Concept 8 of 10
The Opposite Engine: Finding Unknowns
Today's big idea
Find a missing factor by flipping the multiplication into a division.
3 x ? = 534 flips to 534 divided by 3, which is 178.
How it works
👀 What's going on?
You are staring at 3 x ? = 534. You could guess numbers until one fits, which could take all day. Flipping the question turns it into something you can simply work out.
💡 Here's the trick
Look at the flip above. Multiplying and dividing are opposites, so 3 x ? = 534 becomes 534 divided by 3, which is 178. Whenever the missing number is being multiplied, divide to find it. Then multiply back, and you should land on the total you started with.
🪄 How to do it
- 1.Spot which number is the total, as 534 is the total in the picture above.
- 2.Divide that total by the factor you already know.
- 3.Multiply your answer back to check it returns the total.
✍️ Try it
Solve 7 x ? = 210, then solve ? x 9 = 72. Does the flip work the same way both times?
For the first, 210 divided by 7 = 30, and 7 x 30 = 210 checks out. For the second, 72 divided by 9 = 8, and 8 x 9 = 72 checks out. It makes no difference which side the missing number sits on, because you always divide the total.
⚠️ Watch out
You might be thinking the flip of 7 x ? = 210 could be 7 divided by 210. The total always goes first. A quick sense check settles it: 210 shared into 7 groups has to be smaller than 210, and 30 is.
🌟 Why this matters
Working out the price of one item from a total bill, or how many weeks of pocket money buy something, are both missing-number questions. This is also the first real step into algebra, where that box becomes an x.
Exam tip: Multiply your answer back every time, it takes seconds and catches a flipped division.
Watch me solve one
Quanta solves it
“Watch how I think through one of these — then you try.”
THE PROBLEM
“4 × ? = 340 — find the missing factor.”
- 1
Spot the missing piece
One factor is hidden. The total (product) is 340.
- 2
Hit the FLIP switch 🔄
Multiplication flips into division: 340 ÷ 4 = ?
- 3
Divide
340 ÷ 4 = 85
- 4
Missing factor found ✓
4 × 85 = 340 ✓
Now you try one — same steps, different story 👇
See it in action
⚡ Time to play!
Multiplication and division are OPPOSITES! Use the flip to find the missing number.
Problem 1 of 3
FIND THE MISSING NUMBER
6 × ? = 48
Quick check
✅ VERIFICATION
If you are trying to solve X * 4 = 144, what inverse math equation will find X?
⚡ Finish the activity above first to answer.
Keep going!
Up next: Brackets First
Why 4 + 5 x 2 is 14 and not 18.