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Difference between revisions of "1981 AHSME Problems/Problem 4"

(Created page with "==Problem== If three times the larger of two numbers is four times the smaller and the difference between the numbers is 8, the the larger of two numbers is: <math>\text{(A)...")
 
 
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==Problem==  
 
==Problem==  
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If three times the larger of two numbers is four times the smaller and the difference between the numbers is 8, the the larger of two numbers is:
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If three times the larger of two numbers is four times the smaller and the difference between the numbers is <math>8</math>, the the larger of two numbers is
  
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<math>\text{(A)}\quad 16 \qquad \text{(B)}\quad 24 \qquad \text{(C)}\quad 32 \qquad  \text{(D)}\quad 44 \qquad \text{(E)} \quad 52 </math>
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<math>\text{(A)}\ 16 \qquad \text{(B)}\ 24 \qquad \text{(C)}\ 32 \qquad  \text{(D)}\ 44 \qquad \text{(E)} \ 52 </math>
  
 
==Solution==
 
==Solution==
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Let the smaller number be <math>x</math> and the larger number be <math>y.</math> We can use the information given in the problem to write two equations:
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Let <math>x</math> be the smaller number and <math>y</math> be the larger number.  Then <math>3y = 4x</math>, so <math>x = \frac{3}{4}y</math>. Also, <math>y - x = 8</math>, so <math>y - \frac{3}{4}y = \frac{1}{4}y = 8</math>, and <math>y = \boxed{(\textbf{C})\ 32}</math>.
  
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1) Three times the larger of two numbers is four times the smaller.
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==See also==
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<cmath> 3y = 4x </cmath>
 
  
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2) The difference between the number is 8.
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{{AHSME box|year=1981|num-b=3|num-a=5}}
  
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<cmath> y - x = 8 </cmath>
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{{MAA Notice}}
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Solving this system of equation yields <math>x = 24</math> and <math>y = 32.</math> The answer is <math>\text{C}.</math>
 

Latest revision as of 15:57, 18 August 2025

Problem

If three times the larger of two numbers is four times the smaller and the difference between the numbers is $8$, the the larger of two numbers is

$\text{(A)}\ 16 \qquad \text{(B)}\ 24 \qquad \text{(C)}\ 32 \qquad \text{(D)}\ 44 \qquad \text{(E)} \ 52$

Solution

Let $x$ be the smaller number and $y$ be the larger number. Then $3y = 4x$, so $x = \frac{3}{4}y$. Also, $y - x = 8$, so $y - \frac{3}{4}y = \frac{1}{4}y = 8$, and $y = \boxed{(\textbf{C})\ 32}$.

See also

1981 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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