GRE Sample Test 4 Quantitative Reasoning 1

You’re asked to find what $\frac{\mathit{b}}{\mathit{a}}$ could be; that tells you there may be more than one possible value for $\frac{\mathit{b}}{\mathit{a}}$. You’re told the ratio of 2a to b is 8 times the ratio of b to a. That’s awkward to keep track of in English—it’s a little easier to write fractions. The ratio of 2a:b equals $8\left(\frac{\mathit{b}}{\mathit{a}}\right)$. So, $2\left(\frac{\mathit{a}}{\mathit{b}}\right)$ = $8\left(\frac{\mathit{b}}{\mathit{a}}\right)$ , or $\frac{2\mathit{a}}{\mathit{b}}$ = $\frac{8\mathit{b}}{\mathit{a}}$. Cross multiply to get $2{\mathit{a}}^2$ = $8{\mathit{b}}^2$ or ${\mathit{a}}^2$ = $4{\mathit{b}}^2$. Multiply each side of the equation by $\frac{1}{4{\mathit{a}}^2}:\frac{{\mathit{a}}^2}{4{\mathit{a}}^2}$ = $\frac{4{\mathit{b}}^2}{4{\mathit{a}}^2}$.
This is the same as $\frac{1}{4}$ = $\frac{{\mathit{b}}^2}{{\mathit{a}}^2}$.
Take the square root of both sides of the equation: ± $\frac{1}{2}$ = $\frac{\mathit{b}}{\mathit{a}}$.
The ratio of b to a is $\frac{1}{2}$ or $-\frac{1}{2}$.
So, (B) and (D) are the answers. This problem is also a great candidate for Back solving, although since this question could have more than one correct answer, you would need to test all answer choices to see which ones work out.