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:

9a = bc.

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:

bc= 10b + c
 = 10(a - 1) + (9 - (a - 1))
 = 10a - 10 + 10 - a
 = 9a,

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

bc= 100b + c
 = 100(a - 1) + (99 - (a - 1))
 = 100a - 100 + 100 - a
 = 99a.

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

543×999= 1000×542 + (999 - 542)
 = 999×542 + 999
 = 999×543

just by using the distributive law twice.


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