2290.21 – Ratio-Difference-Sum-Ratio

If the ratio of $2 x - y$ to $x + y$ is 2/3, then what is the ratio of $x$ to $y$ ? This sort of problem is known as "symbol pushing" – that is, simply doing algebraic manipulations for no reward other than just being able to do them.

$\dfrac{1}{5}$
$\dfrac{4}{5}$
$1$
$\dfrac{6}{5}$
$\dfrac{5}{4}$

Solution

Proceed as follows:

$\begin{aligned} \dfrac{2 x - y}{x + y} && \dfrac{2}{3} \\2 (x + y) &= 3 (2 x - y) \\2x + 2 y &= 6 x - 3 y \\-4 x &= -5 y \\\dfrac{x}{y} &= \dfrac{5}{4}\end{aligned}$

The answer is e.