Problem 12 · 1992 AJHSME
Medium
Ratios, Rates & Proportions
total-divided
The five tires of a car (four road tires and a full-sized spare) were rotated so that each tire was used the same number of miles during the first 30,000 miles the car traveled. For how many miles was each tire used?
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Answer: C — 24,000 miles.
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Hint 1 of 3
The car drives 30,000 miles, but how many tires are touching the road during ALL of that distance? That's the total amount of "tire wear" to share.
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Hint 2 of 3
Count total tire-miles (tires on the road × distance), then split that workload equally — this "total work ÷ sharers" idea is the heart of the problem.
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Hint 3 of 3
Five tires share the work, but only four are ever rolling at once — so the total work is 4 cars' worth of miles, not 5.
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Approach: total tire-miles of road work, shared equally among 5 tires
- At every moment 4 tires are on the road, and the car covers 30,000 miles. So the total road work is 4 × 30,000 = 120,000 tire-miles of wear.
- By the rotation, all 5 tires share that work equally: 120,000 ÷ 5 = 24,000 miles each.
- Why this transfers: "everyone shares equally" problems all run the same way — add up the total work being done, then divide by the number of sharers. Here the work is 4-at-a-time even though there are 5 tires.
- Sanity check: 24,000 is less than the 30,000 miles driven, which must be true — each tire rests for part of the trip while the spare takes a turn.
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