Problem
Rebuild the dividend with multiplication and addition
A number divided by one decimal leaves a known whole-number quotient and remainder. We need the remainder when the same number is divided by a different decimal.
Number system
Your answer
How to solve
Strategy Work Backwards — The second division cannot start until the number itself is known, and the first division holds it: quotient times divisor plus remainder rebuilds the number exactly.
1STEP 1
Rebuild the number
A division says the number is the divisor times the quotient plus what was left over.
6.4 × 5 = 32, 32 + 1.2 = 33.2
The number is 33.2.
6.NS.B.2Work BackwardsThe number is times plus .
Why?
A quotient of means five whole s were taken out, and taking equal amounts out five times is what multiplying by five undoes.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
The remainder is what was left after those five, so the five parts and the remainder together are the whole number again.
🧱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
Find the new quotient
Count how many whole 9.5s fit inside 33.2 without going over.
9.5 × 3 = 28.5, 9.5 × 4 = 38
Three fit; four would overshoot.
6.NS.B.3Guess And Check3STEP 3
Take the remainder
What is left over is the number minus what the three whole divisors accounted for.
33.2 - 28.5 = 4.7
The remainder is 4.7.
6.NS.B.3Identify SubproblemsAnswer
4.7
Rebuild the second division too: 9.5 × 3 + 4.7 = 33.2, the same number, and 4.7 is less than 9.5 as a remainder must be.
Takeaway
Quotient times divisor plus remainder gives the original number back.
- Rebuild the number
- Find the new quotient
- Take the remainder