Today, as I walked down to the store to pick up groceries for the upcoming week, I thought some more on the math formula that I worked on yesterday. The end result is that I have no figured out a trick to easily multiply two “teen” numbers together.

Of course, it behooves us to define what a “teen” number is then! I’m defining a “teen” number as any number between (and including) 10 and 19. This can be expressed in the following way as well (which will benefit us shortly):

A “teen” number is any two-digit number in the format 1x, where x is a whole number between 0 and 9.

If we multiply two “teen” numbers together, then it is equivalent to the format 1x x 1y where x and y are both integers between 0 and 9, and where x may or may not equal y.

Now that we have it defined, let me give you an example of the trick first, and then I will demonstrate the reasons why the trick works after that. Suppose you wanted to find out what 17 x 15 is. Here’s how the trick works:

1. Begin with 100. This is your “baseline” and will always be the baseline for teen numbers.
2. Take off the beginning “1”s on your numbers (i.e. 17 becomes 7, 15 becomes 5. This gives us 7 & 5).
3. Add those numbers together (i.e. 7 + 5 = 12).
4. These are our “tens”, so multiply the result by 10 by sticking a 0 at the end (i.e., 12 becomes 120).
5. Add that to the baseline 100 (i.e. 100 + 120 = 220).
6. Now multiply the two numbers we got in step 2 together (i. e. 7 x 5 = 35).
7. Add that result to the number we got in 5. (220 + 35 = 255).

That’s your answer. Now at first glance, the above looks difficult. However, as you practice it, you’ll see it makes mental math quite simple. For instance, to figure out what 15 x 13 is, you only need to calculate 5 + 3 and 5 x 3, which are both simple. The rest follows simply too: since 5 + 3 = 8, then you simply add a zero to the end (equivalent to multiplying by 10) to make 80, and add that to your 100 baseline (180), and then add 5 x 3, or 15 and you’ve got 195.

Thus far, I think the method is easier than doing the multiplication the way we’re all taught to do it. However, where this method really shines is when our x and y variables add up to 10. For example, 18 x 12. 8 + 2 = 10. When you have those variables that add up to 10, you do this:

1. Begin with 200 (instead of 100!)
2. Take off the beginning “1”s on your numbers (same as step 2 in the first method)
3. Multiply those numbers together (i.e. 8 x 2 = 16)
4. Add to our baseline. (200 + 16 = 216).

And you’re done. Yup, that easy. So, 17 x 13. Start with 200, add 7 x 3 = 21. 221 is your answer. This even works if you break out of the single-digit mould. For instance, 10 x 20 = 200. You can take your x as 0 and your y as 10, and find out that 200 + 0 x 10 = 200. But of course it becomes a little more difficult to mentally think of 20 as 1A (where A = 10).

However, if you did continue with that, 21 x 9 (where you think of 21 = 1B, (B = 11), and 9 = 1-A, (-A = -1) in the same format: 9 x 21 = 189. So you have 200 + 11 x – 1, or 200 – 11 = 189. So the method still works, but at this point it becomes difficult to do it mentally.

Now on to the reason why this method works. Let’s look at the equation I used yesterday:

n2 = (n – x)(n + x) + x2

The (n –x)(n + x) section of the equation is where we put in our multiplication terms. So what would happen if I wanted to see what (n + a)(n + b) would need?

n2 = (n + x)(n + x) + ?

Once again, we factor:

(n + x)(n + x) = n2 + nx + xn + x2 Or:
(n + x)(n + x) = n2 + x2 + 2nx.

So the ? would have to get rid of everything except for the n2 to satisfy the equation. The ? would therefore be –x2 - 2xn. Therefore:

n2 = (n + x)(n + x) – x2 - 2xn.

But remember last time I made it relative by using (n – a) and (n + b). What would that look like here?

n2 = (n + a)(n + b) + ?
(n + a)(n + b) = n2 + bn + an + ab.

Therefore, we need to subtract bn, an, and ab:

n2 = (n + a)(n + b) – bn – an – ab.

Now let us make these numbers “teen” numbers. In that instance, n = 10:

100 = (10 + a)(10 + b) – 10b – 10a – ab

The formula can be rearranged to show us:

(10 + a)(10 + b) = 100 + 10b + 10a + ab

And there you have the trick I showed above. You start with the baseline of 100. You take the “1” off of the number and get your x and y, which are equivalent to the a & b in the above formula. The rest follows.

The only part that would be tricky to immediately see at this point is the fact that:

10b + 10a = 10(a + b).

However, if you know how to factor, you’ll see that that is the case. And we can demonstrate it by plugging in some values too. For instance, a = 7, b = 9:

(10 x 7) + (10 x 9) = 10(7 + 9)
70 + 90 = 10 x 16
160 = 160.

This also gives us our reason for why, if the digits (a and b, or x and y) add up to 10, we start with 200 instead of 100. If:

(10 x a) + (10 x b) = 10(a + b)

Then if a + b = 10, then we are adding another 100 to the baseline.

Therefore, we can do that from the start and don’t have to worry about adding 100 + 100. We can simply say, if a + b (or x + y in the original formula) = 10, then start with 200 instead of 100.

So there you have it. The general trick to multiplying teen numbers is to start with 100, add the right-hand digits together and put a zero at the end, multiply the right hand digits together, and add all of those together. If a + b = 10, then start with 200 and add the multiple of the right-hand digits to that number.