2005 AMC 12A Problem 6

Question

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

Answer: B

Concepts:distance rate and time ratio and proportion

Difficulty rating: 1270

Small hint:

Distance equals rate times time for each rider

Big hint:

Josh’s distance is

\dfrac{4}{5} \cdot 2 = \dfrac{8}{5}

of Mike’s distance

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.