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Difference between revisions of "1985 AJHSME Problems/Problem 1"

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==Question==
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==Problem==
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<math>\frac{3\times5}{9\times11} \times \frac{7\times9\times11}{3\times5\times7} </math><br><br>
 
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<math>(A) 1 (B) 0 (C) 49 (D) \frac{1}{49} (E) 50</math>
 
  
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==Solution==
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<math>\frac{3\times 5}{9\times 11}\times \frac{7\times 9\times 11}{3\times 5\times 7}=</math>
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We '''could''' go at it by just multiplying it out, dividing, etc, but there is a much more obvious, simpler method.<br>Noticing that multiplying and dividing by the same number is the equivalent of multiplying (or dividing) by 1, we can rearrange the numbers in the numerator and the denominator (commutative property of multiplication) so that it looks like...<br><br><math>\frac{3}{3} \times \frac{5}{5} \times \frac{7}{7} \times \frac{9}{9} \times \frac{11}{11}</math><br><br>Notice that each number is still there, and nothing has been changed - other than the order.<br>Finally, since each fraction is equal to one, we have <math>1\times1\times1\times1\times1</math>, which is equal to 1.
 
  
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Thus, <math>A</math> is the answer.
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<math>\text{(A)}\ 1 \qquad \text{(B)}\ 0 \qquad \text{(C)}\ 49 \qquad \text{(D)}\ \frac{1}{49} \qquad \text{(E)}\ 50</math>
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==Solution 1==
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By the [[associative property]], we can rearrange the numbers in the numerator and the denominator. [katex display=true]\frac{3}{3}\cdot \frac{5}{5}\cdot\frac{7}{7}\cdot\frac{9}{9}\cdot\frac{11}{11}=1\cdot1\cdot1\cdot1\cdot1=\boxed{\text{(A)}  1}[/katex]
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==Solution 2==
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Notice that the <math>9 \times 11</math> in the denominator of the first fraction cancels with the same term in the second fraction, the <math>7</math>s in the numerator and denominator of the second fraction cancel, and the <math>3 \times 5</math> in the numerator of the first fraction cancels with the same term in the denominator second fraction. Then everything in the expression cancels, leaving us with <math>\boxed{\textbf{(A)}~1}</math>.
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~[https://artofproblemsolving.com/wiki/index.php/User:Cxsmi cxsmi]
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== Solution 3 (Brute Force) ==
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(Note: This method is highly time consuming and should only be used as a last resort in math, also please do not do this.)
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<math>3 \times 5 \times 7 \times 9 \times 11 = 10395</math>
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<math>9 \times 11 \times 3 \times 5 \times 7 = 10395</math>
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Thus, the answer is <math>1</math>, or <math>\boxed{\textbf{(A)}\ 1}</math>
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~ lovelearning999 ~[[shunyipanda]] (Minor edit)
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==Solution 4 (Easy, similar to solution 1)==
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Notice the two fractions are multiplied, so we can consider them as 1 fraction. All terms cancel out, leaving us with <math>\boxed{\textbf{(A)}~1}</math>.
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~[[shunyipanda]]
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==Solution 5 (Capital Pi)==
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This fraction can be simplified to <cmath>\frac{\prod_{k=1}^{5}(2k+1)}{\prod_{k=1}^{5}(2k+1)}</cmath> The numerator and denominator are the same, so the answer is <math>\boxed{\textbf{(A)}\ 1}</math>. ~[[shunyipanda]] Note: please can somebody make this look better
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==Video Solution by BoundlessBrain!==
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https://youtu.be/eC_Vu3vogHM
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==See Also==
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{{AJHSME box|year=1985|before=First <br> Question|num-a=2}}
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[[Category:Introductory Algebra Problems]]
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{{MAA Notice}}

Latest revision as of 13:43, 28 October 2025

Problem

$\frac{3\times 5}{9\times 11}\times \frac{7\times 9\times 11}{3\times 5\times 7}=$


$\text{(A)}\ 1 \qquad \text{(B)}\ 0 \qquad \text{(C)}\ 49 \qquad \text{(D)}\ \frac{1}{49} \qquad \text{(E)}\ 50$

Solution 1

By the associative property, we can rearrange the numbers in the numerator and the denominator. [katex display=true]\frac{3}{3}\cdot \frac{5}{5}\cdot\frac{7}{7}\cdot\frac{9}{9}\cdot\frac{11}{11}=1\cdot1\cdot1\cdot1\cdot1=\boxed{\text{(A)} 1}[/katex]

Solution 2

Notice that the $9 \times 11$ in the denominator of the first fraction cancels with the same term in the second fraction, the $7$s in the numerator and denominator of the second fraction cancel, and the $3 \times 5$ in the numerator of the first fraction cancels with the same term in the denominator second fraction. Then everything in the expression cancels, leaving us with $\boxed{\textbf{(A)}~1}$.

~cxsmi

Solution 3 (Brute Force)

(Note: This method is highly time consuming and should only be used as a last resort in math, also please do not do this.) $3 \times 5 \times 7 \times 9 \times 11 = 10395$

$9 \times 11 \times 3 \times 5 \times 7 = 10395$

Thus, the answer is $1$, or $\boxed{\textbf{(A)}\ 1}$

~ lovelearning999 ~shunyipanda (Minor edit)

Solution 4 (Easy, similar to solution 1)

Notice the two fractions are multiplied, so we can consider them as 1 fraction. All terms cancel out, leaving us with $\boxed{\textbf{(A)}~1}$. ~shunyipanda

Solution 5 (Capital Pi)

This fraction can be simplified to \[\frac{\prod_{k=1}^{5}(2k+1)}{\prod_{k=1}^{5}(2k+1)}\] The numerator and denominator are the same, so the answer is $\boxed{\textbf{(A)}\ 1}$. ~shunyipanda Note: please can somebody make this look better

Video Solution by BoundlessBrain!

https://youtu.be/eC_Vu3vogHM

See Also

1985 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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