2015 AMC 10A Problem 6

Let the numbers be $a > b > 0$. The condition says
$$a + b = 5(a - b) = 5a - 5b.$$
Move everything to one side: $6b = 4a$, so
$$\frac{a}{b} = \frac{6}{4} = \frac{3}{2}.$$

Quick check with $a = 3$, $b = 2$: the sum is $5$ and the difference is $1$, and indeed $5 = 5 \cdot 1$.

The answer is $\boxed{\textbf{(B)}\ \frac{3}{2}}$.

A single linear relation between two unknowns cannot determine them, but it always determines their ratio: divide the relation through by $b$ and solve for $a/b$. Any "the ratio of ... to ..." question with one equation works this way.

Test the choices. For $\frac{3}{2}$, take $3$ and $2$: sum $5$, difference $1$, and $5 = 5 \cdot 1$. None of the other choices survives the same test (for example $\frac{5}{4}$ gives sum $9$ and difference $1$).

• Solving for the difference-to-sum ratio, or writing the condition backwards as $5(a + b) = a - b$, which gives a negative ratio.
• Reporting $\frac{2}{3}$ (smaller to larger) after a correct computation; the question asks for larger to smaller.