How to Use Quick Mental Arithmetic to Multiply by 11
In this article we’re going to look at a quick, easy method to multiply by 11 without a calculator. You might need a pen and paper for the bigger numbers, but the method is extremely simple and will make you look like a human calculator
How to Multiply a Two Digit Number by 11
We’re going to begin by multiplying a two digit number by 11 before we extend our method on to numbers with three or more digits.
Let’s try 36 × 11. There are only two steps to this:
- Add together the two digits of our number: 3 + 6 = 9.
- Put this digit in between the two digits (the 3 and the 6) of the original number so:
36 × 11 = 396
It’s as simple as that. Let’s try again with another example:
71 × 11
- Add the two digits together: 7 + 1 = 8
- Put this in between the original two digits: 781
71 x 11 = 781.
But What About If The Sum of the Digits is Greater Than 9?
If the sum of the two digits is greater than 9, there is a very simple fix. We put the units of our sum in the middle as before, and carry the ten over to the left-hand number.
For example, 48 × 11:
- Add the two digits together: 4 + 8 = 12
- Put the 2 in the middle and carry the 1 over to add to the 4
48 × 11 = 528
How to Multiply Numbers With Three or More Digits By 11
If you want to multiply a number with three or more digits by 11, the method is very similar, but maybe slightly more of a faff. This can still be done using pure mental maths, but there is more to remember and a pencil and paper may be handy.
Let’s start with an example, 4271 × 11:
- Add together consecutive pairs of digits: 4 + 2 = 6, 2 + 7 = 9, 7 + 1 = 8
- Place all of these numbers in order between the first and last digits of the original number: 46981
4271 × 11 = 46 981
If you end up with any pairs which add together to make more than 9, then just as before, carry the 1 over to the next column to the left.
For example, 25 724 × 11:
- Add together the consecutive pairs: 2 + 5 = 7, 5 + 7 = 12, 7 + 2 = 9, 2 + 4 = 6
- Carry the 1 over, so we have: 7 + 1 = 8, 2, 9, 6 as our middle digits.
- Put these digits in between the 2 and 4 from the ends of the original number: 282 964.
25 724 × 11 = 282 964
How Does This Work?
To understand how this method works we need to remember that 11 = 10 + 1. Therefore when we multiply by 11, we are multiplying by 10 and then adding one more lot of the number. When we multiply by 10, every digit shifts one column to the left, so when we add the original number to this we are then adding each digit of the original number to the digit that would have been one to its right.
This is easiest to see in an example: 43 526 × 11
- 43 526 × 10 = 435 260 (each digits shifts one to the left and we put an extra 0 in as a placeholder).
- We then add one more lot of 43 526.
- You can see that starting from the right-hand side of 43 526, the 6 adds to the 0 (hence why we always end with the same end digit that we started with), the 2 adds to the 6 in the tens column, the 5 adds to the 2 in the hundreds column and so on until we get to the left-hand side and don’t have anything to add to the 4 of 435 260, hence the left-hand number remains the same as we started with.
If any pairings sum to more than 9, we carry the 1 just as we would in normal column addition (as we travel left, each column is worth ten times the column before it).