2015 AMC 12A Problem 6

Attempt Problem 6 of the 2015 AMC 12A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2015 AMC 12A solutions, or check the answer key.

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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: 1350

Small hint:

Let

p,c

be their current ages:

p-2=3(c-2)

and

p-4=4(c-4)

Big hint:

Solve for

p

and

c,

then find

n

with

p+n = 2(c+n)

Solution:

Let

p

and

c

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

p-2 = 3(c-2)

and

p-4 = 4(c-4).

Solving these gives

p = 20

and

c = 8,

so Pete is

12

years older than Claire.

The ratio is

2 : 1

when Claire is

12,

which is

12 - 8 = 4

years from now.

Thus, the correct answer is B.