Depth · one concept from first principles · step 2 of 4
Reverse the operation order to find the unknown
Find the number that makes correct.
Operations
Your answer
How to solve
Strategy Work Backwards — We know the final result (9) and want the starting hidden number, so we undo the operations in reverse order: the last thing done was subtracting 27, so we add it back first, and we keep undoing each step until only the box is left. We can confirm the result by plugging the number back in (Guess and Check).
1STEP 1
Undo the subtraction
The last operation was subtracting 27. To go backwards, add 27 to both sides. So the part before subtracting must equal 9 + 27 = 36.
72 ÷ (10 - □) × 3 = 9 + 27 = 36
Working backwards turns a take-away into an add-back, just like retracing steps to find where you started.
5.OA.A.1Work BackwardsWorking backwards, the value standing there before 27 was subtracted must be 36.
Why?
Subtracting 27 landed on 9, so adding 27 back to 9 brings back the value that stood there before the subtraction.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
Why?
The unwinding has to start at the very last operation, because that is the only one whose result the equation actually tells us.
Why?
Each operation worked on the result of the one before it, so the earlier values stay hidden until the later ones are peeled off.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
2STEP 2
Undo the multiplication by 3
Dividing undoes the times-3, so 72 divided by (10 - box) is 36 divided by 3 = 12.
72 ÷ (10 - □) = 36 ÷ 3 = 12
Multiplying and dividing are opposites, so dividing by 3 reverses the times-3 step.
3.OA.A.4Work Backwards3STEP 3
Find the value inside the parentheses
72 divided by the parentheses is 12, so the parentheses hold 72 divided by 12 = 6.
10 - □ = 72 ÷ 12 = 6
If a known number divided by something equals 12, that something is the known number divided by 12.
3.OA.A.4Work Backwards4STEP 4
Solve for the box
Finally, 10 - □ = 6, so the box is 10 - 6 = 4.
□ = 10 - 6 = 4
The box is whatever you subtract from 10 to leave 6.
3.OA.A.4Work Backwards5STEP 5
Check by going forwards
Putting 4 back: 10 - 4 = 6, 72 divided by 6 = 12, 12 x 3 = 36, 36 - 27 = 9. It matches.
72 ÷ (10 - 4) × 3 - 27 = 72 ÷ 6 × 3 - 27 = 36 - 27 = 9
Substituting the answer back in and following the rules confirms it really works.
5.OA.A.1Guess And CheckAnswer
4
The box must be less than 10 (so that 10 - □ stays positive for the division), and 4 satisfies that. Checking forwards gives exactly 9, so the answer is consistent.
Takeaway
When you know the answer but not the start, undo each step in reverse order until only the box is left.
- Undo the subtraction
- Undo the multiplication by 3
- Find the value inside the parentheses
- Solve for the box
- Check by going forwards
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested
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The same shape shows up later on the AMC, as Arithmetic Expression Decomposition, Branching State Enumeration, Multiplicative Ratio with Integrality Constraints. Why the shapes repeat →