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

Answer: D

Concepts:distance rate and time system of equations

Difficulty rating: 2100

Small Hint:

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

Big Hint:

If those times are $t_1,t_2,t_3$ , write one $100$ -mile equation for the car, Dick, and Harry

Solution:

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.

1951 AMC 12 Problem 50

Answer: D

Difficulty rating: 2100

Small Hint:

Big Hint:

Solution:

Problem 50 in Other Years