# 1997 AHSME Problems/Problem 4

## Problem

If $a$ is $50\%$ larger than $c$, and $b$ is $25\%$ larger than $c$, then $a$ is what percent larger than $b$? $\mathrm{(A)\ } 20\% \qquad \mathrm{(B) \ }25\% \qquad \mathrm{(C) \ } 50\% \qquad \mathrm{(D) \ } 100\% \qquad \mathrm{(E) \ }200\%$

## Solution

### Solution 1

Translating each sentence into an equation, $a = 1.5c$ and $b = 1.25c$.

We want a relationship between $a$ and $b$. Dividing the second equation into the first will cancel the $c$, so we try that and get: $\frac{a}{b} = \frac{1.5}{1.25}$ $\frac{a}{b} = \frac{150}{125}$ $\frac{a}{b} = \frac{6}{5}$ $a = 1.2b$

In this case, $a$ is $1.2 - 1 = 0.2 = 20\%$ bigger than $b$, and the answer is $\boxed{A}$.

### Solution 2

Arbitrarily assign a value to one of the variables. Since $c$ is the smallest variable, let $c = 100$.

If $a$ is $50\%$ larger than $c$, then $a = 150$.

If $b$ is $25\%$ larger than $c$, then $b = 125$.

We see that $\frac{a}{b} = \frac{150}{125} = 1.2$ So, $a$ is $20\%$ bigger than $b$, and the answer is $\boxed{A}$.

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 