7. 2475 (2-M1) 18. 135 (2-T1)
8. r=6 (2-D1) 19. 18 (2-T1)
9. 54 (2-T1) 20. 9009 (2-M1)
10. 4752 (2-M1) 21. r=6 (2-D1)
11. r=0 (2-D1) 22. 7326 (2-M1)
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Section 3 Practice
3-M1 3-M2 3-M3 3-M4
3,182 384 3,264 2,016
1,512 756 2,016 4,225
4,872 684 2,704 216
2,208 1,068 2,625 2,021
1,824 912 2,304 621
Section 3 Quiz
1. 2,208 (3-M1) 15. 912 (3-M2)
2. 2,625 (3-M3) 16. 5,624 (3-M4)
3. 4,872 (3-M1) 17. 756 (3-M2)
4. 384 (3-M2) 18. 625 (3-M4)
5. 3,182 (3-M1) 19. 3,149 (3-M3)
6. 7,224 (3-M4) 20. 1,495 (3-M1)
7. 3,264 (3-M3) 21. 4,221 (3-M4)
8. 1,512 (3-M1) 22. 1,104 (3-M2)
9. 2,304 (3-M3) 23. 2,021 (3-M4)
10. 7,225 (3-M4) 24. 7,221 (3-M1)
11. 1,824 (3-M1) 25. 708 (3-M2)
12. 684 (3-M2) 26. 1,068 (3-M2)
13. 2,016 (3-M3) 27. 2,704 (3-M3)
14. 2,409 (3-M3) 28. 9,025 (3-M4)
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Section 4 Practice
4-M1 4-M2
8,052 42,624
5,346 62,937
2,365 16,539
3,839 30,858
1,694 47,952
Section 4 Quiz
1. 53,502 (4-M2) 9. 1,122 (4-M1)
2. 4,136 (4-M1) 10. 53,502 (4-M2)
3. 7,667 (4-M1) 11. 4,136 (4-M1)
4. 35,409 (4-M2) 12. 58,608 (4-M2)
5. 2,651 (4-M1) 13. 82,473 (4-M2)
6. 26,307 (4-M2) 14. 5,599 (4-M1)
7. 9,394 (4-M1) 15. 1,122 (4-M1)
8. 72,594 (4-M2) 16. 51,948 (4-M2)
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Section 5 Practice
5-M1 5-M2 5-M3 5-M4 5-M5
1,800 2,100 4,800 4,275 1,320
400 4,250 1,350 5,525 1,440
2,700 3,200 6,150 1,625 180
1,100 3,800 2,550 2,275 360
51 4,900 4,250 1,425 1,140
Section 5 Quiz
1. 1,800 (5-M1) 15. 2,550 (5-M3)
2. 2,100 (5-M2) 16. 1,625 (5-M4)
3. 4,800 (5-M3) 17. 51.0 (5-M1)
4. 1,575 (5-M4) 18. 4,900 (5-M2)
5. 240 (5-M5) 19. 8,40 (5-M5)
6. 4,250 (5-M2) 20. 2,025 (5-M1)
7. 1,350 (5-M3) 21. 2,275 (5-M4)
8. 5,225 (5-M4) 22. 2,850 (5-M2)
9. 2,700 (5-M1) 23. 1,500 (5-M3)
10. 3,200 (5-M2) 24. 1,425 (5-M4)
11. 6,150 (5-M3) 25. 14.0 (5-M1)
12. 5,525 (5-M4) 26. 40.0 (5-M2)
13. 1,100 (5-M1) 27. 6,900 (5-M3)
14. 1,140 (5-M5) 28. 2,475 (5-M4)
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Section 6 Practice
6-M1 6-M2 6-M3
10,608 8,928 9,964
11,130 8,460 9,360
11,227 8,924 9,828
11,544 8,645 10,028
11,984 8,190 10,323
Section 6 Quiz
1. 10,914 (6-M1) 13. 11,639 (6-M1)
2. 7,820 (6-M2) 14. 13,328 (6-M1)
3. 10,290 (6-M3) 15. 8,277 (6-M2)
4. 11,440 (6-M1) 16. 6,384 (6-M2)
5. 8,526 (6-M2) 17. 9,215 (6-M2)
6. 9,570 (6-M3) 18. 8,096 (6-M2)
7. 13,800 (6-M1) 19. 9,936 (6-M3)
8. 11,448 (6-M1) 20. 10,235 (6-M3)
9. 9,024 (6-M2) 21. 9,682 (6-M3)
10. 9,595 (6-M3) 22. 10,192 (6-M3)
11. 12,765 (6-M1) 23. 9,483 (6-M3)
12. 15,000 (6-M1) 24. 9,024 (6-M2)
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Section 7 Practice
7-M1 7-M2 7-M3 7-M4 7-M5
