AMC 8 · 2004 · #1
Grade 6 arithmeticPick an answer.
AMC 8 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
A map scale is a fixed ratio: kilometers per centimeter is the same for every length on the map. That is exactly the setup for Tool #3 (Set Up an Equation) — write a proportion that says "12 cm matches 72 km the same way 17 cm matches x km." Tool #4 (Introduce a Variable) names the unknown distance x so we can solve. The cleanest path is to find the unit rate (km per cm) first, then multiply by 17.
Name the unknown: let x be the distance that 17 cm on the map represents.
Giving the unknown a letter is the Grade 6 "write expressions with variables" move.
6.EE.A.2Use Matrix LogicFind the unit rate: divide 72 km by 12 cm to get 6 km per centimeter.
Dividing the two quantities turns a 12-to-72 ratio into the per-centimeter rate the map uses everywhere.
6.RP.A.2Eliminate PossibilitiesApply the rate to 17 cm: 17 × 6 km/cm = 102 km.
Once you know the per-unit value, scaling up is one multiplication.
6.RP.A.3Eliminate PossibilitiesMatch to a choice: 102 km is answer (B).
The unit-rate method always gives a number that matches one of the choices exactly when the problem is well posed.
6.RP.A.3Eliminate PossibilitiesMap scale problems become easy once you pin down the per-centimeter rate — find it once, then multiply by whatever length you need.