Problem
Recover the unknown, then recompute
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?
Operations
Your answer
Watch out Finding the hidden number 6 isn't the end — you still have to multiply by 5 to get the real answer.
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
Working backwards from 48 ÷ 8, the hidden number is 6.
□ × 8 = 48 → □ = 6
Division exactly undoes multiplication: dividing the product 48 by the known factor 8 gives 6, and six equal groups of 8 really do add up to 48, confirming the number.
Work Backwards3.OA.B.6The 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 6 by 5, the right multiplier, to get 30.
6 × 5 = 30
The hidden number 6 stays the same — only the multiplier changes from 8 to 5. Counting six equal groups of 5 gives the answer the problem originally intended.
Work Backwards3.OA.A.4Answer
30
6 × 5 = 30
Check — 6 × 8 = 48 matches the mistaken calculation, and 6 × 5 = 30 is smaller than 48, which makes sense.
Takeaway
Find the hidden number by working backwards, then redo the correct multiplication.
- □ × 8 = 48 → hidden number 6
- 6 × 5 = 30 is the correct calculation
Where next?
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