Big Friend Formulas: Carrying with 10

Set 8 on the ones rod — heaven bead down, three earth beads up — and try to add 5.

The heaven bead is already used. The Small Friend trick won't help, because there is no 5 left to add. The ones rod is simply full, and the answer (13) needs two digits. The abacus handles this with the second great formula: the Big Friend, which is nothing more than carrying — made visible.

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5

100Hundreds

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1Ones

8

Tap the beads to try it

What "Big Friend" means

Two numbers are Big Friends if they add up to 10. There are five pairs:

• 1 and 9
• 2 and 8
• 3 and 7
• 4 and 6
• 5 and 5

The rule: when a rod can't hold the result, add 10 by pushing one earth bead up on the rod to the left, and take away the Big Friend on the current rod.

• To add 8: +10 on the next rod, −2 on this rod. Because +8 = +10 −2.
• To add 7: +10, −3.
• To add 6: +10, −4.
• To add 5: +10, −5.
• To add 9: +10, −1.

Worked example: 8 + 5

1. Ones rod shows 8.
2. Push one earth bead up on the tens rod: +10. The abacus reads 18.
3. On the ones rod, take away 5 — the Big Friend of 5 — by pushing the heaven bead up. The abacus reads 13.

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5

100Hundreds

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1Ones

13

Tap the beads to try it

Notice that nothing was "carried" in the written sense. The 1 in 13 didn't appear from nowhere — the child put it there, on purpose, as the first move. This is why abacus-trained children rarely make carrying errors later in written arithmetic.

Subtraction: borrowing with Big Friends

Set 12 and subtract 7. The ones rod shows 2, so you can't take 7 from it. Reverse the rule:

• To subtract 7: −10 on the tens rod, then +3 (the Big Friend of 7) on the ones rod. Because −7 = −10 +3.
• To subtract 8: −10, +2. To subtract 9: −10, +1. To subtract 6: −10, +4. To subtract 5: −10, +5.

Worked: 12 − 7. Tens rod: one earth bead down (12 → 2). Ones rod: add 3 (2 → 5). The abacus shows 5.

500

50

5

100Hundreds

10Tens

1Ones

5

Tap the beads to try it

When both friends are needed: the Family formulas

Sometimes the Big Friend's "take away" step is itself impossible with the beads on the rod. Set 6 (heaven bead plus one earth bead) and add 7:

• Big Friend says: +10, then −3. But the ones rod only has one earth bead up. You can't take three.
• So you take the 3 away using a Small Friend in reverse: −3 = −5 +2.

Put together: +7 = +10 −5 +2. Bead by bead: tens rod +1 (→16), ones heaven bead up (→11), ones two earth beads up (→13). The answer is 13.

Abacus teachers call these combined moves Family formulas, because they use both friends at once. Written out as the app shows them:

• Addition family: +6 = +10 −5 +1 · +7 = +10 −5 +2 · +8 = +10 −5 +3 · +9 = +10 −5 +4
• Subtraction family: −6 = −10 +5 −1 · −7 = −10 +5 −2 · −8 = −10 +5 −3 · −9 = −10 +5 −4

How Abacus Kids teaches it

In the Level 2 Learning Path, Big Friend addition and subtraction are chapters 8 and 9, and they are the first chapters that use two rods at once — the app widens the practice abacus to two columns so the carry is impossible to miss. Chapter 10 covers the Family formulas.

Once a child is fluent here, they can add or subtract any numbers of any length on the abacus: there are no more special cases. Everything after Level 2 — speed work, Anzan, and the multiplication and division levels of Abacus Practice — is about doing the same moves faster and, eventually, without the abacus.