The numerator of a fraction is three less than the denominator. The sum of the fraction and four times its reciprocal is 17/2. How do you find the fraction?

Jun 2, 2016

Let the numerator be x.

Explanation:

$\frac{x}{x + 3} + 4 \times \frac{x + 3}{x} = \frac{17}{2}$

$\frac{x}{x + 3} + \frac{4 x + 12}{x} = \frac{17}{2}$

Put on a common denominator.

$\frac{{x}^{2} + \left(4 x + 12\right) \left(x + 3\right)}{\left(x\right) \left(x + 3\right)} = \frac{17}{2}$

$\frac{{x}^{2} + 4 {x}^{2} + 12 x + 12 x + 36}{\left(x\right) \left(x + 3\right)} = \frac{17}{2}$

$2 \left(5 {x}^{2} + 24 x + 36\right) = 17 \left({x}^{2} + 3 x\right)$

$10 {x}^{2} + 48 x + 72 = 17 {x}^{2} + 51 x$

$0 = 7 {x}^{2} + 3 x - 72$

$0 = 7 {x}^{2} - 21 x + 24 x - 72$

$0 = 7 x \left(x - 3\right) + 24 \left(x - 3\right)$

$0 = \left(7 x + 24\right) \left(x - 3\right)$

$x = - \frac{24}{7} \mathmr{and} 3$

The fraction is $\frac{3}{6} = \frac{1}{2}$ or $\frac{8}{1} = 8$

Hopefully this helps.