AMC 10 · 2016 · #4
Easy mode Grade 4Zoey reads 15 books, one after another. The first book takes 1 day, the second takes 2 days, the third takes 3 days, and so on — each book takes one more day than the book before it. She finished the first book on a Monday and the second on a Wednesday. On what day of the week does she finish the 15th book?
Pick an answer.
AMC 10 2016 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: Zoey reads $15$ books in order. Book $1$ takes $1$ day, book $2$ takes $2$ days, and each later book takes one more day than the one before. She finished book $1$ on a Monday and book $2$ on a Wednesday. Find the day of the week on which she finished book $15$.
Givens: Book number $k$ takes exactly $k$ days to read; Book $1$ was finished on a Monday; Book $2$ was finished on a Wednesday; Answer choices: (A) Sunday, (B) Monday, (C) Wednesday, (D) Friday, (E) Saturday
Unknowns: The day of the week on which the $15$th book is finished
Understand
Restated: Zoey reads $15$ books in order. Book $1$ takes $1$ day, book $2$ takes $2$ days, and each later book takes one more day than the one before. She finished book $1$ on a Monday and book $2$ on a Wednesday. Find the day of the week on which she finished book $15$.
Givens: Book number $k$ takes exactly $k$ days to read; Book $1$ was finished on a Monday; Book $2$ was finished on a Wednesday; Answer choices: (A) Sunday, (B) Monday, (C) Wednesday, (D) Friday, (E) Saturday
Plan
Primary tool: #5 Look for a Pattern
Secondary: #7 Identify Subproblems, #3 Eliminate Possibilities
Tracking $15$ separate books day by day is slow, but two patterns make it fast. Tool #5 (Look for a Pattern) handles the days of the week, which repeat every $7$, so only the count of days matters, not the calendar. Tool #7 (Identify Subproblems) splits the work into two clean parts: first find how many total days pass, then turn that count into a weekday. Tool #3 (Eliminate Possibilities) confirms the single choice the cycle points to.
Execute — Answer: B
4.OA.C.5 Step 1 Day each book is finished
- Reading is back to back, so the day a book is finished is the running total of all reading days so far.
- Book $1$ is finished on day $1$, book $2$ on day $1+2=3$, book $3$ on day $1+2+3=6$, and in general book $n$ is finished on day $1+2+\cdots+n$.
- Check it against the clues: day $1$ is Monday, and day $3$ (two days later) is Wednesday — exactly as stated.
💡 Because each book starts right after the last one ends, the finish day is just the total of every reading day used so far.
4.NBT.B.4 Step 2 Total days for 15 books
- The $15$th book is finished on day $1+2+\cdots+15$.
- Add by pairing the ends: $1+15=16$, $2+14=16$, and so on.
- There are $7$ such pairs of $16$ plus the leftover middle number $8$, giving $7\times 16 + 8 = 112 + 8 = 120$.
- So she finishes book $15$ on day $120$.
💡 Pairing the smallest and largest numbers makes every pair the same size, so a long sum collapses into one quick multiplication.
4.NBT.B.6 Step 3 Turn the count into a weekday
- Day $1$ is Monday.
- Day $120$ comes $120-1 = 119$ days after that Monday.
- Since weekdays repeat every $7$ days, only the remainder of $119$ divided by $7$ matters: $119 = 7\times 17$ with remainder $0$.
- A remainder of $0$ means a whole number of weeks has passed, landing back on the same weekday — Monday.
- The other choices are different remainders, so they are ruled out.
- The answer is (B).
💡 Every full block of $7$ days returns to the same weekday, so a remainder of $0$ lands right back on Monday.
4.OA.C.5 Reading is back to back, so the day a book is finished is the running total of a 4.NBT.B.4 The $15$th book is finished on day $1+2+\cdots+15$. Add by pairing the ends: $1+ 4.NBT.B.6 Day $1$ is Monday. Day $120$ comes $120-1 = 119$ days after that Monday. Since w Review
Reasonableness: The setup checks out on the given clues: day $1$ is Monday and day $3$ is Wednesday, matching the problem. The final count $120$ is a multiple of $7$ shifted by $1$ (since $119$ is a multiple of $7$), so finishing back on the starting weekday, Monday, is exactly what the cycle predicts — answer (B).
Alternative: Number the weekdays by their remainder mod $7$, with Monday $=1$ (since day $1$ is Monday). Then the finish day of book $15$, day $120$, has $120 \bmod 7 = 1$, the same remainder as Monday, so book $15$ is finished on a Monday — again (B).
CCSS standards used (min grade 4)
4.OA.C.5Generate a number or shape pattern following a given rule (Seeing that each book's finish day is the running total $1+2+\cdots+n$ because each book takes one more day than the last.)4.NBT.B.4Fluently add and subtract multi-digit whole numbers (Adding $1+2+\cdots+15$ (by pairing) to get the total of $120$ days.)4.NBT.B.6Find whole-number quotients and remainders with up to four-digit dividends (Dividing $119$ by $7$ to get remainder $0$, which fixes the weekday in the $7$-day cycle.)
⭐ Days of the week repeat every $7$, so if the number of days that pass is a multiple of $7$, you land right back on the same weekday.
⭐ Days of the week repeat every $7$, so if the number of days that pass is a multiple of $7$, you land right back on the same weekday.
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