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Difference between revisions of "1993 AJHSME Problems/Problem 1"

(Created page with '==Problem== Which pair of numbers does NOT have a product equal to <math>36</math>? <math> \text{(A)}\ \{-4,-9\}\qquad\text{(B)}\ \{-3,-12\}\qquad\text{(C)}\ \left\{\frac{1}{2},-…')
 
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<math> \text{(A)}\ \{-4,-9\}\qquad\text{(B)}\ \{-3,-12\}\qquad\text{(C)}\ \left\{\frac{1}{2},-72\right\}\qquad\text{(D)}\ \{ 1,36\}\qquad\text{(E)}\ \left\{\frac{3}{2},24\right\} </math>
 
<math> \text{(A)}\ \{-4,-9\}\qquad\text{(B)}\ \{-3,-12\}\qquad\text{(C)}\ \left\{\frac{1}{2},-72\right\}\qquad\text{(D)}\ \{ 1,36\}\qquad\text{(E)}\ \left\{\frac{3}{2},24\right\} </math>
  
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==Solution==
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==Solution 1==
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A.The ordered pair <math>{-4,-9}</math> has a product of <math>-4*-9=36</math>
 
  
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B. The ordered pair <math>{-3,-12}</math> has a product of <math>-3*-12=36</math>
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Let's calculate each of the answer choices and see which one DOES NOT equal <math>36</math>.
  
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C. The ordered pair <math>{1/12, -72}</math> has a product of <math>-36</math>
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<math>A</math> comes out to be <math>-4 \times -9= 36</math>,
  
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D. The ordered pair <math>{1, 36}</math> has a product of <math>1*36=36</math>
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<math>B</math> equals <math>-3 \times -12= 36</math>,
  
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E. The ordered pair <math>{3/2, 24}</math> has a product of <math>3/2*24=36</math>
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<math>C</math> is <math>\frac{1}{2} \times -72= -36</math>,
  
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Since C is the only ordered pair which doesn't equal 36, <math>\boxed{C}</math> is the answer.
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<math>D</math> simplifies to <math>1 \times 36= 36</math>,
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and <math>E</math> equals <math>\frac{3}{2} \times 24= 3 \times 12= 36</math>.
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Thus, our answer is <math>\boxed{C}</math>.
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==Solution 2==
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In order for a product to be positive (<math>36</math>), the numbers should either be both positive or both negative. Looking at our answer choices, the only option that does not fit this description is <math>\boxed{C}</math>.
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==See Also==
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{{AJHSME box|year=1993|before=First<br />Question|num-a=2}}
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{{MAA Notice}}

Latest revision as of 11:05, 27 June 2023

Problem

Which pair of numbers does NOT have a product equal to $36$? $\text{(A)}\ \{-4,-9\}\qquad\text{(B)}\ \{-3,-12\}\qquad\text{(C)}\ \left\{\frac{1}{2},-72\right\}\qquad\text{(D)}\ \{ 1,36\}\qquad\text{(E)}\ \left\{\frac{3}{2},24\right\}$

Solution 1

Let's calculate each of the answer choices and see which one DOES NOT equal $36$.

$A$ comes out to be $-4 \times -9= 36$,

$B$ equals $-3 \times -12= 36$,

$C$ is $\frac{1}{2} \times -72= -36$,

$D$ simplifies to $1 \times 36= 36$,

and $E$ equals $\frac{3}{2} \times 24= 3 \times 12= 36$.

Thus, our answer is $\boxed{C}$.

Solution 2

In order for a product to be positive ($36$), the numbers should either be both positive or both negative. Looking at our answer choices, the only option that does not fit this description is $\boxed{C}$.

See Also

1993 AJHSME (Problems • Answer Key • Resources)
Preceded by
First
Question 		Followed by
Problem 2
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
All AJHSME/AMC 8 Problems and Solutions

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