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Number Bonds and Making Tens

Complements to 10, 100 and 1000 are the layer every mental addition method assumes: what they are, the digit rule with one trap, and how adults install them.

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Key takeaways

  • A number bond is a pair that combines to a landmark number. There are only five pairs to know for ten — 1 and 9, 2 and 8, 3 and 7, 4 and 6, 5 and 5 — and everything else in mental addition leans on them.
  • Making tens means moving a little from one number to the other so one of them becomes round: 8 + 5 becomes 10 + 3 = 13. The mistake to watch for is adding the moved amount without removing it.
  • For complements to 100, subtract each digit from 9 and the last from 10: 100 − 37 gives 9−3 = 6 and 10−7 = 3, so 63. The rule breaks when the final digit is 0, and you handle that case separately.
  • The same bridging works at any friendly anchor — 50, 100, 250, or simply the next multiple of ten — which is what makes three-digit addition tractable without paper.
  • In a list, pairing the numbers that combine to a round total first is legitimate because addition can be reordered freely, and it removes most of the working-memory load.

A number bond is a pair that combines to a landmark number, and for ten there are only five worth naming: 1 and 9, 2 and 8, 3 and 7, 4 and 6, 5 and 5. Knowing them as retrieved facts rather than small calculations is the layer that every other mental addition method quietly assumes, which is why an adult whose arithmetic feels slow is usually slow here rather than anywhere more interesting.

This article covers the bonds themselves, the make-ten move that uses them, the digit rule for complements to 100 and the one case where that rule breaks, and a realistic route for installing the lot as an adult.

Why bonds sit underneath everything else

The mathematics programmes of study for England expect pupils to know number bonds to 20 by the end of year 2 and to be increasingly fluent with number facts thereafter. That sequencing is not arbitrary — the later methods do not work without the earlier facts.

Left-to-right addition assumes you know that 70 + 50 crosses into the hundreds without stopping to work it out. Counting up for subtraction assumes you can see the distance from 47 to 50 instantly. Percentage work assumes complementing to 100 is free, which is why "30% off" and "you pay 70%" feel like the same thought rather than two. When any of those feel effortful, the bottleneck is almost always the bond, not the method.

NRICH, the mathematics education project at the University of Cambridge, describes fluency as efficiency, accuracy and flexibility held together, with accuracy resting partly on knowledge of number facts and other important number relationships. Bonds are the smallest, densest version of that knowledge.

Making tens: move a little, then compensate

The make-ten move is the reason the bonds matter. You take a little from one number so that the other becomes a clean ten, then add what is left.

8 + 5. The 8 needs 2 to reach ten. Take 2 from the 5, leaving 3: 10 + 3 = 13.

7 + 6. The 7 needs 3. Take 3 from the 6, leaving 3: 10 + 3 = 13.

9 + 4. The 9 needs 1. Take 1 from the 4, leaving 3: 10 + 3 = 13.

The pattern is deliberate — all three land on 13, which makes visible what the move is doing. You have not changed the total, only redistributed it.

The mistake to watch for is adding the moved amount without removing it. Taking 2 from the 5 to make 8 into 10 and then adding the original 5 gives 15, which is wrong by exactly the amount you moved. If an answer is out by one or two after a make-ten, this is almost always why. The check is that the two numbers must still sum to what you started with: 10 and 3 is 13, and 8 and 5 is 13.

The move scales without modification. 47 + 8: the 47 needs 3 to reach 50, leaving 5, so 55. 68 + 25: the 68 needs 2 to reach 70, leaving 23, so 93. The Maine Department of Education lists both making friendly numbers and compensation in its addition strategies progression for exactly this reason — they are the same instinct applied at different landmarks.

Complements to 100, and the digit rule

Bonds to 100 come up constantly — change from a note, a discount, a percentage that has to total 100 — and there is a rule that removes all the borrowing.

Subtract every digit from 9 except the last, which you subtract from 10.

100 − 37: 9 − 3 = 6, 10 − 7 = 3, giving 63. 100 − 45: 9 − 4 = 5, 10 − 5 = 5, giving 55. 100 − 8, read as 100 − 08: 9 − 0 = 9, 10 − 8 = 2, giving 92.

The reason it works is worth knowing, because it is what lets you extend it. You are not complementing to 100; you are complementing to 99 and then adding the spare 1. Every digit subtracts from 9 with no borrow anywhere, and the leftover 1 lands on the ones column, which is why the final digit comes from 10.

That generalises directly. 1000 − 428: 9 − 4 = 5, 9 − 2 = 7, 10 − 8 = 2, giving 572. Check it by adding back: 428 + 572 = 1000.

The one case that breaks it

A final digit of zero. Applied mechanically, 100 − 60 gives 9 − 6 = 3 and 10 − 0 = 10, which is not a digit and not an answer.

