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Difference between revisions of "1997 AHSME Problems/Problem 4"
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==Problem==
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If <math>a</math> is <math>50\%</math> larger than <math>c</math>, and <math>b</math> is <math>25\%</math> larger than <math>c</math>, then <math>a</math> is what percent larger than <math>b</math>?
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<math> \mathrm{(A)\ } 20\% \qquad \mathrm{(B) \ }25\% \qquad \mathrm{(C) \  } 50\% \qquad \mathrm{(D) \  } 100\% \qquad \mathrm{(E) \  }200\%  </math>
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==Solution 1==
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Translating each sentence into an equation, <math>a = 1.5c</math> and <math>b = 1.25c</math>.
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We want a relationship between <math>a</math> and <math>b</math>.  Dividing the second equation into the first will cancel the <math>c</math>, so we try that and get:
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<math>\frac{a}{b} = \frac{1.5}{1.25}</math>
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<math>\frac{a}{b} = \frac{150}{125}</math>
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<math>\frac{a}{b} = \frac{6}{5}</math>
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<math>a = 1.2b</math>
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In this case, <math>a</math> is <math>1.2 - 1 = 0.2 = 20\%</math> bigger than <math>b</math>, and the answer is <math>\boxed{B}</math>.
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==Solution 2==
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Arbitrarily assign a value to one of the variables.  Since <math>c</math> is the smallest variable, let <math>c = 100</math>.
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If <math>a</math> is <math>50\%</math> larger than <math>c</math>, then <math>a = 150</math>.
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If <math>b</math> is <math>25\%</math> larger than <math>c</math>, then <math>b = 125</math>.
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We see that <math>\frac{a}{b} = \frac{150}{125} = 1.2</math>  So, <math>a</math> is <math>20\%</math> bigger than <math>b</math>, and the answer is <math>\boxed{B}</math>.
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== See also ==
 
== See also ==
 
{{AHSME box|year=1997|num-b=3|num-a=5}}
 
{{AHSME box|year=1997|num-b=3|num-a=5}}

Revision as of 16:36, 8 August 2011

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 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 $\boxed{B}$.

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 $\boxed{B}$.


See also

1997 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 3 		Followed by
Problem 5
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions
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