Multiplication by 9, 99, 999, etc.

One way to multiply a number by 9 is to multiply by 10 and then subtract the number from the product. There is another way to multiply fast by 9 and as the first one it has an analogue for multiplication by 99, 999 and all such numbers. Let's start with the multiplication by 9.

To multiply a one digit number a by 9, first subtract 1 and form b = a - 1. Next, subtract b from 9: c = 9 - b. Then just write b and c next to each other:

For example, find 6×9 (so that a = 6.) First subtract: 5 = 6 - 1. Subract the second time: 4 = 9 - 5. Lastly, form the product 6×9 = 54.

Next, find 37×99. First, subtract 1: 36 = 37 - 1. Then subtract 63 = 99 - 36. Lastly, form the product: 37×99 = 3663.

Why does this work? For the multiplication by 9, bc = 10b + c:

as required. Similarly, for a 2-digit a:

Do try the same derivation for a three digit number. As an example,

just by using the distributive law twice.