Problem
Recover the unknown, then recompute
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.
Operations
Your answer
Watch out The question asks for the correct result of multiplying by 5, not the hidden number you find along the way.
How to solve
Strategy 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.
1STEP 1
Recover the hidden number
What number times 8 makes 48? Work backwards to recover it: that number is 6.
□ × 8 = 48 → □ = 6
Finding the missing factor of 48 with 8 is the unknown-factor idea: 6 times 8 is 48.
3.OA.B.6Work BackwardsThe hidden number is 6 -- the number that gives 48 when it is multiplied by 8.
Why?
The mistaken step multiplied the hidden number by 8 and landed on 48, so the number is whatever times 8 makes 48.
Why?
To pull the hidden factor back out, divide the product 48 by the factor we already know, 8, and that gives 6.
Why?
Division is built to reverse multiplication, so dividing by 8 exactly cancels the earlier multiply-by-8.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
Why?
6 is the right recovered number because six equal groups of 8 pile up to 48, matching the mistaken result.
Why?
Multiplying 6 by 8 means counting six equal groups of 8, which is repeated adding that reaches 48.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
2STEP 2
Do the correct calculation
Now multiply the recovered number 6 by 5 as the problem intended.
6 × 5 = 30
Once the number is known, the correct answer is just the fact 6 times 5.
3.OA.A.4Work BackwardsAnswer
30
6 × 5 = 30
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.
Takeaway
Work backwards to find the hidden number, then multiply the right way -- only Grade 3 fact families needed!
- Recover the hidden number
- Do the correct calculation
Where next?
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