2015 AMC 10A Problem 6

Below is the professionally curated solution for Problem 6 of the 2015 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2015 AMC 10A solutions, or check the answer key.

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Concepts:linear equation ratio and proportion

Difficulty rating: 900

The sum of two positive numbers is $5$ times their difference. What is the ratio of the larger number to the smaller number?

$\dfrac{5}{4}$

$\dfrac{3}{2}$

$\dfrac{9}{5}$

$2$

$\dfrac{5}{2}$

Solution:

Let $x$ and $y$ be the two numbers. Then we have that $x + y = 5(x - y).$ Note that we are assuming $x \gt y.$ This gives us $x + y = 5x - 5y$ $6y = 4x.$

Dividing through yields $\dfrac{x}{y} = \dfrac{6}{4} = \dfrac{3}{2}.$

Thus, B is the correct answer.

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