1991 AMC 12 Problem 11

Question

Jack and Jill run

10

kilometers. They start at the same point, run

5

kilometers up a hill, and return to the starting point by the same route. Jack has a

10

-minute head start and runs at the rate of

15

km/hr uphill and

20

km/hr downhill. Jill runs

16

km/hr uphill and

22

km/hr downhill. How far from the top of the hill are they when they pass going in opposite directions?

\frac54

km

\frac{35}{27}

km

\frac{27}{20}

km

\frac73

km

\frac{28}{9}

km

Determine Jill’s position when Jack reaches the top

After Jack turns around, use their combined closing speed

Jack reaches the top in

\frac{5}{15}=\frac{1}{3}

hour. Jill has then run for

\frac{1}{3}-\frac{1}{6}=\frac{1}{6}

hour and is

\frac{16}{6}=\frac{8}{3}

km uphill, so they are

5-\frac{8}{3}=\frac{7}{3}

km apart. Their closing speed is

20+16=36

km/hr, so they meet after

\frac{\frac{7}{3}}{36}=\frac{7}{108}

hour. Jack descends

20\left(\frac7{108}\right)=\frac{35}{27}

kilometers.

Thus the correct answer is B.

Problem 11 in other years