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

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==Problem==
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Problem
 
Which pair of numbers does NOT have a product equal to <math>36</math>?
 
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},-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 36.
  
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C. The ordered pair <math>{1/12, -72}</math> has a product of <math>-36</math>
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A comes out to be -4 x -9= 36
  
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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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B equals -3 x -12= 36
  
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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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C is 1/2 x -72= -36
  
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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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D simplifies to 1 x 36= 36
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and E equals 3/2 x 24= 3 x 12= 36
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Thus, our answer is C. It is the only negative result.
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Solution 2
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In order for a product to be positive (36), the numbers should either be both positive or both negative. Looking at our answer choices, this description matches with C since one number is positive and the other is negative.
  
 
==See Also==
 
==See Also==
 
{{AJHSME box|year=1993|before=First<br />Question|num-a=2}}
 
{{AJHSME box|year=1993|before=First<br />Question|num-a=2}}
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{{MAA Notice}}

Revision as of 19:32, 28 January 2022

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 x -9= 36

B equals -3 x -12= 36

C is 1/2 x -72= -36

D simplifies to 1 x 36= 36

and E equals 3/2 x 24= 3 x 12= 36

Thus, our answer is C. It is the only negative result.

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, this description matches with C since one number is positive and the other is negative.

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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