AMC 8 · 2010 · #21
Grade 6 arithmeticProblem
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?
Pick an answer.
AMC 8 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: Hui reads the book over four days. On day $1$ she reads $\tfrac{1}{5}$ of the whole book plus $12$ more pages. On day $2$ she reads $\tfrac{1}{4}$ of what was left plus $15$ more. On day $3$ she reads $\tfrac{1}{3}$ of what was left plus $18$ more. After that there are $62$ pages left, which she finishes on day $4$. Find the total number of pages.
Givens: Day $1$: read $\tfrac{1}{5}$ of the total pages, plus $12$ extra; Day $2$: read $\tfrac{1}{4}$ of the remaining pages, plus $15$ extra; Day $3$: read $\tfrac{1}{3}$ of the remaining pages, plus $18$ extra; After day $3$, exactly $62$ pages remain (read on day $4$); Answer choices: (A) $120$, (B) $180$, (C) $240$, (D) $300$, (E) $360$
Unknowns: The total number of pages in the book
Understand
Restated: Hui reads the book over four days. On day $1$ she reads $\tfrac{1}{5}$ of the whole book plus $12$ more pages. On day $2$ she reads $\tfrac{1}{4}$ of what was left plus $15$ more. On day $3$ she reads $\tfrac{1}{3}$ of what was left plus $18$ more. After that there are $62$ pages left, which she finishes on day $4$. Find the total number of pages.
Givens: Day $1$: read $\tfrac{1}{5}$ of the total pages, plus $12$ extra; Day $2$: read $\tfrac{1}{4}$ of the remaining pages, plus $15$ extra; Day $3$: read $\tfrac{1}{3}$ of the remaining pages, plus $18$ extra; After day $3$, exactly $62$ pages remain (read on day $4$); Answer choices: (A) $120$, (B) $180$, (C) $240$, (D) $300$, (E) $360$
Plan
Primary tool: #11 Work Backwards
Secondary: #6 Guess and Check
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.
Execute — Answer: C
4.OA.A.3 Step 1 - Start at the end.
- After day $3$, exactly $62$ pages were left for day $4$.
- Day $3$'s reading was "$\tfrac{1}{3}$ of the morning pile, plus $18$ more." Undo the $+18$ first: before those $18$ extra pages were read, day $3$ had $62 + 18 = 80$ pages still unread.
💡 Undoing a "$+18$" is just subtracting $18$ — a Grade 4 multi-step word-problem move.
6.RP.A.3 Step 2 - Those $80$ pages are what was left after Hui read $\tfrac{1}{3}$ of day $3$'s morning pile, so $80$ is $\tfrac{2}{3}$ of that pile.
- Scale up by $\tfrac{3}{2}$ to recover the full pile she faced on day $3$ morning.
💡 If a known fraction of a pile equals a known number, multiplying by the reciprocal gives the whole pile — Grade 6 ratio reasoning.
4.OA.A.3 Step 3 - Now those $120$ pages are the leftover at the end of day $2$.
- Day $2$'s reading was "$\tfrac{1}{4}$ of the morning pile, plus $15$ more." Add back the $15$ extra pages: $120 + 15 = 135$ pages were left right after the $\tfrac{1}{4}$ chunk.
💡 Same undoing trick as before — peel off the "$+15$" first, then handle the fraction.
6.RP.A.3 Step 4 - Those $135$ pages are $\tfrac{3}{4}$ of day $2$'s morning pile (since Hui ate $\tfrac{1}{4}$ off the top).
- Scale up by $\tfrac{4}{3}$ to find that pile, which is also what was left at the end of day $1$.
💡 Same Grade 6 ratio move: known part $\div$ known fraction $=$ whole pile.
6.RP.A.3 Step 5 - Finally, $180$ pages is what was left after day $1$.
- Day $1$'s reading was "$\tfrac{1}{5}$ of the whole book, plus $12$ more." Add back the $12$ extras: $180 + 12 = 192$ pages were left after just the $\tfrac{1}{5}$ chunk.
- That $192$ is $\tfrac{4}{5}$ of the entire book.
- Scale up by $\tfrac{5}{4}$ to get the total.
💡 One more Work-Backwards loop — undo $+12$, then scale $\tfrac{4}{5} \to 1$ — and the whole book pops out.
4.OA.A.3 Start at the end. After day $3$, exactly $62$ pages were left for day $4$. Day $ 6.RP.A.3 Those $80$ pages are what was left after Hui read $\tfrac{1}{3}$ of day $3$'s mo 4.OA.A.3 Now those $120$ pages are the leftover at the end of day $2$. Day $2$'s reading 6.RP.A.3 Those $135$ pages are $\tfrac{3}{4}$ of day $2$'s morning pile (since Hui ate $\ 6.RP.A.3 Finally, $180$ pages is what was left after day $1$. Day $1$'s reading was "$\tf Review
Reasonableness: Run the story forward with $240$ pages to double-check. Day $1$: $\tfrac{1}{5} \times 240 + 12 = 48 + 12 = 60$ pages read, $180$ left. Day $2$: $\tfrac{1}{4} \times 180 + 15 = 45 + 15 = 60$ pages read, $120$ left. Day $3$: $\tfrac{1}{3} \times 120 + 18 = 40 + 18 = 58$ pages read, $62$ left. Day $4$: $62$. Total $= 60 + 60 + 58 + 62 = 240$. Everything lines up.
Alternative: Tool #6 (Guess and Check) directly on the choices. Try (A) $120$: day $1$ leaves $\tfrac{4}{5}(120) - 12 = 84$; day $2$ leaves $\tfrac{3}{4}(84) - 15 = 48$; day $3$ leaves $\tfrac{2}{3}(48) - 18 = 14 \neq 62$. Try (C) $240$: $\tfrac{4}{5}(240) - 12 = 180$; $\tfrac{3}{4}(180) - 15 = 120$; $\tfrac{2}{3}(120) - 18 = 62$. Match — answer is (C). The other choices miss $62$, confirming the result.
CCSS standards used (min grade 6)
4.OA.A.3Solve multi-step word problems with whole numbers (Undoing each day's "$+k$ extra pages" step — adding $18$, $15$, and $12$ back in turn to recover the pile size right after the fraction chunk.)5.NF.B.6Solve real-world problems involving multiplication of fractions (Reading each day's $\tfrac{1}{n}$-of-remaining as a fraction of a pile and tracking how much is left after that fractional chunk is taken off the top.)6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Recovering each morning's pile from its leftover: $\tfrac{2}{3}$ pile $= 80 \Rightarrow$ pile $= 120$, $\tfrac{3}{4}$ pile $= 135 \Rightarrow$ pile $= 180$, and $\tfrac{4}{5}$ book $= 192 \Rightarrow$ book $= 240$.)
⭐ When 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.
⭐ When 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.
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