Problem
Recover the dividend from quotient and remainder
A number divided by 8 gives quotient 9 and remainder 1. We first recover that number, then divide it by 5 to find the new quotient and remainder.
Operations
Your answer
Watch out Recovering the number (73) isn't the final answer — you still need to divide it by 5, so don't stop at 73.
How to solve
Strategy Work Backwards — Rebuild the original number from quotient times divisor plus remainder, then carry out the new division by 5.
1STEP 1
Recover the number
The number equals 8 times the quotient 9, plus the remainder 1.
8 × 9 + 1 = 72 + 1 = 73
Multiplying the quotient by the divisor and adding the remainder undoes the division.
3.OA.B.6Work BackwardsThe original number equals the divisor times the quotient plus the remainder, so rebuilds it.
Why?
Dividing the number by 8 split it into 9 full groups of 8 with 1 left over, so gathering those groups back and adding the leftover restores the exact number.
Why?
Nine full groups of 8 brought back together make , because multiplying is just counting several equal-sized groups.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
The number is only its two parts — the grouped-up eights and the 1 left over — so adding those parts back gives the whole number and nothing extra.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
2STEP 2
Divide by 5
Now divide 73 by 5: five fits in fourteen times (using 70), leaving 3.
73 ÷ 5 = 14 … 3
Since 5 times 14 is 70 and 73 minus 70 is 3, the quotient is 14 and the remainder 3 (less than 5).
3.OA.C.7Guess And CheckAnswer
Quotient 14, remainder 3
73 ÷ 5 = 14 … 3
Check: 5 times 14 plus 3 equals 73, the recovered number, and the remainder 3 is less than the divisor 5, so the answer is valid.
Takeaway
This only needs Grade 3 division: multiply back to find the number, then divide it again!
- Recover the number
- Divide by 5
Where next?
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