how to multiply any number by 11 in one step

Multiply 11 by 11.

Method, imagine a zero in frontof the first number.

011

This is the number to be multiplied by eleven.

Work from left to right.

Add 0 to 1 = 1

Move to the next number.

Add 1 to 1 = 2

Now we have reached the last digit, write it down: 1

The answer is 121.

That was an extremely easy example, so let's try a longer one:

434211234 X 11

Imagine 0 in front:

0434211234

0+4=4

4+3=7

3+4=7

4+2=6

2+1=3

1+1=2

1+2=3

2+3=5

3+4=7

4

The answer is 4776323574

Now you can do this in-line. Start at the left, add the number to the right, move along until you run out of digits and write the last one down.

Multiply

31231234 x 11

031231234

343543574 = answer

These have been particularly easy because each digit is less than 5.

Here is an example for 555:

0555

5[5+5][5+5]5

I was forced to group the double digit result as [5+5=10]. Clearly this is an odd way to write a number. It's unconventional and confusing. 555 x 11 = 6105. We need to supplement the rules.

What we are really doing is this:

 555 +

5550
----
6105

Let's examine a longer example:

 555555 +
5555550
-------
6111105

I'll write this out in full so you can see what is going on. After a short time, you will be able to just write down the answer in one hit.

We will use the sawtooth method to explain this.

0   1   2   3   4   5   6   7   8   9
|  /|  /|  /|  /|  /|  /|  /|  /|  /|   This is the sawtooth. 
| / | / | / | / | / | / | / | / | / |   At each point, add digits
1   3   5   7   9  11  13  15  17   9   
|   |   |   |   | / | / | / | / |   |   Sawtooth again for double 
|   |   |   |   |/  |/  |/  |/  |   |   digits.
1   3   5   7  10   2   4   6   7   9
|   |   |   | / |   |   |   |   |   |
|   |   |   |/  |   |   |   |   |   |
1   3   5   8   0   2   4   6   7   9   This is the answer.

This sawtooth method explains the process in detail.As you get used to the method, you can look ahead at the digits to the right to see if you need to add one extra.

Another (easier) way is to start at the right and work left. This is easier because you don't need to look-ahead. If a one needs carrying, then you can note it immediately. The advantage of mastering the left-to-right method allows you to 'speak' the answer in order which is more impressive to an audience. However, have a go at the right-to-left method and compare:

Again, by example:

(This time I will make a deliberate error for you to spot.)

12378 X 11

Add the leading zero and drop down the rightmost digit.

012378
     |
     8

8+7=15 Use the 5 and make a mental note to carry the one.

012378
     |
    58
3 + 7 (plus the carry) = 11. Use the 1 and make another mental note to carry the one.


012378
     |
   158

2 + 3 = 5. Use the 5 and nothing to carry.


012378
     |
  5158

1 + 2 = 3


012378
     |
 35158

0 + 1 = 1. The purpose of the leading zero is to force you to do this
           last step.


012378
     |
135158     This is the answer. Or IS it?

Check your answer by casting out 9s!

Can you spot (and fix) the error?