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 $b$ ?

$\mathrm{(A)\ } 20\% \qquad \mathrm{(B) \ }25\% \qquad \mathrm{(C) \ } 50\% \qquad \mathrm{(D) \ } 100\% \qquad \mathrm{(E) \ }200\%$

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 $\boxed{A}$ .

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 $\boxed{B}$ .

1997 AHSME (Problems • Answer Key • Resources)
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