Multiply by 9

The current state of the art in multiplication by 9, 99 or 999 is that you multiply the number by the nearest power of 10 and subtract the original number from that.

For example, 24 x 9 would be done as 24 x 10 – 24. That is currently, the most widely known algorithm to solve this problem.

The motivation for a newer algorithm stems from the fact that the subtraction involved in the original algorithm is kind of a messy operation. As N grows larger, the process of subtraction becomes harder and eventually it becomes difficult to perform this problem in your head. For example, suppose N = 134. Subtracting 134 from 1340 while still simple, may be hard to do mentally. The algorithm I describe in this post replaces heavy subtraction with lightweight subtraction and only involves mechanical manipulation of numbers.There are a number of many other shortcuts out there but I shall be talking about a different one here.

Starting with 2 digit numbers AB, for AB × 9,

Step 1: Subtract A + 1 from AB.

Step 2: Then append 10 − B to what you get in step 1.

Example

With 68 × 9, simply do 68 − 7 = 61, then append 10 − 8 = 2 to get 612. For 94 × 9, you merge 94 − 10 = 84 with 10 − 4 = 6 to obtain 846.

The underlying algebra is that for a given number 10A + B, with 0 < B ≤ 9, we are computing 10[(10A + B) − (A + 1)] + (10 − B) = 90A + 9B = 9(10A + B).

Consequently, this can be applied to larger numbers. For instance, to do 123 × 9, 123 − 13 = 110 is merged with 10 − 3 = 7 to get 1107.

A similar approach can be taken when multiplying by 99. For AB x 99, subtract A+1 from 10 times AB, and then append 10 – B.

Example

With 87 x 99, simply do 870 – 9 = 861, and then append 10 – 7 = 3 to get 8613.
For 37 x 99, you merge 370 – 4 = 366 with 10 – 7 = 3 to obtain 3663.

The underlying algebra is that for a given number 10A + B, with 0 < B <9, we are computing 10[10(10A + B) – (A + 1)] + (10 – B) = 990A + 99B = 99(10A + B).

And finally, for multiplying with 999, for AB x 999, subtract A+1 from 100 times AB, and then append 10 – B.

Example

With 46 x 999, simply do 4600 – 5 = 4595, and then append 10 – 6 = 4 to get 45954.  For 77 x 999, you merge 7700 – 8 = 7692 with 10 – 7 = 3 to obtain 76923.

The underlying algebra is that for a given number 10A + B, with 0 < B <9, we are computing 10[100(10A + B) – (A + 1)] + (10 – B) = 9990A + 999B = 999(10A +B).