1955 AMC 12 Problem 35

Question

Three boys agree to divide a bag of marbles in the following manner. The first boy takes one more than half the marbles. The second takes a third of the number remaining. The third boy finds that he is left with twice as many marbles as the second boy. The original number of marbles:

is none of the following

cannot be determined from the given data

is

20

or

26

is

14

or

32

is

8

or

38

Answer: B

Concepts:fractions of a remainder integer constraints underdetermined problem

Difficulty rating: 1490

Small hint:

Let the original number be

n

and express the remainder after the first boy

Big hint:

Translate the last two boys’ share condition into an equation and see whether it determines

n

uniquely

Solution:

The first boy takes

\frac{n}{2}+1,

leaving

\frac{n}{2}-1.

The second takes one third of that remainder, and the third receives the other two thirds, automatically twice the second boy’s share. Thus the share condition imposes no unique value of

n.

It only requires

\frac{n}{2}-1

to be a nonnegative multiple of

3,

so many values such as

8,14,20,26,\ldots

work.

Therefore the original number cannot be determined, and the correct answer is B.