AMC 10 · 2010 · #5
Easy mode Grade 4A month has 31 days. In this month, the number of Mondays is the same as the number of Wednesdays. The 1st of the month could land on any weekday. On how many of the seven weekdays could the 1st fall and still have this be true?
Pick an answer.
AMC 10 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: A calendar month has exactly $31$ days, and in that month the number of Mondays equals the number of Wednesdays. Looking at all seven possibilities for which weekday the 1st of the month lands on, we want to count how many of those starting weekdays make the Mondays and Wednesdays come out equal.
Givens: The month has exactly $31$ days.; The number of Mondays in the month equals the number of Wednesdays.; The first day of the month can be any one of the seven weekdays.; Answer choices: (A) $2$, (B) $3$, (C) $4$, (D) $5$, (E) $6$.
Unknowns: How many of the seven possible starting weekdays give a month with equally many Mondays and Wednesdays.
Understand
Restated: A calendar month has exactly $31$ days, and in that month the number of Mondays equals the number of Wednesdays. Looking at all seven possibilities for which weekday the 1st of the month lands on, we want to count how many of those starting weekdays make the Mondays and Wednesdays come out equal.
Givens: The month has exactly $31$ days.; The number of Mondays in the month equals the number of Wednesdays.; The first day of the month can be any one of the seven weekdays.; Answer choices: (A) $2$, (B) $3$, (C) $4$, (D) $5$, (E) $6$.
Plan
Primary tool: #2 Make a Systematic List
Secondary: #7 Identify Subproblems, #5 Look for a Pattern, #3 Eliminate Possibilities
The clean way to be sure of the count is to write out all seven possibilities for the first day of the month (Tool #2, Make a Systematic List) and test each one. Before listing, Tool #7 (Identify Subproblems) breaks $31$ days into "$4$ whole weeks" plus "$3$ extra days," which is the key structural fact. Tool #5 (Look for a Pattern) turns that into the rule "the $3$ extra days are three consecutive weekdays that slide as the start day slides," so the only thing that ever changes is which three weekdays get a bonus $5$th appearance. Tool #3 (Eliminate Possibilities) then gives the crisp test for each row: Mondays equal Wednesdays exactly when both, or neither, of Monday and Wednesday sit in that $3$-day block.
Execute — Answer: B
4.NBT.B.6 Step 1 Split 31 into weeks plus leftovers
- Divide the $31$ days into whole weeks.
- Seven days make a week, and $31 \div 7 = 4$ with remainder $3$.
- So the month is $4$ complete weeks (each weekday appears once per week, $4$ times total) plus $3$ extra days hanging off the end.
💡 Chopping the calendar into full weeks first means every weekday starts out perfectly tied at $4$ apiece; only the leftovers can break the tie.
4.OA.C.5 Step 2 Find who gets the bonus day
- The $3$ leftover days are the last three dates, but by weekday they line up with the very first three days of the month: the starting weekday and the two weekdays right after it.
- Those three consecutive weekdays each occur a $5$th time; every other weekday stays at $4$.
- As the first day slides, this block of three slides with it.
💡 The remainder of $3$ always lands on three weekdays in a row starting at day $1$, so knowing the first day tells you exactly who gets the bonus.
4.OA.A.3 Step 3 State the equal-count test
- Monday and Wednesday have equal counts exactly when they are treated the same: either both are in the $3$-day bonus block (both appear $5$ times) or both are outside it (both appear $4$ times).
- If exactly one of them is in the block, one gets $5$ and the other $4$, so they are unequal and that starting day is ruled out.
💡 Equal counts means Monday and Wednesday must share the same fate; a split decision always breaks the tie.
4.OA.A.3 Step 4 List all 7 starting days
- Go through each possible first day and write its $3$-day bonus block, then apply the test.
- Sunday: {Sun, Mon, Tue} — Monday in, Wednesday out, unequal.
- Monday: {Mon, Tue, Wed} — both in, equal.
- Tuesday: {Tue, Wed, Thu} — Wednesday in, Monday out, unequal.
- Wednesday: {Wed, Thu, Fri} — Wednesday in, Monday out, unequal.
- Thursday: {Thu, Fri, Sat} — both out, equal.
- Friday: {Fri, Sat, Sun} — both out, equal.
- Saturday: {Sat, Sun, Mon} — Monday in, Wednesday out, unequal.
💡 Writing out every case leaves no gap to worry about — you can literally see which rows pass.
4.OA.A.3 Step 5 Count the winners
- Three starting weekdays pass the test: Monday, Thursday, and Friday.
- That is $3$ of the seven days, so the answer is $\textbf{(B)}\ 3$.
💡 Just tally the rows that passed; three of them do.
4.NBT.B.6 Divide the $31$ days into whole weeks. Seven days make a week, and $31 \div 7 = 4.OA.C.5 The $3$ leftover days are the last three dates, but by weekday they line up with 4.OA.A.3 Monday and Wednesday have equal counts exactly when they are treated the same: e 4.OA.A.3 Go through each possible first day and write its $3$-day bonus block, then apply 4.OA.A.3 Three starting weekdays pass the test: Monday, Thursday, and Friday. That is $3$ Review
Reasonableness: Double-check the three winners by actually counting dates. If the 1st is Monday, Mondays fall on $1, 8, 15, 22, 29$ ($5$ of them) and Wednesdays on $3, 10, 17, 24, 31$ ($5$ of them) — equal. If the 1st is Thursday, Mondays are $5, 12, 19, 26$ ($4$) and Wednesdays $7, 14, 21, 28$ ($4$) — equal. If the 1st is Friday, Mondays are $4, 11, 18, 25$ ($4$) and Wednesdays $6, 13, 20, 27$ ($4$) — equal. A failing case confirms the test too: a Sunday start gives Mondays $2, 9, 16, 23, 30$ ($5$) but Wednesdays $4, 11, 18, 25$ ($4$) — unequal. Exactly $3$ starting days work, matching (B).
Alternative: Instead of listing all seven, reason directly with Tool #3 (Eliminate Possibilities). Monday and Wednesday are two weekdays apart with Tuesday between them. The only $3$-in-a-row block that contains both is exactly {Mon, Tue, Wed}, which happens only when the month starts on Monday. Otherwise they must both stay out of the block, so the block must avoid both Monday and Wednesday: the only such blocks are {Thu, Fri, Sat} and {Fri, Sat, Sun}, i.e. starting on Thursday or Friday. That is the same $3$ answers — Monday, Thursday, Friday.
CCSS standards used (min grade 4)
4.NBT.B.6Find whole-number quotients and remainders with one-digit divisors (Dividing $31$ days by $7$ to get $4$ weeks with a remainder of $3$ leftover days.)4.OA.C.5Generate and describe a number or shape pattern that follows a given rule (Recognizing that the $3$ leftover days always form three consecutive weekdays starting at day $1$, and this block slides with the starting weekday.)4.OA.A.3Solve multistep word problems and interpret remainders (Turning the leftover-days count into the equal-count test and systematically checking all seven starting days to count the winners.)
⭐ Split $31$ days into $4$ full weeks plus $3$ leftover days: only three weekdays in a row get a bonus $5$th appearance, so just check whether Monday and Wednesday are both bonus days or both not — that works out for $3$ of the $7$ starting days.
⭐ Split $31$ days into $4$ full weeks plus $3$ leftover days: only three weekdays in a row get a bonus $5$th appearance, so just check whether Monday and Wednesday are both bonus days or both not — that works out for $3$ of the $7$ starting days.
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