AMC 10 · 2015 · #8

Grade 8 rate-ratio
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
systems-of-equationslinear-equations-two-var convert-to-algebra ↑ Prerequisites: linear-equations-two-var
📏 Medium solution 💡 2 insights
Problem
Two years ago Pete was three times as old as his cousin Claire. Four years ago (two years before that) Pete was four times as old as Claire. Starting from now, in how many years will Pete's age to Claire's age be the ratio 2 : 1?

Pick an answer.

(A)
2
(B)
4
(C)
5
(D)
6
(E)
8

AMC 10 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

The puzzle mixes two snapshots in the past with a question about the future, so the safest move is to anchor everything to one fixed moment: now. Tool #4 (Introduce a Variable) names Pete's and Claire's current ages as P and C; every other age in the story is then just P or C shifted by 2 or 4 years. Tool #13 (Convert to Algebra) turns the two word-facts into two equations, which lock down P and C. Once the present ages are known, the future ratio question becomes one more short equation to solve for the number of years.

1STEP 1

Name the present ages

Let P and C be Pete's and Claire's ages now; past ages are P and C minus the shift.

P = Pete now, C = Claire now
2STEP 2

Translate the two facts

The two facts become two equations: P-2 = 3(C-2) and P-4 = 4(C-4).

P-2 = 3(C-2) and P-4 = 4(C-4)
3STEP 3

Solve the system for the ages

Solving both for P and setting them equal gives C = 8, so P = 20.

3C-4 = 4C-12 → C = 8, P = 20
4STEP 4

Write the future-ratio equation

In x years Pete is 20+x, Claire is 8+x; a 2:1 ratio means 20 + x = 2(8 + x).

(20+x)/(8+x) = 2/1 → 20 + x = 2(8+x)
5STEP 5

Solve for the number of years

Solving gives x = 4 years — choice (B).

20 + x = 16 + 2x → x = 4 → (B)
Answer
4
Check the two past facts against P=20, C=8. Two years ago: Pete 18, Claire 6, and 18 = 3 × 6. Four years ago: Pete 16, Claire 4, and 16 = 4 × 4. Both hold. For the future, in 4 years Pete is 24 and Claire is 12, and 24 = 2 × 12, a clean 2:1. Everything matches, so x=4 is confirmed.
💡Key takeaway

Pin every age to today as P and C, let the two clues fix the present ages, then ask how many years make Pete exactly twice Claire.

  • Name the present ages
  • Translate the two facts
  • Solve the system for the ages
  • Write the future-ratio equation
  • Solve for the number of years

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