Problem 10 · 2011 AMC 8
Medium
Fractions, Decimals & Percents
piecewise-rate
The taxi fare in Gotham City is $2.40 for the first 12 mile and additional mileage charged at the rate $0.20 for each additional 0.1 mile. You plan to give the driver a $2 tip. How many miles can you ride for $10?
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Answer: C — 3.3 miles.
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Hint 1 of 2
Strip away the costs that don't depend on distance first: the $2 tip and the $2.40 flat charge for the first half-mile. Whatever's left is the only money that buys extra distance.
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Hint 2 of 2
Simplify the awkward rate before dividing: $0.20 per 0.1 mile is the same as $2 per whole mile — much friendlier numbers.
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Approach: peel off the fixed costs, then divide the leftover by the per-mile rate
- Take out the fixed pieces: $10 − $2 tip − $2.40 for the first half-mile = $5.60 left for extra distance.
- Turn the rate into per-mile: $0.20 per 0.1 mile = $2 per mile. So $5.60 ÷ $2 = 2.8 extra miles.
- Add back the half-mile you already paid for: 0.5 + 2.8 = 3.3 miles.
- Worth keeping: in any "flat fee plus a per-unit rate" problem, subtract every fixed charge first — only the remainder gets divided by the rate.
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