AMC 10 · 2016 · #4
Grade 4 rate-ratioPick an answer.
AMC 10 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Day each book is finished
Back-to-back reading means book n's finish day is the running total 1+2+…+n. Check: day 1 Monday, day 3 Wednesday — matches the clues.
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.OA.C.5Look For A PatternTotal days for 15 books
Book 15 finishes on day 1+2+…+15. Pair the ends (1+15=16, seven such pairs) plus the middle 8: 7×16+8 = 120 days.
Pairing the smallest and largest numbers makes every pair the same size, so a long sum collapses into one quick multiplication.
4.NBT.B.4Identify SubproblemsTurn the count into a weekday
Day 1 is Monday, so day 120 is 119 days later. As 119 = 7×17 (remainder 0), whole weeks pass and it lands back on Monday — (B).
Every full block of 7 days returns to the same weekday, so a remainder of 0 lands right back on Monday.
4.NBT.B.6Look For A PatternDays 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.
- Day each book is finished
- Total days for 15 books
- Turn the count into a weekday