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Difference between revisions of "2021 AMC 10B Problems/Problem 5"

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The ages of Jonie's four cousins are distinct single-digit positive integers. Two of the cousins' ages multiplied together give <math>24</math>, while the other two multiply to <math>30</math>. What is the sum of the ages of Jonie's four cousins?
  
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why u tryna cheat??
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<math>\textbf{(A)} ~21 \qquad\textbf{(B)} ~22 \qquad\textbf{(C)} ~23 \qquad\textbf{(D)} ~24 \qquad\textbf{(E)} ~25</math>
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==Solution 1==
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First look at the two cousins' ages that multiply to <math>24</math>. Since the ages must be single-digit, the ages must either be <math>3 \text{ and } 8</math> or <math>4 \text{ and } 6.</math>
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Next, look at the two cousins' ages that multiply to <math>30</math>. Since the ages must be single-digit, the only ages that work are <math>5 \text{ and } 6.</math> Remembering that all the ages must all be distinct, the only solution that works is when the ages are <math>3, 8</math> and <math>5, 6</math>.
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We are required to find the sum of the ages, which is <cmath>3 + 8 + 5 + 6 = \boxed{(B) \text{ } 22}.</cmath>
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-PureSwag

Revision as of 16:39, 11 February 2021

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 1

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

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