2005 AMC 12A Problems/Problem 6

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Problem

Josh and Mike live 13 miles apart. Yesterday, Josh started to ride his bicycle toward Mike's house. A little later Mike started to ride his bicycle toward Josh's house. When they met, Josh had ridden for twice the length of time as Mike and at four-fifths of Mike's rate. How many miles had Mike ridden when they met?

$(\mathrm {A}) \ 4 \qquad (\mathrm {B}) \ 5 \qquad (\mathrm {C})\ 6 \qquad (\mathrm {D}) \ 7 \qquad (\mathrm {E})\ 8$

Solution

Let $D_J, D_M$ be the distances traveled by Josh and Mike, respectively, and let $r,t$ be the time and rate of Mike. Using $d = rt$ , we have that $D_M = rt$ and $D_J = \left(\frac{4}{5}r\right)\left(2t\right) = \frac 85rt$ . Then $13 = D_M + D_J = rt + \frac{8}{5}rt = \frac{13}{5}rt$ $\Longrightarrow rt = D_M = 5\ \mathrm{(B)}$ .

2005 AMC 12A (Problems • Resources)
Preceded by Problem 5	Followed by Problem 7
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2005 AMC 12A Problems/Problem 6

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