2013 CEMC Gauss (Grade 8) Problems/Problem 4

Problem

Ahmed is going to the store. One quarter of the way to the store, he stops to talk with Kee. He then continues for $12$ km and reaches the store. How many kilometers does he travel altogether?

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$\textbf{(A)}\ 15 \qquad\textbf{(B)}\ 16 \qquad\textbf{(C)}\ 24 \qquad\textbf{(D)}\ 48 \qquad\textbf{(E)}\ 20$

Solution 1

Let $d$ be the total distance that Ahmed travels. From the second sentence of the problem, he traveled $\frac{1}{4}d$ going from his starting location to Kee.

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We can now see that:

$\frac{1}{4}d + 12 = d$

Solving for $d$ , we have:

$12 = d - \frac{1}{4}d = \frac{4 - 1}{4}d = \frac{3}{4}d$

$d = 12 \div \frac{3}{4} = 12 \times \frac{4}{3} = \boxed {\textbf {(B) } 16}$ km

~anabel.disher

Solution 2

Because Ahmed traveled $\frac{1}{4}$ of the distance from his starting location to Kee, we know that he traveled $1 - \frac{1}{4} = \frac{4 - 1}{4} = \frac{3}{4}$ . This is $3 \times \frac{1}{4}$ , so he traveled $3$ times as much from Kee to the store compared to where he started to Kee.