AMC 10 · 2010 · #5
Grade 4 number-theoryPick an answer.
AMC 10 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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 5th 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.
Split 31 into weeks plus leftovers
Seven days make a week, and 31 ÷ 7 = 4 remainder 3 — so the month is 4 whole weeks plus 3 extra days.
Chopping the calendar into full weeks first means every weekday starts out perfectly tied at 4 apiece; only the leftovers can break the tie.
Chopping the month into full weeks first ties every weekday at the same count.
▸ Why?
After seven days everything returns to the same weekday, so each full week gives one of each.
▸ Why?
Only the leftover days after those full weeks can break the tie.
Find who gets the bonus day
By weekday the 3 leftovers repeat the first three days: the start weekday and the two after it appear 5 times; everyone else 4.
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.C.5Look For A PatternState the equal-count test
Counts match only if Monday and Wednesday share a fate: both inside the 3-day bonus block (5 each) or both outside it (4 each).
Equal counts means Monday and Wednesday must share the same fate; a split decision always breaks the tie.
4.OA.A.3Eliminate PossibilitiesList all 7 starting days
Slide the block over all seven starts: Monday gives {Mon,Tue,Wed} with both in; Thursday and Friday leave both out; the other four split.
Writing out every case leaves no gap to worry about — you can literally see which rows pass.
4.OA.A.3Make A Systematic ListCount the winners
Three starting weekdays pass — Monday, Thursday, and Friday — so the count is 3, answer (B).
Just tally the rows that passed; three of them do.
4.OA.A.3Make A Systematic ListSplit 31 days into 4 full weeks plus 3 leftover days: only three weekdays in a row get a bonus 5th 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 into weeks plus leftovers
- Find who gets the bonus day
- State the equal-count test
- List all 7 starting days
- Count the winners