Ever wondered if there’s a quick way to multiply two number in the teens, problems like: 14 x 17, or 18 x 16?

Well, there is. And if you just stick around for two minutes, you’ll learn the trick. Let’s start wih 18 x 16.

First, add up the two digits in the one’s place. Here that’s the 8 and the 6. 8 + 6 = 14.

Next take that sum (14) and add 10 to it. 14 + 10 = 24.

Then tack a zero on at the end. 24 becomes 240.

Finally multiply the two numbers in the one’s place, and add the sum to the 240.

8 x 6 = 48, and 240 + 48 = 288.

That’s your answer. This may seem tricky at first, but it gets pretty easy if you try it a few times. Trust me …

O.K., don’t trust me. But just try it one more time, with 14 x 17, and see for yourself.

4 + 7 = 11. 11 + 10 = 21.

21 becomes 210.

4 x 7 = 28, and 210 + 28 = 238.

That’s all there is to it.

Now try these:

a) 13 x 16

b) 12 x 17

c) 14 x 19

d) 12 x 19

e) 13 x 14

f) 17 x 18

g) 19 x 17

h) 15 x 19

j) 16 x 17

k) 18 x 19

Answers:

a) 13 x 16 = 208

b) 12 x 17 = 204

c) 14 x 19 = 266

d) 12 x 19 = 228

e) 13 x 14 = 182

f) 17 x 18 = 306

g) 19 x 17 = 323

h) 15 x 19 = 285

j) 16 x 17 = 272

k) 18 x 19 = 342

Wonderful trick!

I wonder if there is a proof for the validity of this method?

Thanks!

Yes, there is a proof.

To understand this proof, it helps to view the multiplication of any two 2-digit numbers in an interesting new way.

Take a problem like 18 x 16.

First, view each number as the sum of its two digits. In other words, view 18 as

[10 + 8], and view 16 as [10 + 6].

Once you do that, you can view 18 x 16 as the same as [10 + 8] x [10 + 6]

O.K., now if you recall the concept of F.O.I.L, from Algebra 1, you’ll remember that it gives you a way to multiply using the pattern: Firsts, Outers, Inners, Lasts. Sound vaguely familiar?

Using this pattern of F.O.I.L, from algebra, you can now look at the product of

[10 + 8] x [10 + 6] as a four-part process:

Firsts give you 10 x 10 = 100

Outers give you 10 x 6 = 60

Inners give you 10 x 8 = 80

Lasts give you 6 x 8 = 48

Add up those four sub-products, and you get 248, the answer to 18 x 16.

O.K., fine you say. But what does that have to do with the trick I offered?

Well, everything, if you think about it.

Think for a moment about the part of the trick that asks you to add the 6, 8, and 10 to get 24, then tack on a 0, to get 240. In terms of F.O.I.L, this is the same as the first three steps: the Firsts give you 10 x 10 = 100; the Outers give you 10 x 6 = 60; and the lasts give you 10 x 8 = 80. Add up those products and you get 240, which is the same thing you get in the trick: 24 with a 0 at the end: 240.

Then the last step, multiplying the 6 x 8, gives you the 48, and that is the last step of F.O.I.L: the step where you multiply the “Lasts,” the 6 x 8.

So essentially, the trick I have explained is just a shortened form of doing F.O.I.L. for multiplication with teen numbers.