The denominator of a fraction is one more than twice its numerator. If the sum of the fraction and its reciprocal is $\frac{34}{15}$ , find the fraction.

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Solution

Let the fraction be $\frac{x}{y}$
Given $y = 2 x + 1$
Sum of fraction and reciprocal $= \frac{34}{15}$

$\frac{x}{2 x + 1} + \frac{2 x + 1}{x} = \frac{34}{15}$

$\frac{x}{2 x + 1} + 2 + \frac{1}{x} = \frac{34}{15}$

$\frac{x^{2} + 2 x + 1}{(2 x + 1) (x)} = \frac{4}{15}$

$15 x^{2} + 30 x + 15 = 4 (2 x^{2} + x)$
$15 x^{2} + 30 x + 15 = 8 x^{2} + 4 x$
$15 x^{2} - 8 x^{2} + 30 x - 4 x + 15 = 0$
$7 x^{2} + 26 x + 15 = 0$
$7 x^{2} + 21 x + 5 x + 15 = 0$
$7 x (x + 3) + 5 (x + 3) = 0$
$(x + 3) (7 x + 5) = 0$

$x + 3 = 0$ or $7 x + 5 = 0$
$x = - 3$ or $x = - \frac{5}{7}$
$∴ F r a c t i o n = \frac{- 3}{- 2 (3) + 1} = \frac{3}{5}$

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