Reasoning · Grade 3-2 Applications of Division

Problem

Finding the day of the week

January 1 falls on a Tuesday and the year is not a leap year. Weekdays repeat every 7 days, so only the remainder shifts the day. Forward moves the weekday forward, backward moves it back. Find the weekday of May 5, of 100 days before January 1, and of next January 1.
January Sun Mon Tue Wed Thu Fri Sat 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
Your answer
How to solve
Strategy Look for a Pattern — Weekdays repeat every 7 days, so the key is the number of days between two dates divided by 7. I count the days from January 1 to the target date, take the remainder after dividing by 7, and step that many weekdays forward (or backward). Trying a tiny count first (a few days) shows how the remainder maps to a weekday.
1STEP 1

Count days from Jan 1 to May 5

From January 1 to May 5 is 124 days.

31 + 28 + 31 + 30 + 5 = 125 (May 5 is the 125th day); 125 - 1 = 124 days after Jan 1
2STEP 2

Use the 7-day cycle for May 5

The remainder 5 shifts Tuesday to Sunday.

124 ÷ 7 = 17 remainder 5; Tue → Sun
3STEP 3

Go 100 days before Jan 1

Going back, the remainder 2 also lands on Sunday.

100 ÷ 7 = 14 remainder 2; Tue → Sun
4STEP 4

Jump a full year to next Jan 1

365 leaves 1, so next January 1 is Wednesday.

365 ÷ 7 = 52 remainder 1; Tue → Wed
Answer
May 5 Sunday, 100 days before Sunday, next January 1 Wednesday
Each answer used a remainder between 0 and 6, which is the only valid range for a 7-day cycle. Quick check on (3): a common year always pushes the same date forward one weekday, and Tue + 1 = Wed, which matches. For (1), 124 = 119 + 5 and 119 = 17 x 7, so the +5 step is sound; Tue + 5 = Sun.
Takeaway

Weekdays loop every 7 days, so just divide the number of days by 7 and step the leftover days forward or back -- only Grade 4 division-with-remainder needed!

  • Count days from Jan 1 to May 5
  • Use the 7-day cycle for May 5
  • Go 100 days before Jan 1
  • Jump a full year to next Jan 1