Depth · one concept from first principles · step 2 of 4
The last forward step divided by 5 to reach 14, so its inverse is multiply by 5. Multiplying 14 by 5 recovers the value before that step.
Multiplying undoes dividing, so we climb back up from 14 to 70.
3.OA.A.4Work BackwardsMultiplying the final result 14 by 5 gives 70, the value the number had just before the divide-by-5 step.
The divide-by-5 was the last thing done to reach 14, so undoing that one step gives back exactly the number that went into it.
Undoing a step means doing its opposite, and the opposite of dividing by 5 is multiplying by 5.
Multiplying and dividing by the same number are opposite operations that reverse each other.
Before dividing, 50 had been added. The inverse of adding 50 is subtracting 50, so take 50 away from 70.
Subtraction reverses addition, peeling off the 50 that was put on.
3.OA.D.8Work BackwardsThe first forward step multiplied the start by 4. The inverse is divide by 4, so divide 20 by 4 to recover the original number.
Dividing undoes multiplying, returning us to the very first number.
3.OA.A.4Work BackwardsTo find a hidden starting number, walk the steps backward and flip each operation — divide becomes multiply, add becomes subtract!
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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 →