2015 AMC 10A Problem 8

Question

Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be

2:1?

2

4

5

6

8

Answer: B

Concepts:ages system of equations

Difficulty rating: 1280

Small hint:

Translate the two age statements using current ages

Big hint:

Solve for current ages before setting the future ratio

Solution:

Let

p

and

c

be Pete’s and Claire’s current ages respectively.

Then we have that

p - 2 = 3(c - 2)

and

p - 4 = 4(c - 4).

Simplifying both equations gives us

p = 3c - 4

and

p = 4c - 12.

Setting them equal, we have

3c - 4 = 4c - 12

c = 8.

This means that

p = 3 \cdot 8 - 4 = 20.

Now, we need to find the number of years (

y

) until

20 + y = 2(8 + y),

which gives us

20 + y = 16 + 2y

y = 4.

Thus, B is the correct answer.