Recover the unknown, then recompute

3.OA.A.43.OA.B.6 · 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 $5$ , but by mistake it was multiplied by $8$ , giving $48$ . What is the result of the correct calculation?

Show solution

Understand

A number should have been multiplied by 5, but it was multiplied by 8 instead, giving 48. We must first find that number, then multiply it by 5 the correct way.

Givens

The wrong calculation multiplied the number by 8 and got 48.
The correct calculation should multiply the number by 5.

Unknowns

The hidden number.
The result of the correct calculation (number x 5).

Constraints

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

Plan

#11 Work Backwards

The wrong result is given, so work backwards from 48 to recover the hidden number, then go forward to do the correct multiplication by 5.

Execute

#11 Work Backwards 3.OA.B.6

Since the number times 8 gives 48, find the number by asking what times 8 makes 48. That is 6.

\square \times 8 = 48 \Rightarrow \square = 6

Finding the missing factor of 48 with 8 is the unknown-factor idea: 6 times 8 is 48.

#11 Work Backwards 3.OA.A.4

Now multiply the recovered number 6 by 5 as the problem intended.

6 \times 5 = 30

Once the number is known, the correct answer is just the fact 6 times 5.

Answer: 30

Review

The hidden number 6 checks out because 6 x 8 = 48, matching the wrong result, and the correct answer 30 is smaller than 48, which makes sense since multiplying by 5 instead of 8 gives less.

You could divide directly: 48 divided by 8 is 6, then 6 x 5 = 30, using division to undo the multiplication.

Standards · min grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding the correct product 6 x 5 once the number is known.
3.OA.B.6 Understand division as an unknown-factor problem — Recovering the hidden number from box x 8 = 48 as a missing factor.

💡 Work backwards to find the hidden number, then multiply the right way -- only Grade 3 fact families needed!