AMC 10 · 2015 · #6
Grade 8 rate-ratioPick an answer.
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.
Name the present ages
Two letters name the present ages.
Anchoring every age to one moment turns a confusing timeline into simple add-or-subtract shifts.
6.EE.B.6Introduce A VariableTranslate the two facts
Each fact is written at its own time.
Each English sentence about ages becomes one equation once you write the shifted ages in symbols.
6.EE.A.2Convert To AlgebraSolve the system for the ages
Solving the pair gives both ages.
Two true facts about the same two people leave exactly one pair of ages that fits both.
Two true facts about the same two people leave exactly one pair of ages that fits both.
▸ Why?
Doing the same thing to both sides keeps each equation true, so the two can be combined safely.
▸ Why?
Every other pair fails at least one of the two facts, so only one candidate survives.
Write the future-ratio equation
The future ratio is one more equation.
A ratio of 2:1 is just the statement that one quantity equals twice the other.
7.RP.A.2Convert To AlgebraSolve for the number of years
Solving gives 4 years, choice (B).
Solving the single equation pins down the one future moment when the ratio works.
8.EE.C.7Convert To AlgebraPin 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