47 times 99. 52 times 11. 63 times 999.
Each of those takes most people a pen and half a minute. Each takes about two seconds once you know the method, and there are only two methods to learn: one for 11 and its cousin 12, one for any string of nines. Here they are, with the carries and the awkward cases included, and the reason each one works.
Multiplying by 11: the neighbour trick
Take 52 times 11. Write the 5 and the 2 as they are, and put their sum in the middle.
52 × 11
- First digit stays: 5
- Middle is the two digits added: 5 + 2 = 7
- Last digit stays: 2
- Answer: 572
That is the whole method. Every digit of the answer is a digit of the number added to its right-hand neighbour, with nothing to the left of the first digit and nothing to the right of the last.
When the sum is ten or more
87 times 11. 8 stays, 8 + 7 is 15, 7 stays: 8, 15, 7. The 15 does not fit in one box, so keep the 5 and carry the 1 to the left: 9, 5, 7. Answer 957.
Longer numbers work the same way, from the right. 678 times 11: 6, then 6 + 7 is 13, then 7 + 8 is 15, then 8. Working from the right: 8 stays; 15 gives 5, carry 1; 13 plus the carry is 14, gives 4, carry 1; 6 plus the carry is 7. Answer 7458.
And with no carries at all it is almost embarrassing: 4321 times 11 is 4, 4 + 3, 3 + 2, 2 + 1, 1, which is 47531, written left to right without pausing.
Why it works
Eleven is ten plus one. So a number times 11 is the number shifted one place to the left, plus the number itself:
520
+ 52
────
572
Look at the middle column: 2 sits under 5, so they get added. Every digit lands beside its neighbour. The trick is that addition done one column at a time in your head, which is why a child who understands column addition understands this in a minute.
The cousin: multiplying by 12
Twelve is ten plus two, so the rule changes by one word: double each digit, then add its right-hand neighbour. Work from the right.
34 times 12: 4 doubled is 8, nothing to its right, write 8. 3 doubled is 6, plus its neighbour 4, is 10: write 0, carry 1. Then the invisible 0 to the left of 3: 0 doubled plus neighbour 3 plus the carry is 4. Answer 408.
123 times 12: 6; 4 + 3 is 7; 2 + 2 is 4; 0 + 1 is 1. Answer 1476. This is a small case of the sutra Ūrdhva-tiryagbhyam, vertically and crosswise, which handles any pair of numbers; the full method is taught here.
Multiplying by 9, 99, 999: one less than the one before
The sutra is Ekanyūnena Pūrvena, by one less than the one before. It handles any multiplier that is a string of nines, and the answer comes in two parts.
47 × 99
- Left part: the number less one. 47 − 1 = 46
- Right part: what the number needs to reach 100. 100 − 47 = 53
- Answer: 4653
The right part is the old Nikhilam rule, all from 9 and the last from 10: 4 from 9 is 5, 7 from 10 is 3. If you have read the Nikhilam post, you already know how to find it without subtracting.
It scales in both directions. 8 times 9: 7 on the left, 10 − 8 is 2 on the right, 72. 63 times 999: 62 on the left, 1000 − 63 is 937 on the right, 62937. 25 times 99: 24 and 75, 2475. The right part always has as many digits as the multiplier has nines, so pad with zeros if it comes up short: 7 times 99 is 6 and 93, 693.
Why it works, in one line
A number times 99 is the number times 100, minus the number once. Write that as (N − 1) hundreds plus (100 − N), and you have the two parts: the hundreds are the left part, the remainder is the right part. Nothing mystical, and it is worth showing a child the line, because a method with a reason is a method they will still trust in an exam hall.
When the number is longer than the nines
The two-part form needs the number to have no more digits than the multiplier has nines: two for 99, three for 999. For 123 times 99 the short form does not apply, so use the algebra directly: shift two places and subtract once. 12300 − 123 = 12177.
The same shift-and-subtract handles any number by 9: 34 times 9 is 340 − 34 = 306. Faster than long multiplication, and the same idea underneath.
Check it in five seconds
Add the digits of each number down to a single digit and multiply those; the answer's digits should reduce to the same thing. 47 reduces to 2, 99 reduces to 9 which counts as 0, and 2 times 0 is 0; the answer 4653 reduces to 4 + 6 + 5 + 3 = 18, then 9, then 0. It checks. The digit-sum check catches most slips and takes less time than reading this sentence.
Which trick, when
| The sum | Use |
|---|---|
| Anything × 11 or 12 | The neighbour trick, above |
| Anything × 9, 99, 999 | One less than the one before, above |
| Both numbers near 10, 100 or 1000 | Nikhilam |
| A number times itself | The squaring methods |
| Any other pair | Vertically and crosswise |
Knowing which to reach for is half the skill. A child who has all five, and has been shown why each one works, does the arithmetic sections of a school paper with time left over, which is the whole point.
Learning the rest of the sutras
Ms. Ishita Parikh holds the International Vedic Maths Teachers' Training certification at Grade A, a six-month programme approved under Skill India and the NSDC. Classes cover all sixteen sutras with the reasoning underneath each one, taught to children as methods they can explain, not tricks they have to trust.
See how the course is structured, or book the free first class and try these live.
Questions people ask
How do you multiply by 11 in your head?
Write the first and last digits of the number as they are, and between them write each pair of neighbouring digits added together. For 52 times 11, the 5 and the 2 stay and their sum, 7, goes between: 572. If a sum is 10 or more, keep its units digit and carry the 1 to the left, so 87 times 11 gives 8, 15, 7, which becomes 957.
Why does the multiply-by-11 trick work?
Eleven is ten plus one, so a number times 11 is the number shifted one place left plus the number itself. When you add a number to its own shifted copy, every digit lands next to its neighbour and the two get added. The trick is that addition done in your head, one column at a time.
How do you multiply any number by 99?
For a two-digit number, subtract 1 from it for the left part and subtract it from 100 for the right part. 47 times 99: 46 on the left, 53 on the right, 4653. For a longer number, shift it two places and subtract the number: 123 times 99 is 12300 minus 123, which is 12177.
What is Ekanyunena Purvena?
One of the sixteen Vedic Maths sutras. It means by one less than the one before, and it describes multiplication by a string of nines: the left part of the answer is the number reduced by one, the right part is what remains to reach the next power of ten. It is the Nikhilam idea, all from 9 and the last from 10, applied to a multiplier that is itself all nines.
Does the 99 trick work when the number has more digits than the nines?
Not in its short form. The short form needs the number to have no more digits than the multiplier has nines: two digits for 99, three for 999. Beyond that, use the shift and subtract: multiply by 100 by adding two zeros, then subtract the number once. It is still faster than long multiplication, and it is the same algebra.