2005 AMC 10A Problem 7

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 linear equation

Difficulty rating: 1240

Small hint:

Distance equals rate times time; write Josh’s distance in terms of Mike’s

Big hint:

Josh’s distance is

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

of Mike’s, and the two distances add to

13

Solution:

Let Mike ride

m

miles. Josh rides

\frac{4}{5}

the rate for

2

times the time, so Josh’s distance is

\frac{8}{5}m.

Together they cover

13,

so

m + \frac{8}{5}m = \frac{13}{5}m = 13,

giving

m = 5.

Thus, the correct answer is B.