Problem 17 · 1994 AJHSME
Hard
Algebra & Patterns
partial-sums
Pauline can shovel snow at the rate of 20 cubic yards for the first hour, 19 cubic yards for the second, 18 for the third, and so on, always shoveling one cubic yard less per hour than the previous hour. If her driveway is 4 yards wide, 10 yards long, and covered with snow 3 yards deep, then the number of hours it will take her to shovel it clean is closest to
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Answer: D — 7.
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Hint 1 of 2
Two separate jobs hide here. First, how much snow is there at all? It's a box: width × length × depth.
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Hint 2 of 2
Her rate drops each hour, so you can't just divide. Keep a running total — 20, then 20+19, then +18, … — and stop the moment it reaches the goal volume.
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Approach: accumulate the decreasing hourly amounts
- Volume of snow = 4 × 10 × 3 = 120 cubic yards (it's a rectangular box).
- Now stack up her hourly amounts and watch the running total: 20, 39, 57, 74, 90, 105, 119. After 7 hours she's cleared 119 — just 1 cubic yard short of 120, which she finishes a sliver into hour 8.
- Since 119 is essentially the whole job, the time is closest to 7 hours. Don't let choice (E) 12 fool you — that's the hour her rate would hit zero (20−12+1=9... she'd actually stall), but the snow runs out long before then.
- Lesson: when a rate changes step by step, accumulate term-by-term rather than dividing total by a single rate — and read the question's wording ('closest to') to know you can stop at 'almost there.'
Another way — estimate with an average rate:
- Over the first several hours her rate averages roughly the middle of 20 and the low teens — about 17 yds/hr. Then 120 ÷ 17 ≈ 7, pointing straight at 7 hours without summing every term.
- A quick average is a great gut-check before (or instead of) the exact running total.
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