Mental Math: Multiplying by numbers that end in 9

The following equation mathematically explains the title:

yx = x(y+1) - x, where y (modulo 10) ≡ 9

The last part is basically saying that we will use this for numbers that end in 9. Why 9?

Here's an example that should make it apparent why it works best with digits that end in 9:

19 ⋅ 5, so we're multiplying by 19...a number that ends in 9.

Applying the above formula, yx = x(y+1)-x, we substitute as follows:

y = 19 (since 19 (modulo 10) ≡ 9 is true)
x = 5
 
19⋅5 = 5(19+1) - 5
19⋅5 = 5(20) - 5

Now here is why this works best with numbers that end in 9. 19 + 1 = 20, and multiplying by 20 in your head or any other multiple of 10 is much easier than multiplying by 19 in your head.

19⋅5 = 100 - 5
19⋅5 = 95

So after multiplying by y+1, you then need to subtract the extra element (the +1) you added in the previous calculation, which in the above example, is 5 (basically you need to add -x * 1 to the first product)

Here's another example:

29⋅7 = 7(29+1) - 7
29⋅7 = 7(30) - 7
29⋅7 = 210 - 7
29⋅7 = 203