Problem 21 · 2010 AMC 8
Hard
Algebra & Patterns
work-backward
Hui is an avid reader. She bought a copy of the best seller Math is Beautiful. On the first day, Hui read 1/5 of the pages plus 12 more, and on the second day she read 1/4 of the remaining pages plus 15 pages. On the third day she read 1/3 of the remaining pages plus 18 pages. She then realized that there were only 62 pages left to read, which she read the next day. How many pages are in this book?
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Answer: C — 240 pages.
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Hint 1 of 3
The only number you actually know for sure is the 62 pages left at the very end. So run the story in reverse — start from 62 and undo each day.
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Hint 2 of 3
Undoing a day: if she read 1/3 of the pile plus 18, then 2/3 of the pile minus 18 is what's left. First add the 18 back, then the leftover is 2/3 of the pile — so multiply by 3/2 to recover the start of that day.
Still stuck? Show hint 3 →
Hint 3 of 3
Repeat the same undo for day 2, then day 1, each time recovering the pile at the start of that day.
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Approach: work backwards, undoing one day at a time
- Start from the end: 62 pages remain after day 3. She'd read 1/3 of that day's pile plus 18, leaving 2/3 of it minus 18. So (2/3)·pile − 18 = 62 ⇒ (2/3)·pile = 80 ⇒ pile = 120 at the start of day 3.
- Undo day 2 the same way: (3/4)·pile − 15 = 120 ⇒ (3/4)·pile = 135 ⇒ pile = 180 at the start of day 2.
- Undo day 1: (4/5)·pile − 12 = 180 ⇒ (4/5)·pile = 192 ⇒ pile = 240 — the whole book.
- Why this transfers: when a process ‘take a fraction, then a fixed amount’ repeats and you only know the final state, working backward is far cleaner than one giant forward equation — you invert each step in turn (add the constant back, then scale up).
Another way — forward algebra in one variable:
- Let the book be N pages. After day 1, remaining = N − (N/5 + 12) = (4/5)N − 12.
- After day 2 (read 1/4 of that plus 15), remaining = (3/4)[(4/5)N − 12] − 15 = (3/5)N − 24.
- After day 3 (read 1/3 of that plus 18), remaining = (2/3)[(3/5)N − 24] − 18 = (2/5)N − 34.
- Set that to 62: (2/5)N = 96 ⇒ N = 240 — matching the backward method.
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