AMC 10 · 2002 · #8
Grade 4 arithmeticPick an answer.
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
A 31-day month is four weeks plus 3 days, so only the weekdays of the 1st, 2nd and 3rd appear five times.
Four weeks cover 28 days evenly, so only the three leftover days hand out a fifth copy of their weekday.
4.NBT.B.6Draw A DiagramPin down where July can start
Five Mondays therefore force July to start on a Monday, Sunday or Saturday.
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 starts 31 days later, advancing the weekday by 3: Thursday, Wednesday or Tuesday.
Thirty-one days is three days past a whole number of weeks, so the weekday just slides forward by three.
Thirty-one days is three days past whole weeks, so the weekday of the first slides forward by three.
▸ Why?
The weekdays repeat every seven days, so any whole number of weeks lands back on the same weekday.
▸ Why?
Splitting 31 into whole weeks and a leftover is possible in exactly one way, and only the leftover moves the weekday.
Keep the day common to all cases
Intersecting the three sets of five-time days leaves only Thursday, 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