How to solve a ratio question

Question

Studying for the GRE. In the GRE guide, it says that

If the ratio is $2x:5y$, and this equals the ratio $3:4$, what is the ratio of $x:y$?

I tried cross multiplying but I don't get the answer. It says the answer is $15:8$. I get $8:15$. Which step am I missing?

Answer 1

We are given: $$\dfrac {2x}{5y} = \frac 34$$

$$2x\cdot (4) = 5y \cdot (3)\tag{1}$$ $$ \iff 8x = 15 y\tag{2: cross-multiplied}$$ $$\iff \frac {8x}{y} = 15\tag{divide by y}$$ $$ \iff \frac xy = \frac{15}{8}\tag{divide by 8}$$

It seems as though you went from $(2)$ to $\dfrac {8x}{15y} = 1$, concluding the ratio is $8:15$. But we want $x: y$ which is the value of $\dfrac xy$, so $$\frac {8x}{15y} = 1 \iff \dfrac{8x}{15y}\cdot \dfrac{15}{8} = 1\cdot \dfrac{15}{8} \iff \dfrac xy = \dfrac{15}{8}$$

Rather than cross-multiplying, it makes more sense in this problem to start from the given $$\frac {2x}{5y} = \frac 34 \iff \frac {2x}{5y}\cdot \frac 52 = \frac 34 \cdot \frac 52 \iff \frac xy = \frac{15}{8}$$

Answer 2

You can construct the following equation:

Given that

$\dfrac{2x}{5y} = \dfrac{3}{4}$

Multiply both sides by $\dfrac52$.

$\dfrac{x}{y} = \dfrac{15}{8}$