The first method I'm going to look at is for single digits which are both 5 or above. If one of the digits is below 5, you gain nothing over just multiplying them together, or at best, very little. If both digits are 5 or above, all you need to know is how to subtract and times tables up to 5 x 5.
The Method
As an example, we'll multiply 8 and 6.
First we subtract both numbers from 10 to get 2 and 4, and write the answer in the corresponding right hand column below.
| Numbers to Multiply | 10 minus left hand column |
|---|---|
8
|
2
|
6
|
4
|
Subtract the number diagonally opposite. It doesn't matter which pair we use as we get the same answer. The pairs are shown in red and blue.
|
Multiply the numbers in this column
|
8-4 or 6-2 = 4
|
2 x 4 = 8
|
Multiply this by 10
| |
4 x 10 = 40
|
8
|
Finally, we add the two numbers.
| |
40 + 8 = 48
| |
So when we multiply 8 and 6 we get 48.
Sometimes when you multiply the two new numbers together, we get a number which is two digits, as shown in the example below where we multiply 6 by 6. This doesn't make any difference to the method.
| Numbers to Multiply | 10 minus left hand column |
|---|---|
6
|
4
|
6
|
4
|
6 - 4 or 6 - 4 = 2
|
4 x 4 = 16
|
2 x 10 = 20
|
16
|
| 20 + 16 = 36 | |
So 6 x 6 = 36.
How does it work?
With some algebra, we can see that the method above really does give the right answer.
Let the two numbers to be multiplied be a and b.
Let the two numbers to be multiplied be a and b.
| Numbers to Multiply | 10 minus left hand column |
|---|---|
a
|
10-a
|
b
|
10-b
|
a - (10 - b) = a -10 + b
= a + b -10 |
(10 - a)(10 - b)
|
Multiply by 10.
|
Multiply out the brackets.
|
10(a + b -10)=10a + 10b -100
|
(10 - a)(10 - b) = 100 - 10a - 10b + ab
|
We now add the two expression above.
| |
10a + 10b -100 + 100 - 10a - 10b + ab = (-100 + 100) + (10a - 10a) + (10b - 10b) +ab = ab
| |
As we get ab, this means the method does indeed give us the correct answer when we multiply a and b.
| |
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