2005 AMC 12A Problem 6

Below is the professionally curated solution for Problem 6 of the 2005 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2005 AMC 12A solutions, or check the answer key.

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Concepts:distance rate and time ratio and proportion

Difficulty rating: 1270

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?

$4$

$5$

$6$

$7$

$8$

Solution:

Since distance is rate times time, Josh rode $\dfrac{4}{5} \cdot 2 = \dfrac{8}{5}$ as far as Mike.

Let $m$ be the miles Mike rode. Then $13 = m + \dfrac{8}{5}m = \dfrac{13}{5}m,$ so $m = 5.$

Thus, the correct answer is B.

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