Competition · AMC preparation · step 4 of 4
AMC 8 · 2010 · #21
Grade 6 arithmeticPick an answer.
AMC 8 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The problem describes the book being eaten away day by day, and the last day's leftover (62 pages) is fully known. That is exactly when Tool #11 (Work Backwards) shines: start at the known end and undo each day in reverse. On each day, undoing has two clean moves — add back the "+k extra pages" first, then recognize what's left as a known fraction of that morning's pile and scale up to find the pile. Tool #6 (Guess and Check) is kept as a backup: plug each of the five choices into the forward story and see which one ends with exactly 62 pages left.
Work backward from the end
Start from the end: 62 pages remain after day 3. Day 3 was " plus 18," so undo the +18 first, leaving 80 pages just after the chunk.
Undoing a "+18" is just subtracting 18 — a Grade 4 multi-step word-problem move.
4.OA.A.3Work BackwardsUndo the day 3 fraction
Those 80 pages are of day 3's morning pile, so multiply by the reciprocal to get 120 pages that morning.
If a known fraction of a pile equals a known number, multiplying by the reciprocal gives the whole pile — Grade 6 ratio reasoning.
The 80 unread pages are 2/3 of the pile Hui faced on day 3 morning, so that morning pile held 120 pages.
▸ Why?
Hui read 1/3 of the morning pile off the top, so the leftover is the other 2/3: the pile splits with no gaps or overlaps into the third she read and the part she left, and those two parts add back to the whole pile, so what is left must be 1 - 1/3 = 2/3 of it.
▸ Why?
If two equal thirds of the pile come to 80 pages, then one third is 40 and the whole pile is three such thirds, 120 pages.
▸ Why?
Two equal thirds making 80 means 80 is split into 2 equal parts to get one third, and dividing into equal parts undoes the doubling, so each third is 80 ÷ 2 = 40.
▸ Why?
The full pile is three of those equal 40-page thirds stacked together, which is three equal groups of 40, or 3 × 40 = 120.
Add back the day 2 leftover
Now 120 is day 2's leftover. Day 2 was " plus 15," so add the 15 back to get 135 pages just after the chunk.
Same undoing trick as before — peel off the "+15" first, then handle the fraction.
4.OA.A.3Work BackwardsUndo the day 2 fraction
Those 135 pages are of day 2's morning pile, so multiply by to get 180 — the start of day 2, which is also day 1's leftover.
Same Grade 6 ratio move: known part ÷ known fraction = whole pile.
6.RP.A.3Work BackwardsUndo the day 1 fraction
Finally 180 is day 1's leftover; add 12 back to get 192, which is of the whole book, so multiply by to get 240 → (C).
One more Work-Backwards loop — undo +12, then scale 4/5 → 1 — and the whole book pops out.
6.RP.A.3Work BackwardsWhen a problem hands you the very last leftover, walk the story backwards — undo the "+ extras" first, then scale the fraction up to the whole pile. This AMC 8 question only needs Grade 6 ratio reasoning to crack.
- Work backward from the end
- Undo the day 3 fraction
- Add back the day 2 leftover
- Undo the day 2 fraction
- Undo the day 1 fraction
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