1996 AMC 12 Problem 13

Attempt Problem 13 of the 1996 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1996 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

13.

Sunny runs at a steady rate, and Moonbeam runs

m

times as fast, where

m

is a number greater than

1.

If Moonbeam gives Sunny a head start of

h

meters, how many meters must Moonbeam run to overtake Sunny?

hm

\frac{h}{h+m}

\frac{h}{m-1}

\frac{hm}{m-1}

\frac{h+m}{m-1}

Let Sunny’s speed be

v

and Moonbeam’s speed be

mv

If Moonbeam runs

x

meters, express Sunny’s distance during the same time in terms of

x

Suppose Moonbeam runs

x

meters. The elapsed time is

\frac{x}{mv},

so Sunny runs

\frac{v\cdot x}{mv}=\frac{x}{m}

meters after the start. At the catch,

x=h+\frac{x}{m}.

Solving gives

x=\frac{hm}{m-1},

so the correct answer is D.

Problem 13 in other years