2,323 12,423 12,012 789,789 1,279,278
1,212 22,725 26,026 822,822 3,629,626
5,656 56,560 52,052 567,567 5,327,322
3,535 35,956 37,037 379,379 3,769,766
4,646 42,016 89,089 987,987 1,806,805
Section 7 Quiz
1. 231,231 (7-M4) 11. 5,252 (7-M1)
2. 34,845 (7-M2) 12. 2,357,355 (7-M5)
3. 6,464 (7-M1) 13. 3,434 (7-M1)
4. 54,054 (7-M3) 14. 21,412 (7-M2)
5. 124 x 101 (7-M2) 15. 580,580 (7-M4)
6. 12,524 (7-M5) 16. 45,753 (7-M2)
7. 23,023 (7-M3) 17. 341,341 (7-M4)
8. 47,167 (7-M2) 18. 78,078 (7-M3)
9. 87,870 (7-M42) 19. 23,432 (7-M2)
10. 9,797 (7-M1) 20. 12,524 (7-M2)
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Section 8 Practice
8-M1 8-M2
7,722 41,958
2,871 66,933
3,663 33,966
3,267 51,948
4,851 26,973
Section 8 Quiz
1. 4,455 (8 – M1) 11. 7, 524 (8 – M1)
2. 77,922 (8 – M2) 12. 42, 957 (8 – M2)
3. 9,702 (8 – M1) 13. 7, 227 (8 – M1)
4. 75,924 (8 – M2) 14. 2, 772 (8 – M1)
5. 33, 966 (8 – M2) 15. 61, 938 (8 – M2)
6. 2,178 (8 – M1) 16. 8, 316 (8 – M1)
7. 55, 944 (8 – M2) 17. 3, 663 (8 – M1)
8. 8, 712 (8 – M1) 18. 83, 916 (8 – M2)
9. 2, 574 (8 – M1) 19. 63, 936 (8 – M2)
10. 53, 946 (8 – M2) 20. 4, 752 (8 – M1)
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Section 9 Practice
9-A1 9-A2
30 25
420 225
240 144
930 625
1,640 1,600
Section 9 Quiz
1. 240 (9 – A1) 6. 625 (9 – A2)
2. 100 (9 – A2) 7. 1,640 (9 – A1)
3. 930 (9 – A1) 8. 2,070 (9 – A1)
4. 225 (9 – A2) 9. 2,050 (9 – A2)
5. 2,550 (9 – A1) 10. 900 (9 – A2)
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Bibliography
RAPID CALCULATION
Cutler, Ann, and Rudolph McShane. The Trachtenberg Speed System of Basic
Mathematics. New York: Doubleday, 1960.
Devi, Shakuntala. Figuring: The Joys of Numbers. New York: Basic Books, 1964.
Tirtha, Jagadguru Swami Bharati Krishna, Shankaracharya of Govard-
hana Pitha. Vedic Mathematics or “Sixteen Simple Mathematical For- mulae from the
Vedas.” Banaras, India: Hindu University Press, 1965.
Menninger, K. Calculator’s Cunning. New York: Basic Books, 1964.
MEMORY
Lorayne, Harry, and Jerry Lucas. The Memory Book. New York: Ballan- tine Books,
1974.
RECREATIONAL MATHEMATICS
Gardner, Martin. Magic and Mystery. New York: Random House, 1956. ———.
Mathematical Carnival. Washington, D.C.: Mathematical Associa-
tion of America, 1965.
Stewart, Ian. Game, Set, and Math: Enigmas and Conundrums. New York:
Penguin Books, 1989.
ADVANCED MATHEMATICS (BY ARTHUR BENJAMIN)
Benjamin, Arthur T., and Jennifer J. Quinn. Proofs That Really Count: The Art of
Combinatorial Proof. Washington: Mathematical Association of America, 2003.
Benjamin, Arthur T., and Kan Yasuda. “Magic ‘Squares’ Indeed!,” The American
Mathematical Monthly 106, no. 2 (February 1999): 152–56.
Useful Links


