AMC 10 · 2002 · #8
Grade 4 arithmeticSuppose July of year N has five Mondays. Which of the following must occur five times in the August of year N? (Note: Both months have 31 days.)
Pick an answer.
AMC 10 2002 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: In some year, July has five Mondays, and both July and August have 31 days. Determine which single day of the week is forced to occur five times in that August, no matter how the calendar happens to fall.
Givens: July of year $N$ has five Mondays.; July has $31$ days and August has $31$ days.; Answer choices are days of the week: (A) Monday, (B) Tuesday, (C) Wednesday, (D) Thursday, (E) Friday.
Unknowns: The one day of the week that must occur five times in August.
Understand
Restated: In some year, July has five Mondays, and both July and August have 31 days. Determine which single day of the week is forced to occur five times in that August, no matter how the calendar happens to fall.
Givens: July of year $N$ has five Mondays.; July has $31$ days and August has $31$ days.; Answer choices are days of the week: (A) Monday, (B) Tuesday, (C) Wednesday, (D) Thursday, (E) Friday.
Plan
Primary tool: #2 Make a Systematic List
Secondary: #1 Draw a Diagram, #3 Eliminate Possibilities
The word 'must' means the answer has to survive every July that fits the clue, not just a lucky one. There are only a few such Julys, so Tool #2 (Make a Systematic List) enumerates them all. Tool #1 (Draw a Diagram) turns each month into a simple calendar grid of four weeks plus three leftover days, which is what decides who gets a fifth appearance. Tool #3 (Eliminate Possibilities) then keeps only the day that shows up five times in every one of those Augusts and throws out the days that fail in some case.
Execute — Answer: D
4.NBT.B.6 Step 1 Only three weekdays get a fifth day
- A 31-day month is four full weeks (28 days) plus 3 extra days: $31 = 4\times 7 + 3$.
- Every weekday lands at least four times inside the 28-day block.
- The 3 extra days are the 29th, 30th, and 31st, whose weekdays match the 1st, 2nd, and 3rd.
- So exactly the weekdays of the 1st, 2nd, and 3rd of the month occur five times.
💡 Four weeks cover 28 days evenly, so only the three leftover days hand out a fifth copy of their weekday.
4.OA.C.5 Step 2 Pin down where July can start
- For July to have five Mondays, Monday must be one of those first three weekdays — the 1st, 2nd, or 3rd.
- That leaves exactly three possible Julys: July 1 is a Monday, or a Sunday (so Monday is the 2nd), or a Saturday (so Monday is the 3rd).
- No other starting weekday produces five Mondays.
💡 The fifth Monday can only come from a leftover day, so Monday has to sit in the first three days of July.
4.NBT.B.6 Step 3 Find where each August starts
- August 1 comes 31 days after July 1.
- Since $31 = 4\times 7 + 3$, the weekday advances by 3.
- Sliding each July start forward three days: a Monday July gives a Thursday August, a Sunday July gives a Wednesday August, and a Saturday July gives a Tuesday August.
💡 Thirty-one days is three days past a whole number of weeks, so the weekday just slides forward by three.
4.OA.A.3 Step 4 Keep the day common to all cases
- In each August the five-time days are the weekdays of the 1st, 2nd, and 3rd — three days in a row starting at August 1.
- The three cases give $\{\text{Thu, Fri, Sat}\}$, $\{\text{Wed, Thu, Fri}\}$, and $\{\text{Tue, Wed, Thu}\}$.
- Only Thursday appears in all three lists, so Thursday is the day that must occur five times: choice (D).
- Friday and Wednesday each miss one case, and Monday and Tuesday miss two, so all of them are eliminated.
💡 A guarantee is whatever survives every case, so keep only the weekday shared by all three Augusts.
4.NBT.B.6 A 31-day month is four full weeks (28 days) plus 3 extra days: $31 = 4\times 7 + 4.OA.C.5 For July to have five Mondays, Monday must be one of those first three weekdays 4.NBT.B.6 August 1 comes 31 days after July 1. Since $31 = 4\times 7 + 3$, the weekday adv 4.OA.A.3 In each August the five-time days are the weekdays of the 1st, 2nd, and 3rd — th Review
Reasonableness: Test the middle case directly: if July 1 is a Sunday, the Mondays fall on the 2nd, 9th, 16th, 23rd, and 30th — five Mondays, as required. Then August 1 is a Wednesday, and August's five-time days are Wednesday, Thursday, and Friday, so Thursday is present. Running the other two Julys the same way, Thursday appears every single time while no other day does, matching choice (D).
Alternative: Track the actual Monday dates instead of the start weekdays. Five Mondays force them onto $\{1,8,15,22,29\}$, $\{2,9,16,23,30\}$, or $\{3,10,17,24,31\}$. In each case read July 31's weekday, step one day into August, and list August's first three weekdays. This reproduces the Augusts that begin on Thursday, Wednesday, and Tuesday, and Thursday is again the only day guaranteed to occur five times.
CCSS standards used (min grade 4)
4.NBT.B.6Find whole-number quotients and remainders with up to four-digit dividends (Writing $31 = 4\times 7 + 3$ to see that a 31-day month is four weeks plus three leftover days, and that August's start advances three weekdays past July's.)4.OA.C.5Generate a number or shape pattern following a given rule (Using the repeating 7-day week to list the three July start days (Monday, Sunday, Saturday) that produce five Mondays.)4.OA.A.3Solve multi-step word problems using four operations with whole numbers (Combining the three cases and keeping only the weekday that occurs five times in every one of them.)
⭐ A 31-day month is four weeks plus three extra days, so only the weekdays of the 1st, 2nd, and 3rd get a fifth turn — and the day that must repeat is the one shared by every possible calendar.
⭐ A 31-day month is four weeks plus three extra days, so only the weekdays of the 1st, 2nd, and 3rd get a fifth turn — and the day that must repeat is the one shared by every possible calendar.
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