# Multiplication Trick #4: Multiplying Teen Numbers

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

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

## 2 Responses to “Multiplication Trick #4: Multiplying Teen Numbers”

1. Ha says:

Wonderful trick!
I wonder if there is a proof for the validity of this method?
Thanks!

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2. joshturtle says:

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.

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