Reverse the operations to find the start

3.OA.D.83.OA.A.4 · take · grade 3

Archetype: Work Backwards to Recover a Start Value · step in a 9-type progression

Mia thought of a number. She multiplied it by $4$ , added $50$ , and then divided the result by $5$ , which gave $14$ . What number did Mia think of first?

(The diagram shows the starting number going through $\times 4$ , then $+50$ , then $\div 5$ to reach $14$ , drawn as arrows. Below each arrow is the inverse operation used when working backward: $\div\square$ , $-\square$ , $\times\square$ .)

Show solution

Understand

A starting number was multiplied by 4, then 50 was added, then the result was divided by 5, ending at 14. We must find the original starting number by reversing each step.

Givens

The forward steps are: multiply by 4, then add 50, then divide by 5
The final result after all three steps is 14
The flow diagram shows each forward arrow with its inverse below: divide by, subtract, multiply by

Unknowns

The starting number Mia first thought of

Constraints

The operations must be undone in reverse order: undo divide-by-5 first, then undo add-50, then undo multiply-by-4

Plan

#11 Work Backwards · also uses: #6 Guess and Check

The end result (14) is given and the start is unknown, which is the classic signal to work backwards. The diagram even labels each inverse operation, so we apply the opposite of each step in reverse order. A quick forward check confirms the answer.

Execute

#11 Work Backwards 3.OA.A.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.

14 \times 5 = 70

Multiplying undoes dividing, so we climb back up from 14 to 70.

#11 Work Backwards 3.OA.D.8

Before dividing, 50 had been added. The inverse of adding 50 is subtracting 50, so take 50 away from 70.

70 - 50 = 20

Subtraction reverses addition, peeling off the 50 that was put on.

#11 Work Backwards 3.OA.A.4

The first forward step multiplied the start by 4. The inverse is divide by 4, so divide 20 by 4 to recover the original number.

20 \div 4 = 5

Dividing undoes multiplying, returning us to the very first number.

Answer: 5

Review

Run it forward to check: 5 times 4 is 20, plus 50 is 70, divided by 5 is 14 — exactly the given result. So 5 is correct.

Guess and check (tool 6): try a start of 5, push it through multiply 4, add 50, divide 5, and you land on 14, confirming the answer without reversing.

Standards · min grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Inverting the multiply-by-4 and divide-by-5 steps
3.OA.D.8 Solve two-step word problems using four operations within 100 — Reversing the chain of four-operation steps in order

💡 To find a hidden starting number, walk the steps backward and flip each operation — divide becomes multiply, add becomes subtract!