1998 AMC 12 Problem 21

Question

In an

h

-meter race, Sunny is exactly

d

meters ahead of Windy when Sunny finishes the race. The next time they race, Sunny sportingly starts

d

meters behind Windy, who is at the starting line. Both runners run at the same constant speed as they did in the first race. How many meters ahead is Sunny when Sunny finishes the second race?

\frac dh

0

\frac{d^2}{h}

\frac{h^2}{d}

\frac{d^2}{h-d}

Answer: C

Concepts:rates algebra

Difficulty rating: 1630

Small hint:

Express Windy’s speed as a fraction of Sunny’s speed using the first race

Big hint:

In the second race Sunny travels

h+d

meters before finishing

Solution:

If Sunny’s speed is

r,

Windy’s is

\frac{r(h-d)}{h}.

Sunny needs time

\frac{h+d}{r}

in the second race, during which Windy runs

\frac{r(h-d)}h\cdot\frac{h+d}{r} =h-\frac{d^2}{h}.

Sunny finishes at position

h,

so the lead is

\frac{d^2}{h}.

Thus C is correct.