Finding Optimal Bundle and Change in Satisfaction after Changes in Budget Constraint I am looking at the following exercise and struggling with the solution proposed by my microeconomics book.


  A consumer spends all his income on two goods, X and Y. The prices he paid and the quantities he consumed last year are as follows: PX = 15, X = 20, PY = 25, and Y = 30. 
  We assume the consumer's preferences follow the 5 properties for preference ordering (complete, transitive, more-is-better, convex, continuous).
  
  If the prices next year are PX = 6 and PY = 30, and the consumer's income is 1,020, will he be better or worse off than he was in the previous year? (Assume that his tastes do not change.)


For the Solution we first find the consumer's budget constraint of last year:

M = PX * X + PY * Y = 15 * 20 + 25 * 30 = 1050


We need to ask can he buy last year’s bundle given the new prices?

PXnew * X + PYnew * Y = 6 * 20 + 30 * 30 = 1020


Given that the consumer can still afford last year's bundle, he must be (in my opinion) at least as well off. I am working with a economics book and the solution proposed in the book claims, that the consumer must be better off using the following figure as explanation:



I am wondering, couldn't the indifference curve that is tangent to the bundle at X = 20, Y = 30 for the last year, also be tangent to a point on the new budget constraint and equal the highest indifference curve that can be reached? 

Is there an explanation, given the 5 properties of preference ordering that the consumer MUST be better off.

EDIT:

I have added the image below to clarify my question.



I am wondering whether the indifference curve in red in the image could look like this and imply that the satisfaction level is not increasing as the consumer will stay on the same indifference curve but consume a different combination of X and Y.
I understand what you though, the old indifference curve sure can touch but cannot conclude to be the optimal one. 

As the evidence of the money he can pay 1050 of the last year budget but for this year he could only use just 1020 to buy the same combination of those goods with the same utility so that means he still can buy and consume more with the relative purchasing power. 

I will just use 2 assumptions that are "more is better", "no bliss point". It is said that the budget which has shown or given here is not significant to the level that the consumer will keep and not spend so that as he can spend he will and with "no bliss point" that means this consumption is too small and yet not to the level that he will have negative marginal utility.

Therefore, as he can buy the old combination with the lesser price so he can allocate the leftover budget for either of two goods.