World Problem Math Algebra Fraction

Question

The denominator of a fraction in simplest form is greater than the numerator by $3$. If $7$ is added to the numerator, and $5$ added to the denominator, then the fraction itself is increased by $\dfrac 1 2$. Find the original fraction.

I got the equation $\dfrac x {x+3} = \dfrac {2x+14} {x+8}$.

When I solve this, I get a weird answer. My equation might be wrong. Can someone help me?

Answer 1

Suppose the fraction (in simplest form) is $\frac{a}{b}$.

Then from the first statement $$b = a+3$$

From the second statement $$\frac{a+7}{b+5} = \frac{1}{2}+\frac{a}{b}$$

You know have a simple system of equations which you can solve.

Answer 2

Let $x$ be the numerator. Then:

$\frac{x}{x+3}+\frac{1}{2}=\frac{x+7}{x+8}$

$\frac{1}{2}(x+8)(x+3)+x(x+8)=(x+3)(x+7)$

Then solve the quadratic equation.