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Combinatorial Proof. Washington: Mathematical Association of America, 2003.
Benjamin, Arthur T., and Kan Yasuda. “Magic ‘Squares’ Indeed!,” The American
Mathematical Monthly 106, no. 2 (February 1999): 152–56.
Useful Links
Number Description Website address
1. Vibba https://Vibba.com
2. TMSCA https://www.tmsca.org
3. Mathematical Olympiad https://moems.org
4. The International Mathematical Olympiad https://www.imo-official.org
(IMO)
5. Lemonade With Ice Charity https://lemonadewithice.org
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
Index
A Nord Anglia Education School .................................................................................... 3
Albert Einstein ........................................................................................................ 38, 77
Arti Venkatesh ............................................................................................................ 2, 4
Aryabhata ................................................................................................................... 101
Bill Delbrugge ................................................................................................................. 8
Book series .................................................................................................................. 16
Charles Caleb Colton ................................................................................................... 95
Cindy Proske .................................................................................................................. 9
Doug Bell ........................................................................................................................ 3


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Doug Bell ........................................................................................................................ 3
Georg Cantor ......................................................................................................... 55, 88
Ice Venkatesh .................................................................................................... 1, 2, 8, 9
James Joseph Sylvester .............................................................................................. 49
Mental Mathematics ............................................................................................... 10, 13
Path to Expert .............................................................................................................. 17
Shakuntala Devi ......................................................... 29, 38, 49, 55, 68, 77, 88, 95, 101
Singapore Mathematics .................................................................................... 10, 12, 13
Stefan Banach .............................................................................................................. 29
The Village School ......................................................................................................... 3


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The Village School ......................................................................................................... 3
Vedic Mathematics ..................................................................................... 10, 11, 12, 14
Vibba LLC .................................................................................................................. 1, 2
William Paul Thurston .................................................................................................. 68
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ELEMENTARY MATHEMATICS: CORE SKILLS (BOOK 1)
About The Author
Ice Venkatesh, a recognized state medalist, has
excelled in her own academic and professional
pursuits. She also mentored numerous students to
achieve top honors in prestigious mathematics
competitions and examinations. Her unwavering
dedication, passion, and expertise make her an
invaluable asset to the educational community, where
she strives to ensure every student receives the
highest quality instruction and support needed to
succeed.
Ice has combined her profound knowledge of
Singapore, Vedic, and Mental Math into this
comprehensive book series. This initiative is designed
to provide students with a strong foundation in
essential mathematical concepts that are crucial for
their academic journey. The first book in the series focuses on the fundamental principles
of math, aligning with the curriculum requirements and preparing students for competitive
math events such as the AMC (American Mathematics Competitions), TMSCA (Texas
Math and Science Coaches Association), and the Texas Math League.
Ice is helping The Village School as a Teacher’s Assistant (TA) for the mathematics
department. She loves teaching math, cultivating problem-solving skills, and building the


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Math and Science Coaches Association), and the Texas Math League.
Ice is helping The Village School as a Teacher’s Assistant (TA) for the mathematics
department. She loves teaching math, cultivating problem-solving skills, and building the
confidence necessary for students to excel in academic areas. Her work in this book
series is an extension of her lifelong mission to nurture young minds and guide them
toward success.
Ice is deeply committed to advancing educational opportunities for young girls,
particularly in the fields of STEM, recognizing the importance of empowering young girls
to pursue their academic goals. She founded a non-profit organization, Lemonade with
Ice, dedicated to this cause. Through this organization, she has established a
comprehensive scholarship program that provides financial support, resources, and
mentorship to young girls, ensuring they have access to quality education. This
scholarship program is designed to assist with educational expenses and encourage girls
to pursue careers in STEM, fostering a new generation of female leaders. Ice’s work
through the non-profit reflects her lifelong commitment to gender equality in education
and her belief in the transformative power of learning.


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to pursue careers in STEM, fostering a new generation of female leaders. Ice’s work
through the non-profit reflects her lifelong commitment to gender equality in education
and her belief in the transformative power of learning.
To learn more about this impactful initiative and how you can support the cause, please
visit LemonadewithIce.org.
For more information about personalized math learning and the tailored educational
programs offered, please visit Vibba.com
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