Competition · AMC preparation · step 4 of 4
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 distance
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.2Introduce A VariableFind the kilometers per centimeter
Find 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 centimeters
Apply the rate to 17 cm: 17 × 6 km/cm = 102 km.
Once you know the per-unit value, scaling up is one multiplication.
On this map, the real distance matching a 17-centimeter length equals the map's fixed kilometers-per-centimeter rate multiplied by 17.
▸ Why?
The map is drawn to a single scale, so one centimeter on it always stands for the same fixed number of real kilometers, no matter where it is measured.
▸ Why?
That fixed rate is 6 kilometers per centimeter: twelve equal centimeters together stand for 72 kilometers, and dividing 72 by 12 undoes that multiplication to give one centimeter's share.
▸ Why?
A 17-centimeter length is 17 of those equal one-centimeter pieces laid end to end, so its real distance is 17 equal groups of one centimeter's kilometers — which is exactly what multiplying the rate by 17 counts.
Match the value to a choice
Match 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.
- Name the unknown distance
- Find the kilometers per centimeter
- Apply the rate to 17 centimeters
- Match the value to a choice
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