The cause is that there was no borrow to undo in the first place. When the ones digit is 0, drop it and take the tens complement directly: 10 − 6 = 4, so 40. Same at the next scale: 1000 − 750 is 250, found by complementing 75 to 100 and reattaching the zero.

Any shortcut is worth testing on its awkward case before you trust it under pressure, and for this rule the awkward case is a trailing zero.

Bridging at bigger anchors

Ten is not the only useful landmark. Fifty, 100 and 250 all behave the same way, and so does "the next multiple of ten", which is the one you will reach for most.

236 + 87. Bridge at the next ten: 236 needs 4 to reach 240, leaving 83, giving 323. Or bridge at 300: 236 needs 64, leaving 23, giving the same 323. Both are legitimate; the first involves smaller adjustments and the second reaches a rounder waypoint, and which one is easier depends on the numbers.

247 + 168, anchored at 250: 247 needs 3, leaving 165, giving 415.

48 + 37, anchored at 50: 48 needs 2, leaving 35, giving 85.

Choosing the anchor is the skill. The rule of thumb is to move the smaller distance — push whichever number is nearer to its landmark.

Pairing in a list

Addition can be reordered freely, and in a list of numbers that freedom is worth using deliberately: pair the ones that combine to something round before adding anything else.

7 + 3 + 8 + 2 + 6. Pair 7 and 3 for 10, pair 8 and 2 for 10, then add the 6: 26.

14 + 26 + 35 + 5 + 12. The 14 and 26 make 40; the 35 and 5 make another 40; 80 plus 12 gives 92.

The benefit is not fewer additions — it is fewer awkward intermediate values to hold. Holding 40, then 80, then 92 is much lighter work than holding 14, 40, 75, 80, 92, and reducing what you have to keep in mind is most of what mental arithmetic technique is for.

Installing them as an adult

The good news is that adults are rarely starting from zero. The facts are half-known; what is missing is retrieval speed, and that responds quickly.

Three things work. Short and frequent beats long and rare — a few minutes on most days will do more than an hour on a Sunday. Practise both directions, because "what goes with 7 to make 10" and "10 − 7" should be the same retrieved fact rather than two separate calculations. And practise the complement to 100 as its own drill, since it is the one that pays off outside arithmetic, in discounts, tips and any percentage that has to total a whole.

IQ Test & Brain Training Games is our own app, built by Reign Creative, and its technique library carries both the make-ten move and the friendly-numbers approach with anchors at 50, 100 and 250 — each with steps, a worked example and a named common mistake, plus a practice link into Addition Sprint, a 60-second drill that runs from single-digit work up through two- and three-digit settings and mixed chains. The single-digit setting is the one that drills bonds directly; the others test whether they have actually become automatic.

Once the bonds are solid, the method that uses them most is left-to-right addition, and the place they pay off fastest outside pure arithmetic is percentages in your head. The three moves underneath almost every mental method are set out in our mental math practice guide; more methods sit in the mental math hub and across the education and brain category, and the app itself is described on the IQ Test & Brain Training Games app page.

Good to know

Frequently asked questions

What exactly is a number bond?
A pair of numbers that combines to a landmark value, usually 10, 20, 100 or 1000. Saying that 3 and 7 are a bond to ten means both that 3 + 7 = 10 and that 10 − 7 = 3, held as one fact rather than two. Treating the pair as a single retrieved object, rather than as a calculation to perform, is what makes it useful under time pressure.
Why do complements to 100 use nine for every digit but the last?
Because you are really finding the complement to 99 and then adding one. Splitting 100 as 99 + 1 removes every borrow: each digit subtracts cleanly from 9, and the spare 1 goes on the end, which is why the last digit comes from 10 instead. For 100 − 45 that gives 9 − 4 = 5 and 10 − 5 = 5, so 55.
What is the trap in the complement rule?
A final digit of zero. Applying the rule to 100 − 60 mechanically gives 9 − 6 = 3 and 10 − 0 = 10, which is nonsense. When the ones digit is 0 there is no borrow to undo, so take the tens complement directly: 10 − 6 = 4, giving 40. The same applies at 1000 − 750, which is 250.
Is it worth learning number bonds as an adult?
It is the highest-return thing on the list, because almost every other mental method silently assumes them. Left-to-right addition assumes you know that 70 + 50 crosses into the hundreds; percentage work assumes you can complement to 100 without thinking. An adult does not need flash cards for months — a few short sessions mostly closes the gap, because the facts are already half-known.

Sources

  1. National curriculum in England: mathematics programmes of study (opens in a new tab)Department for Education (GOV.UK) — accessed
  2. Developing Number Fluency — What, Why and How (opens in a new tab)NRICH, University of Cambridge — accessed
  3. Addition Strategies Progression (opens in a new tab)Maine Department of Education — accessed
  4. Algorithms — Everyday Mathematics (opens in a new tab)University of Chicago School Mathematics Project — accessed

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