Every formula you need to master the abacus
Small Friends are used when you need to add a number but there aren't enough earth beads on the rod. The trick is to subtract the complement of 5 and add 5 (push the heaven bead down).
Set 3 on the abacus (3 earth beads up). Now add 4 — but we only have 1 earth bead left!
Use Small Friends: +4 = −1 +5. Remove 1 earth bead (now 2 up), push heaven bead down (+5).
Result: heaven bead (5) + 2 earth beads = 7. Correct!
Big Friends are used when the current rod is full (all beads used) and you need to add more. The trick is to subtract the complement of 10 from the current rod and add 1 to the next rod on the left (which represents adding 10).
Set 6 on the abacus. Now add 9 — but the ones rod only goes to 9!
Use Big Friends: +9 = −1 +10. Remove 1 from ones (now showing 5), add 1 to tens rod.
Result: tens rod shows 1 (=10) + ones rod shows 5 = 15. Correct!
These advanced formulas combine both Small Friends and Big Friends in a single operation. They're used when you need to add a number that requires adjusting both the heaven bead and carrying to the next rod.
| Multiply by | Trick | Example (8 ×) |
|---|---|---|
| ×2 | Double the number | 8×2 = 16 |
| ×3 | Double, then add once more | 8×2=16, +8 = 24 |
| ×4 | Double twice | 8→16→32 |
| ×5 | Half it, then ×10 | 8÷2=4, ×10 = 40 |
| ×6 | ×5 plus the number | 8×5=40, +8 = 48 |
| ×7 | ×5 plus double | 8×5=40, +16 = 56 |
| ×8 | Double three times | 8→16→32→64 |
| ×9 | ×10 minus the number | 8×10=80, −8 = 72 |
| ×11 | ×10 plus the number | 8×10=80, +8 = 88 |
| Divide by | Trick | Example |
|---|---|---|
| ÷2 | Half the number | 48÷2 = 24 |
| ÷3 | How many 3s fit? | 27÷3 = 9 |
| ÷4 | Half, then half again | 48→24→12 |
| ÷5 | ÷10, then ×2 | 80÷10=8, ×2 = 16 |
| ÷6 | ÷2, then ÷3 | 48÷2=24, ÷3 = 8 |
| ÷8 | Half three times | 72→36→18→9 |
| ÷9 | ÷10, add quotient | 90÷10=9, +1 = 10 |
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