# 2005 AMC 10A Problems/Problem 7

## 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? $\mathrm{(A) \ } 2\qquad \mathrm{(B) \ } 5\qquad \mathrm{(C) \ } 3\qquad \mathrm{(D) \ } 6\qquad \mathrm{(E) \ } 4$

## Solution

Let $m$ be the distance in miles that Parthib rode.

Since Wahida rode for twice the length of time as Parthib and at four-fifths of Parthib's rate, he rode $2\cdot\frac{4}{5}\cdot m = \frac{8}{5}m$ miles.

Since their combined distance was $13$ miles, $\frac{8}{5}m + m = 13$ $\frac{13}{5}m = 13$ $m = 5 \Longrightarrow \mathrm{(B)}$

## Video Solution

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