Recover the original number from a wrong product

3.OA.A.43.OA.C.7 · take · grade 3

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

A certain number was supposed to be multiplied by $7$ , but by mistake $7$ was subtracted instead, giving $409$ . What is the result of the correct calculation?

Show solution

Understand

A number should have been multiplied by 7, but instead 7 was subtracted, giving 409. Find what the correct answer (number times 7) should be.

Givens

The correct operation was: the number times 7.
By mistake, 7 was subtracted from the number, giving 409.

Unknowns

The original number, and then the correct result (number x 7).

Constraints

The same starting number is used in both the wrong and the correct calculation.

Plan

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

We know the result of the wrong operation (subtract 7 gives 409). Undo the subtraction by adding 7 to recover the original number, then do the correct operation (multiply by 7).

Execute

#11 Work Backwards 3.OA.A.4

The mistake subtracted 7 to get 409. To find the original number, add 7 back.

409 + 7 = 416

Adding undoes subtracting, so we work backward to the start.

#6 Guess and Check 3.OA.C.7

The number should have been multiplied by 7. Multiply 416 by 7.

416 \times 7 = 2800 + 70 + 42 = 2912

Break 416 into 400 + 10 + 6, multiply each by 7, then add.

Answer: 2912

Review

The original number 416 is a bit over 400, and 400 x 7 = 2800, so the correct answer should be near 2900; 2912 fits, and checking: 416 - 7 = 409 matches the wrong result.

Set up the relationship as 'number - 7 = 409' to solve number = 416, then compute 416 x 7 = 2912.

Standards · min grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Working backward to find the original number from the wrong result.
3.OA.C.7 Fluently multiply and divide within 100 — Multiplying the recovered number by 7.

💡 Undo the mistake first, then do it right -- working backwards is a Grade 3 superpower!