1961 AMC 12 Problem 37

Question

In racing over a distance

d

at uniform speed,

A

can beat

B

by

20

yards,

B

can beat

C

by

10

yards, and

A

can beat

C

by

28

yards. Then

d,

in yards, equals:

not determined by the given information

58

100

116

120

Translate each winning margin into a ratio of speeds

Multiply the ratios

\frac{v_B}{v_A}

and

\frac{v_C}{v_B}

to get

\frac{v_C}{v_A}

The margins give

\begin{aligned} \frac{v_B}{v_A}&=\frac{d-20}{d},\\ \frac{v_C}{v_B}&=\frac{d-10}{d},\\ \frac{v_C}{v_A}&=\frac{d-28}{d}. \end{aligned}

Therefore

\frac{(d-20)(d-10)}{d^2}=\frac{d-28}{d}.

Expanding and simplifying yields

2d=200,

so

d=100.

Thus, the correct answer is C.

Problem 37 in other years