1985 AJHSME Problems/Problem 1

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

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

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: Unable to save thumbnail to destination

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