Problem 1 · 2004 AMC 8
Easy
Ratios, Rates & Proportions
unit-rate
On a map, a 12-centimeter length represents 72 kilometers. How many kilometers does a 17-centimeter length represent?
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Answer: B — 102 km.
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Hint 1 of 2
Before reaching for 17, find what just one centimeter is worth. Once you know the value of a single unit, any length is just a multiplication away.
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Hint 2 of 2
This is the unit-rate idea: collapse any proportion to a 'per one' value, then scale. It beats cross-multiplying because the 'per one' number is reusable.
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Approach: unit rate (value of one cm)
- Shrink the scale to one unit: 72 km spread over 12 cm means each cm is worth 72 ÷ 12 = 6 km. That single number is the whole problem.
- Now any length scales instantly: 17 cm × 6 km/cm = 102 km.
- Sanity check: 17 cm is a bit more than 12 cm, so the answer should be a bit more than 72 — and 102 is. You'll see this 'find the per-one value first' move in every map, recipe, and speed problem.
Another way — proportion (cross-multiply):
- Set equal ratios: 12/72 = 17/x, so 12x = 72 · 17.
- x = 72 · 17 / 12 = 6 · 17 = 102 km.
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