2006 AMC 12A Problem 9

Question

Oscar buys

13

pencils and

3

erasers for

\$1.00.

A pencil costs more than an eraser, and both items cost a whole number of cents. What is the total cost, in cents, of one pencil and one eraser?

10

12

15

18

20

Group as

3

sets of one pencil and one eraser, plus

10

extra pencils

If

s

is the cost of one pencil plus one eraser, then

3s + 10p = 100,

so

3s

is a multiple of

10

Let

p

be a pencil’s cost and

s

the cost of one pencil plus one eraser, in cents. Then

13p + 3e = 3s + 10p = 100,

so

3s

is a multiple of

10

less than

100.

Hence

s \in \{10, 20, 30\},

with

p = 7, 4, 1

respectively.

Since a pencil costs more than an eraser,

p \gt \tfrac{s}{2},

which holds only for

s = 10

(pencil

7,

eraser

3

). So one pencil and one eraser cost

10

cents.

Thus, the correct answer is A.

Problem 9 in other years