Vedic maths › Multiplying by 9, 99, 999
Multiplying by 9, 99 or 999
Ekanyūnena Pūrveṇa, “by one less than the one before”, multiplies a number by a row of nines in two quick steps, as long as the number has no more digits than there are nines.
The method
- Left part: one less than the number: 437 − 1 = 436.
- Right part: take the left part from the nines: 999 − 436 = 563. (That is all from 9 and the last from 10 of 437.)
- Put them side by side: 436563.
- If the number has fewer digits than the nines, put zeros in front first: 45 × 999 is 045 × 999.
Worked examples
437 × 999
- Left part: one less than the number.437 − 1 = 436
- Right part: take that from 999, digit by digit.999 − 436 = 563
437 × 999 = 436 | 563 = 436563
72 × 99
- Left part: one less than the number.72 − 1 = 71
- Right part: take that from 99, digit by digit.99 − 71 = 28
72 × 99 = 71 | 28 = 7128
45 × 999
- 45 has fewer digits than 999, so think of it as 045.045
- Left part: one less than the number.45 − 1 = 44
- Right part: take that from 999, digit by digit.999 − 044 = 955
45 × 999 = 44 | 955 = 44955
Why it works
Multiplying by 999 is multiplying by 1000 and taking the number away once: 437 × 999 = 437,000 − 437 = 436,000 + (1000 − 437) = 436,000 + 563.
Watch out
- When the number has more digits than the nines, this simple form doesn’t apply. The book has a longer version, but 1234 × 99 is easier as 1234 × 100 − 1234 = 123,400 − 1234 = 122,166.
- The right part keeps as many digits as there are nines: 98 × 999 is 097 | 902.
Try these
Work each one out, then tap it to see the answer.
64 × 99
63 | 36 = 6336
238 × 999
237 | 762 = 237762
7 × 99
6 | 93 = 693
506 × 999
505 | 494 = 505494
85 × 99
84 | 15 = 8415
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Source: This method is from Bharati Krishna Tirtha’s Vedic Mathematics (1965). The explanation, examples and questions here are our own.
More Vedic maths
- Squares ending in 5: 75²: 7 × 8 = 56, then 25: 5625.
- All from 9, the last from 10: 1000 − 357: 9 − 3, 9 − 5, 10 − 7: 643.
- Multiplying near 100: 97 × 96: 97 − 4 = 93, 3 × 4 = 12: 9312.
- Squaring near 100: 96²: 96 − 4 = 92, 4² = 16: 9216.
- Vertically and crosswise: 23 × 41: 2 × 4 | 2 × 1 + 3 × 4 | 3 × 1: 943.
- Last digits adding to 10: 63 × 67: 6 × 7 = 42, 3 × 7 = 21: 4221.
- Multiplying by 11: 253 × 11: 2 | 2 + 5 | 5 + 3 | 3: 2783.
- Checking with digit sums: 437 × 26 = 11362? 5 × 8 → 4, and 11362 → 4: it passes.
- Dividing by 9: 2312 ÷ 9: 2, 2 + 3, 5 + 1: 256, remainder 6 + 2 = 8.
- Squaring with the duplex: 43²: 4² | 2 × 4 × 3 | 3² = 16 | 24 | 9: 1849.
Where Vedic maths comes from, all 16 sutras, and our sources: Vedic maths, explained.