2021 AMC 10B Problems/Problem 5

Problem

The ages of Jonie's four cousins are distinct single-digit positive integers. Two of the cousins' ages multiplied together give $24$ , while the other two multiply to $30$ . What is the sum of the ages of Jonie's four cousins?

$\textbf{(A)} ~21 \qquad\textbf{(B)} ~22 \qquad\textbf{(C)} ~23 \qquad\textbf{(D)} ~24 \qquad\textbf{(E)} ~25$

Solution

First look at the two cousins' ages that multiply to $24$ . Since the ages must be single-digit, the ages must either be $3 \text{ and } 8$ or $4 \text{ and } 6.$

Next, look at the two cousins' ages that multiply to $30$ . Since the ages must be single-digit, the only ages that work are $5 \text{ and } 6.$ Remembering that all the ages must all be distinct, the only solution that works is when the ages are $3, 8$ and $5, 6$ .

We are required to find the sum of the ages, which is $3 + 8 + 5 + 6 = \boxed{(B) \text{ } 22}.$

-PureSwag

Video Solution by OmegaLearn (Using Factors)

https://youtu.be/oe7lnbDO8bo

~ pi_is_3.14

Video Solution by TheBeautyofMath

https://youtu.be/gLahuINjRzU?t=857

~IceMatrix

Video Solution by Interstigation

https://youtu.be/DvpN56Ob6Zw?t=358

~Interstigation

2021 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 4	Followed by Problem 6
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 AMC 10 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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