1951 AMC 12 Problem 50

Question

Tom, Dick, and Harry started out on a

100

-mile journey. Tom and Harry went by automobile at the rate of

25

mph, while Dick walked at the rate of

5

mph. After a certain distance, Harry got off and walked on at

5

mph, while Tom went back for Dick and got him to the destination at the same time that Harry arrived. The number of hours required for the trip was:

5

6

7

8

None of these answers

Separate the car’s motion into the first forward trip, the return for Dick, and the final forward trip

If those times are

t_1,t_2,t_3

, write one

100

-mile equation for the car, Dick, and Harry

Let

t_1,t_2,t_3

be the car’s times before Harry leaves it, while it returns for Dick, and while it carries Dick forward. The car, Dick, and Harry each cover

100

miles, giving

\begin{aligned} 25t_1-25t_2+25t_3&=100,\\ 5t_1+5t_2+25t_3&=100,\\ 25t_1+5t_2+5t_3&=100. \end{aligned}

Dividing by

5

and solving yields

t_1=3,\ t_2=2,\ t_3=3.

Therefore the common travel time is

t_1+t_2+t_3=8

hours.

Thus, the correct answer is D.

Problem 50 in other years