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The denominator of a fraction is one more than twice the numerator. If the sum of the fraction and its reciprocal is $\frac{16}{21}$, find the fraction.
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Let numerator be $x$,
then denominator be $(2x + 1)$
Fraction = $\frac{x}{2x+1}$
A.T.Q.
$\frac{x}{2x+1} + \frac{2x+1}{x} = \frac{58}{21}$
$\Rightarrow 11x^2 - 26x - 21 = 0$
$\Rightarrow (x - 3)(11x + 7) = 0$
$x \neq -\frac{7}{11}$ So, $x = 3$
$\therefore$ Fraction = $\frac{3}{7}$
then denominator be $(2x + 1)$
Fraction = $\frac{x}{2x+1}$
A.T.Q.
$\frac{x}{2x+1} + \frac{2x+1}{x} = \frac{58}{21}$
$\Rightarrow 11x^2 - 26x - 21 = 0$
$\Rightarrow (x - 3)(11x + 7) = 0$
$x \neq -\frac{7}{11}$ So, $x = 3$
$\therefore$ Fraction = $\frac{3}{7}$