Problem
Work backwards from the result
Bags A and B hold 100 marbles between them. First 36 move from A to B, then 45 move from B to A. After that A holds four times B. Find how many A started with.
Reasoning · Grade 3-1 Applications of Multiplication and Division
Your answer
Watch out It is natural to undo in the original order, but working back means undoing the later move first.
How to solve
Strategy Work Backwards — The ending is described ('A is 4 times B' with total 100), so I can find the final amounts first, then undo each move in reverse: undo move 2, then undo move 1, to recover the start. Working backwards turns the puzzle into simple add/subtract steps.
1STEP 1
Find the final amounts
With 100 shared four-to-one, A is 80 and B is 20.
100 ÷ 5 = 20, A = 80, B = 20
Splitting 100 into 5 equal shares (one for B, four for A) is the natural way to read '4 times as many'.
3.OA.A.3Guess And Check2STEP 2
Undo move 2 (45 went B to A)
Undoing the later move takes 45 off A, leaving A at 35.
A: 80 - 45 = 35, B: 20 + 45 = 65
Reversing a move just swaps add and subtract, so undoing '+45 to A' means '-45 from A'.
3.OA.C.7Work BackwardsUndoing a move means swapping its add for a subtract, so the 45 that went from B to A is taken back off A.
Why?
Subtracting a number reverses exactly what adding it did, so running a move backwards restores what it changed.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
Why?
Nothing leaves the two bags, so whatever one bag loses the other gains and the grand total stays fixed throughout.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
3STEP 3
Undo move 1 (36 went A to B)
Undoing the first move puts 36 back: A is 71.
A: 35 + 36 = 71, B: 65 - 36 = 29
Undoing the first move last returns each bag to its original count.
3.OA.C.7Work Backwards4STEP 4
Read off the start
B is 29, and 71 + 29 = 100 checks out.
A_start = 71
Once both moves are reversed, the numbers left are exactly the starting amounts.
3.OA.A.3Work BackwardsAnswer
71 marbles
Run it forward from A=71, B=29 (total 100). Move 36 A to B: A=35, B=65. Move 45 B to A: A=80, B=20. Then 80 = 4 x 20, which matches 'A is 4 times B'. The start total 71 + 29 = 100 is also correct.
Takeaway
When you know the ending, just rewind each step (swap add and subtract) to find the start — pure Grade 3 work-backwards!
- Find the final amounts
- Undo move 2 (45 went B to A)
- Undo move 1 (36 went A to B)
- Read off the start