Problem 14 · 2006 AMC 8
Easy
Ratios, Rates & Proportions
time-difference
Problems 14, 15 and 16 involve Mrs. Reed's English assignment. A Novel Assignment. The students in Mrs. Reed's English class are reading the same 760-page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in 20 seconds, Bob reads a page in 45 seconds and Chandra reads a page in 30 seconds. If Bob and Chandra both read the whole book, Bob will spend how many more seconds reading than Chandra?
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Answer: B — 11,400 seconds.
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Hint 1 of 2
You're asked for a DIFFERENCE, so don't compute two big times and subtract giants. Find how much longer Bob takes on a SINGLE page first.
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Hint 2 of 2
Factor before you multiply: 760·45 − 760·30 = 760·(45−30). Pulling out the common 760 turns two big multiplications into one small one.
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Approach: per-page gap, then scale up
- On each page Bob is slower by 45 − 30 = 15 seconds.
- Over all 760 pages that gap accumulates: 760 × 15 = 11,400 seconds.
- Why this beats brute force: computing 760×45 = 34,200 and 760×30 = 22,800 and subtracting works, but factoring out the shared 760 first (the distributive property) skips the big numbers entirely. Difference questions love this — subtract the rates, then multiply once.
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