Why does this multiplication trick

[BMAD]

This trick is an alternative method for multiplying two single digit numbers from 9 to 5. It assumes the user can multiply single digit numbers up to 4.

Here is how it works:

List 9 to 5 in two lines.

Underline the numbers one wants to multiply

Everything in the column under the line is worth ten

Every digit in the column above the lines are counted.

Multiply the counted digits.

Add the tens to the products of the digits

Voila

Why does this work?

For example:

9 x 7

9 9

_ 8

8 7

7 _

6 6

5 5

1d 3d

_

10

10 _

10 10

10 10

60 + 1 x 3 = 63

------------------------

6x5

9 9

8 8

7 7

6 6

_ 5

5 _

4d 5d

_

10

10 + 4 x 5 = 30

[k-man]

Let's say we're multiplying x*y where x and y are between 5 and 9. Then the number of digits above the line can be expressed as 10-x and 10-y respectively, while the number of digits below the line are x-5 and y-5.

 

The trick employs this formula for calculation (x-5+y-5)*10+(10-x)*(10-y).

Opening the parenthesis we get 10x+10y-100+100-10x-10y+xy. After cancelling identical terms all that's remaining is xy.