Problem 16 · 2006 AMC 8
Medium
Ratios, Rates & Proportions
page-time-ratio
Before Chandra and Bob start reading, Alice says she would like to team read with them. If they divide the book into three sections so that each reads for the same length of time, how many seconds will each have to read?
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Answer: E — 7200 seconds.
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Hint 1 of 2
"Read for the same length of time" means in that one shared block of time, each person reads pages at their own speed. The fast reader simply gets through more pages.
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Hint 2 of 2
Pages read in equal time are in proportion to reading SPEED (pages per second), which is the inverse of seconds-per-page. Build the speed ratio first, split the 760 pages by it, then find one person's time.
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Approach: split pages by speed ratio, then read off the shared time
- Speeds (pages/sec): Alice 1/20, Chandra 1/30, Bob 1/45. To clear fractions, scale by 180: 9 : 6 : 4. So in equal time Alice reads 9 parts, Chandra 6, Bob 4.
- The parts total 9 + 6 + 4 = 19, and they cover all 760 pages, so each part is 760 ÷ 19 = 40 pages. Alice 360, Chandra 240, Bob 160.
- Now find the shared time from any one reader — Bob: 160 pages × 45 sec/page = 7200 seconds. (Check: Alice 360 × 20 = 7200, Chandra 240 × 30 = 7200 — all match.)
- The big idea: equal time ⇒ pages divide in proportion to speed. The fact that 760 ÷ 19 came out to a whole number (40) is the contest hinting you're on the intended path.
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