Problem
Remainder must be less than the divisor
A number (the blank) divided by 8 gives quotient 11 and some remainder (the triangle). We want the greatest possible value of the blank, so we make the remainder as large as it can be.
Operations
Your answer
Watch out It's tempting to let the remainder equal the divisor itself, but a remainder of 8 is not allowed — it must be strictly less than the divisor, so the largest valid remainder is 7.
How to solve
Strategy Work Backwards — Use dividend = 8 times quotient plus remainder. To make the dividend largest while keeping quotient 11, choose the biggest allowed remainder, which is 7.
1STEP 1
Pick the largest remainder
A remainder must be smaller than the divisor 8, so the greatest it can be is 7.
▲ less than 8 → ▲ = 7
The remainder is always less than the divisor, so 7 is the biggest value that keeps the quotient at 11.
3.OA.B.6Guess And CheckWhen is divided by 8 to give quotient 11, the remainder can be at most 7.
Why?
The remainder is whatever is left after taking out every whole group of 8 that the quotient counts, so it must be smaller than 8, and the largest whole number below 8 is 7.
Why?
If the leftover were 8 or more, one more whole group of 8 could still be pulled out of it, which would push the quotient past 11 instead of staying behind as a remainder.
Why?
Dividing by 8 sorts the whole amount into equal groups of 8, and every complete group of 8 is counted by the quotient, never left in the remainder.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
2STEP 2
Build the greatest dividend
The dividend equals 8 times the quotient 11, plus the remainder 7.
8 × 11 + 7 = 88 + 7 = 95
Multiplying the divisor by the quotient and adding the largest remainder gives the largest number that still divides to give 11.
3.OA.C.7Work BackwardsAnswer
95
8 × 11 + 7 = 95
Check: 95 divided by 8 is 11 remainder 7, since 8 times 11 is 88 and 95 minus 88 is 7, and 7 is less than 8. Trying 96 would give quotient 12, too big, so 95 is the greatest.
Takeaway
This only needs Grade 3 division: the remainder maxes out at one less than the divisor, then multiply back!
- Pick the largest remainder
- Build the greatest dividend
Where next?
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