A number bond is a pair of numbers that add to a given total. The bonds to ten are 1 and 9, 2 and 8, 3 and 7, 4 and 6, and 5 with itself. That is the whole idea, and it takes about thirty seconds to explain. What takes longer to appreciate is how much mental arithmetic runs through them. Adding 47 and 8 without writing anything down means knowing that 47 needs 3 to reach 50. Subtracting 63 from 100 without borrowing means knowing the bond to 100. Every bridging method, every shortcut, every quick estimate leans on this one small set of facts.
Two parts and a whole. 7 and 3 are the parts, 10 is the whole, and the bond holds in every direction: 7 + 3 = 10, 3 + 7 = 10, 10 − 7 = 3 and 10 − 3 = 7. Four facts for the price of one, which is exactly why bonds are taught as pairs rather than as separate sums.
They are usually drawn as a small diagram — the whole at the top with two lines running down to the parts — which is where the name comes from. The diagram is a teaching aid, not the maths. The maths is the pairing.
The same structure scales. Bonds to 20, bonds to 100, bonds to 1,000 are all the same idea with a different whole, and the ones to 100 have a pattern that makes them nearly free once you spot it.
There are six of them, counting zero, and they are the foundation of everything that follows.
0 + 10 = 10
1 + 9 = 10
2 + 8 = 10
3 + 7 = 10
4 + 6 = 10
5 + 5 = 10
Each one also runs backwards, so 10 − 4 = 6 is the same fact as 4 + 6 = 10. A child who knows the bond in one direction and not the other has not finished learning it, and the subtraction direction is the one that shows up in change from a note and in borrowing.
Five and five is the anchor. Everything else is a step away from it: 6 and 4 is five and five with one moved across, 7 and 3 is two moved across.
Structurally these are the bonds to 10 with a ten hidden inside them. 13 + 7 = 20 because 3 + 7 = 10, and the ten that was already in the 13 makes the second ten.
11 + 9, 12 + 8, 13 + 7, 14 + 6, 15 + 5
16 + 4, 17 + 3, 18 + 2, 19 + 1, 10 + 10
Seeing them as the ten bonds shifted is what stops them being a second list to memorise. If 4 + 6 = 10 is instant, then 14 + 6 = 20 needs no separate learning at all.
These look like ninety-nine separate facts and are really one rule. For any two-digit number, the units digits add to 10 and the tens digits add to 9.
So 63 pairs with 37: 6 + 3 = 9 in the tens, and 3 + 7 = 10 in the units. 28 pairs with 72, because 2 + 7 = 9 and 8 + 2 = 10. 45 pairs with 55.
The one exception is a round number. 60 pairs with 40, where the tens add to 10 rather than 9, because there is no unit to carry. Once you have noticed that, every bond to 100 is available in about a second.
Bonds to 100 are what let you subtract from 100 without borrowing. 100 − 63 is not a column subtraction; it is a bond you already know.
Bridging through ten
47 + 8 is awkward as written and easy as a bridge: 47 needs 3 to reach 50, that leaves 5 of the 8, so the answer is 55. The only thing that makes it quick is knowing instantly that 47 needs 3.
Subtraction without borrowing
Working out change is the everyday case. From £20, a £13.40 bill leaves £6.60, and the fast route there is bonds rather than a column subtraction: 40 needs 60 to reach 100, and 13 needs 6 to reach 19.
Splitting bigger sums
38 + 27 becomes 38 + 2 + 25: bridge to 40, then add the rest. Every mental addition method of this kind is a bond plus a leftover.
Estimating
Knowing that 63 and 37 make 100 is what lets you look at a column of numbers and pair them off into hundreds instead of adding them one by one.
There are very few of these facts. Six bonds to 10, ten to 20, and one rule for the hundred. That is a smaller set than a single times table row, which is why it is worth getting to genuinely instant rather than merely reliable.
Learn them in pairs, not as separate sums. 7 + 3 and 3 + 7 are one fact.
Practise the subtraction direction as much as the addition one — 10 − 6 as often as 4 + 6.
Drill shuffled. In order, 1 + 9, 2 + 8, 3 + 7 becomes a chant rather than recall. Number Bonds shuffles them.
If you are counting on your fingers, drop to the smaller whole. Counting practises counting.
Write a worksheet once a fortnight for a record of what has stuck.
Adults who never got these to automatic recall usually do not know it — the workaround happens fast enough to feel like knowing. The giveaway is reaching for a phone to work out change.
A number bond is a pair of numbers that add to a given total. The bonds to 10 are 0+10, 1+9, 2+8, 3+7, 4+6 and 5+5, and each one also works backwards as a subtraction.
0 and 10, 1 and 9, 2 and 8, 3 and 7, 4 and 6, and 5 with itself. Each also gives a subtraction: 10 − 3 = 7 is the same fact as 3 + 7 = 10.
For any two-digit number, the partner has units digits adding to 10 and tens digits adding to 9. So 63 pairs with 37, and 28 pairs with 72. Round numbers are the exception: 60 pairs with 40.
They are what makes bridging work. 47 + 8 is quick if you know 47 needs 3 to reach 50, and subtracting from 100 stops needing borrowing once the hundred bonds are automatic.
Bonds to 10 are usually introduced in the first couple of years of primary school, with bonds to 20 and then 100 following. They are equally worth learning as an adult — the set is small and it takes days, not years.
They describe the same thing from different angles. A number bond is the pair; the fact family is the four sums that pair produces — two additions and two subtractions.
Number bonds are the smallest set of facts in arithmetic with the largest return. Six to learn for ten, ten more for twenty, and one pattern that covers the hundred. Drill them shuffled on [Number Bonds](/games/number-bonds), print a [sheet](/tools/math-worksheet), and watch how much mental addition stops needing effort.