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Cosmic Abacus
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Big friends practice: the 10-complement rules

The carry, done the abacus way: when a digit will not fit on its rod, take ten from the rod to its left and give back the difference.

  • 4 rows × 2 digits, add & subtract
  • About 8 min
  • Free, no account

Sample problems

Real problems from this drill, with answers. Nothing to install, no account, no email — work them on paper, on a physical abacus, or in your head.

  1. 1
    98
    − 93
    + 48
    − 44
    ?
  2. 2
    36
    + 46
    + 89
    − 69
    ?
  3. 3
    85
    − 68
    + 47
    − 46
    ?
  4. 4
    65
    + 62
    + 72
    + 49
    ?
  5. 5
    43
    − 39
    + 26
    + 46
    ?
  6. 6
    78
    + 20
    − 95
    + 53
    ?
  7. 7
    65
    + 74
    − 39
    + 75
    ?
  8. 8
    93
    + 56
    − 58
    + 28
    ?
  9. 9
    44
    − 13
    + 56
    − 19
    ?
  10. 10
    44
    − 26
    + 38
    − 51
    ?
Full answer key
  1. 98 − 93 + 48 − 44 = 9
  2. 36 + 46 + 89 − 69 = 102
  3. 85 − 68 + 47 − 46 = 18
  4. 65 + 62 + 72 + 49 = 248
  5. 43 − 39 + 26 + 46 = 76
  6. 78 + 20 − 95 + 53 = 56
  7. 65 + 74 − 39 + 75 = 175
  8. 93 + 56 − 58 + 28 = 119
  9. 44 − 13 + 56 − 19 = 68
  10. 44 − 26 + 38 − 51 = 5

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About this drill

Big friends are the pairs that add to ten: 1–9, 2–8, 3–7, 4–6, 5–5. To add 7 to a rod showing 6, you cannot — so you add one bead on the rod to the left (worth ten) and remove three on the right, because 7 is 10 minus 3. On paper this is called carrying. On the abacus it is a single two-handed gesture, and it is the last mechanical rule the method needs.

These drills stack four two-digit numbers with mixed signs, which is where big friends and small friends start appearing in the same problem. That combination is the real difficulty of this level — not either rule alone, but switching between them without pausing.

Look again at that example, because it quietly contains the hardest move on the level. The rod was showing 6 — the five-bead plus a single earth bead — and three had to come off it. There are not three raised earth beads to take. So the removal is itself a small-friend move: take away five, give back two, which leaves 3, and the ten still goes on the rod to the left. 6 + 7 = 13 is two complements nested inside one addition, and if a child has been taught big friends as a single self-contained rule, this is the sum where it comes apart. The walkthrough at kani-math.com/learn/big-friends works this case frame by frame, and it is the chapter worth watching twice.

Knowing which sums are the nested ones makes practice targeted instead of hopeful. When the complement to remove is 4 or less, count the earth beads standing under the five-bead: if the complement is larger than that count, a small friend has to fire inside the carry. A rod showing 6 has one earth bead raised, so 6 + 7 (remove 3) and 6 + 8 (remove 2) both need it; a rod showing 7 has two raised, so 7 + 6 (remove 4) needs it as well — while 8 + 7 does not, because a rod showing 8 has exactly the three earth beads that come off. A complement of 5 is never a problem either: that is just the five-bead going back up. A child still unsure which bead carries which value should spend a session at kani-math.com/abacus/how-to-use before drilling any of this.

How to practise it

  1. 1Memorise the five pairs to ten before starting.
  2. 2Left hand on the tens rod, right hand on the units rod.
  3. 3Add the ten first, then remove the complement — always that order.
  4. 4Watch for problems needing both a small and a big friend in one row.
Where this sits

Level 4 — Elementary B

Big friends — the 10-complement rules — on 2-digit problems.

Teaches: 10-complement (big friends) technique, 2-digit operations

Find your level with the free placement exam →

Questions

Is a big friend the same as carrying?
It produces the same result, but the procedure is different. Written carrying defers the ten to a later step; the abacus moves it immediately, so there is never anything held in reserve.
Why do both hands move?
Because two rods change at once. Training both hands from the start is what keeps speed up later — one-handed abacus work hits a ceiling fast.
My child mixes up small and big friends.
Normal at this stage. The cue is which bead is blocked: if the five-bead can solve it, it is a small friend; if the whole rod is full, it is a big friend.
How many digits should we practise on?
Two is right for learning the rule. Widen to three only after the two-rod carry is automatic, or the child ends up managing place value instead of the technique.
My child can do 9 + 3 but gets stuck on 6 + 7.
That is the nested case, and it is the difficulty of the whole level rather than a gap in the rule. In 9 + 3 the complement to remove is 7, and a rod showing 9 carries the five-bead and four earth beads, so 7 lifts off directly. In 6 + 7 the complement is 3, and a rod showing 6 has only one earth bead raised — so removing 3 is itself a small-friend move, nested inside the same carry. Nothing new is being asked; two known rules are being used in one breath. Slow the pace right down and have the child say both aloud until the pair becomes one gesture. It is worked step by step at kani-math.com/learn/big-friends.
Our speed dropped since starting this level. Is that normal?
Yes, and at this point a drop is the correct sign. Until now every blocked digit had a one-rod answer; here some of them need two rods, both hands, and occasionally both rules, so the child is doing more work per row than on the previous level. Accuracy is the metric that matters for now — a column worked slowly and correctly beats a fast one with a carry missed somewhere in the middle, because a wrong answer gives no clue about which row went wrong. Speed comes back once the two-rod gesture stops being two separate decisions.
Learn the techniqueBig friends (10-complements)Walk through it on the beads →