## Mental Arithmetic: Multiplying a two-digit number by 11

This problem seems like it has limited utility, but that’s fine. We’ll just learn some mental math tricks one at a time and then when they happen to occur, it feels all the more boss that you have a secret weapon for solving them. Over time, the number of tricks and tools we have will grow and the special cases will give way to more general techniques.

What is 36 × 11? One way to do it is the most general and long way, where we go (36 × 10) + (36 × 1) = 360 + 36 = 396. We want a faster way, and to do that we start with the general algebraic problem:

$\displaystyle (10m + n) \times 11$

where $m$ is the tens digit and $m$ is the ones digit. Expanding it out gives

$\displaystyle 100m + 10(m + n) + n$

and the trick begins to emerge. The heuristic is to “expand” out the two-digit number and in the middle put the sum of the two digits. So in the 36 example, we expand it out to 3_6 and in the middle put the sum 3 + 6 which gives 396. If the middle number exceeds ten, we carry a one over to the hundreds place, like in 86 × 11 = 8 (8 + 6) 6 = 8 (14) 6 = 946.

In practice, you want to do the middle summation first to know whether there’s a carry over, since if there is you can immediately starting reciting the answer and work the rest through in your head as you talk (if you’re verbally giving the answer). This has the effect of making you seem even faster than you are, since if somebody happens to ask you “What’s 59 × 11?” you can almost immediately respond “Six hundred… forty-nine,” working out the latter part of the problem after giving the first digit. This seems a lot quicker and a lot more impressive than if you spend a second or two working the problem out to completion first. Might as well show off a bit…

You can try following the procedure above for a three digit number $\displaystyle 100x + 10y + z$ to see how the heuristic can be extended. Perhaps you can begin to see a pattern with the following examples:

1. 345 × 11 = 3795
2. 562 × 11 = 6182
3. 293 × 11 = 3223