Difference between revisions of "1997 AHSME Problems/Problem 4"

Latest revision as of 13:12, 5 July 2013

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 $\boxed{A}$ .

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

@@ Line 5: / Line 5: @@
 <math> \mathrm{(A)\ } 20\% \qquad \mathrm{(B) \ }25\% \qquad \mathrm{(C) \  } 50\% \qquad \mathrm{(D) \  } 100\% \qquad \mathrm{(E) \  }200\%  </math>
-==Solution 1==
+__TOC__
+== Solution ==
+===Solution 1===
 Translating each sentence into an equation, <math>a = 1.5c</math> and <math>b = 1.25c</math>.
@@ Line 19: / Line 21: @@
 <math>a = 1.2b</math>
-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>.
+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{A}</math>.
-==Solution 2==
+===Solution 2===
 Arbitrarily assign a value to one of the variables.  Since <math>c</math> is the smallest variable, let <math>c = 100</math>.
@@ Line 29: / Line 31: @@
 If <math>b</math> is <math>25\%</math> larger than <math>c</math>, then <math>b = 125</math>.
-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>.
+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{A}</math>.
 == See also ==
 {{AHSME box|year=1997|num-b=3|num-a=5}}
+[[Category:Introductory Algebra Problems]]
+{{MAA Notice}}