AMC 10 · 2002 · #8
Grade 4 arithmeticPick an answer.
AMC 10 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Only three weekdays get a fifth day
31 = 4 × 7 + 3, so every weekday lands four times in the 28-day block and only the weekdays of the 1st, 2nd, and 3rd get a fifth.
Four weeks cover 28 days evenly, so only the three leftover days hand out a fifth copy of their weekday.
Four weeks cover twenty-eight days evenly, so only the leftover days hand out a fifth copy of their weekday.
▸ Why?
The weekdays repeat exactly every seven days, so each full week contributes one of every day.
▸ Why?
Any month splits into whole weeks plus a short remainder, and that split is unique.
Pin down where July can start
Monday must sit on the 1st, 2nd, or 3rd, so July 1 is a Monday, a Sunday, or a Saturday — exactly three possible Julys.
The fifth Monday can only come from a leftover day, so Monday has to sit in the first three days of July.
4.OA.C.5Make A Systematic ListFind where each August starts
August 1 falls 31 days after July 1, so its weekday slides forward 3: the three Augusts start on Thursday, Wednesday, or Tuesday.
Thirty-one days is three days past a whole number of weeks, so the weekday just slides forward by three.
4.NBT.B.6Draw A DiagramKeep the day common to all cases
The five-time days are August 1, 2, 3: {Thu, Fri, Sat}, {Wed, Thu, Fri}, {Tue, Wed, Thu} — only Thursday is in all three, choice (D).
A guarantee is whatever survives every case, so keep only the weekday shared by all three Augusts.
4.OA.A.3Eliminate PossibilitiesA 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.
- Only three weekdays get a fifth day
- Pin down where July can start
- Find where each August starts
- Keep the day common to all cases