# The denominator of a fraction is one more than twice the numerator. If the sum of the fraction and its reciprocal is 2 2/6 find the fraction.

87 views

closed

The denominator of a fraction is one more than twice the numerator. If the sum of the fraction and its reciprocal is $2\frac{16}{21}$ find the fraction.

+1 vote
by (38.0k points)
selected

Let the numerator = x

then denominator = 2x + 1

Then the fraction = $\frac{x}{2x+1}$

Its reciprocal = $\frac{2x+1}{x}$

105x2 + 84x + 21 = 116x2 + 58x

11x2 – 26x – 21 = 0

11x2 – 33x + 7x – 21 = 0

11x (x – 3) + 7 (x – 3) = 0

(x – 3) (11x + 7) = 0

⇒ x – 3 = 0 (or) 11x + 7 = 0

⇒ x = 3 (or) $\frac{-7}{11}$

∴ x = 3 Numerator = 3;

Denominator = 2 × 3 + 1 = 7

Fraction = $\frac{3}{7}$