1978 AMC 12 Problem 5

Question

Four boys bought a boat for

\$60.

The first boy paid one half of the sum of the amounts paid by the other boys; the second boy paid one third of the sum of the amounts paid by the other boys; and the third boy paid one fourth of the sum of the amounts paid by the other boys. How much did the fourth boy pay?

\$10

\$12

\$13

\$14

\$15

Answer: C

Concepts:linear equation ratio and proportion money

Difficulty rating: 1280

Small hint:

Replace each “sum paid by the other boys” by

60

minus that boy’s payment

Big hint:

Solve separately for the first three payments, then subtract their sum from

60

Solution:

Let the first three payments be

w,

x,

and

y.

Then

\begin{aligned} w&=\frac12(60-w),\\ x&=\frac13(60-x),\\ y&=\frac14(60-y). \end{aligned}

Hence

w=20,

x=15,

y=12.

The fourth payment is

60-20-15-12=13

dollars.

Therefore, the correct answer is C.