2015 AMC 10A Problem 6

Turn the sentence into an equation, then find the ratio.

1. Call the larger number $a$ and the smaller number $b$. "The sum is $5$ times the difference" says
$$a + b = 5(a - b).$$
2. Multiply out the right side: $a + b = 5a - 5b$.
3. Gather the $a$'s on one side and the $b$'s on the other. Subtract $a$ from both sides and add $5b$ to both sides:
$$6b = 4a.$$
4. Divide both sides by $4b$ to get the ratio the question wants:
$$\frac{a}{b} = \frac{6}{4} = \frac{3}{2}.$$
5. Check with actual numbers $a = 3$, $b = 2$: the sum is $5$, the difference is $1$, and $5$ is indeed $5 \times 1$.

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

One equation with two unknowns cannot tell you both numbers, but it can tell you how they compare. Any doubled pair, like $6$ and $4$, fits too. So when a problem gives one condition and asks for a ratio, collect terms and divide.

Test the choices with real numbers. For $\frac{3}{2}$, use $3$ and $2$: sum $5$, difference $1$, and $5 = 5 \times 1$. It works. For $\frac{5}{4}$, use $5$ and $4$: sum $9$, difference $1$, and $9$ is not $5$. The other choices fail the same way.

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