Put 3 more sweets into the jar, then change your mind and take those same 3 back out — the jar is exactly as full as before. That everyday "undo" is why subtraction lets you check your adding and fill in missing numbers.

is the inverse of — it undoes it. If you add b and then take the same b away again, you are right back where you started.

So whenever you know an addition fact:

you can run it backwards to get two subtraction facts:

Watch how the marker hops right by b to add, then hops left by the same b to subtract — and lands exactly where it began. The left hops perfectly undo the right hops. Replay it: each time it uses fresh numbers.

You bake 8 cookies and eat 3. How many are left? Take three away: 8 - 3 = 5. Not sure you got it right? Put the eaten ones back in your head: 5 + 3 = 8 — yes, back to eight. Adding the answer back always checks a subtraction, because adding undoes taking away.

The part–whole bar

Here is a picture that makes all of this obvious: the part–whole bar. One long bar is the whole. Split it into two parts. Now the whole bar and its two pieces are stuck together forever:

• Put the two parts together and you get the whole — that is adding.
• Start from the whole and take one part away, and what's left is the other part — that is subtracting.

Take the whole 8 split into the parts 5 and 3. The two parts add to the whole, and the whole minus one part leaves the other:

Cover the 5 with your thumb and the bar asks "3 plus what makes 8?" — find the missing part. Subtraction is just finding a missing part of a whole you already know.

One bar makes a whole fact family

Those three numbers a, b and c make a little team called a fact family. From one bar you can write four true number sentences — two adding and two subtracting:

Press Refresh for a brand-new family, and step through to see the same bar give its addition fact and both of its subtractions.

For example, the family 3, 4, 7 gives:

Two superpowers

First, you can check a subtraction by adding back: if c - b = a is right, then a + b should bring you back to c.

Second, you can subtract by thinking addition. To work out 12 - 5, don't count back — instead ask "5 plus what makes 12?". Since 5 + 7 = 12, you know 12 - 5 = 7. The missing number is the answer.

Worked examples

• 8 - 3. You already know 5 + 3 = 8, so the missing part is 5: 8 - 3 = 5. Check by adding back: 5 + 3 = 8. ✓
• 10 - 6. Ask "6 plus what makes 10?" The answer is 4, so 10 - 6 = 4. Check: 4 + 6 = 10. ✓
• Missing part: \square + 7 = 15. The whole is 15 and one part is 7, so the other part is 15 - 7 = 8.

Khan Academy explains how addition and subtraction relate here:

9 ducks are on the pond. Some swim behind the reeds and now you can only see 4. How many are hidden? You don't have to count them — they are the missing part. The whole is 9, one part is 4, so the hidden part is 9 - 4 = 5. Thinking "4 plus what makes 9?" gives the same 5.

A toy costs 12 coins and you hand over 20. Your change is 20 - 12. Counting back from 20 eight times is slow and easy to muddle. Shopkeepers count up instead: "12… 13, 14, 15, 16, 17, 18, 19, 20" — that's 8 coins of change. They are using the inverse: 12 + 8 = 20, so 20 - 12 = 8.