Learn two tips that will make multiplying any number by eleven easy to do in your head.
In the last article, we talked about how to multiply quickly in your head with the help of the distributive property. As luck would have it, I ran into a problem a few days ago that provides a beautiful example of exactly this technique…and it also helps to explain something that appears otherwise inexplicable. The problem I’m referring to is this: Find the answer to 111,111,111 x 111,111,111. Besides being a mouthful to say, the problem doesn’t seem like it would be very interesting. But, as you’ll see today, the answer is!
The “Mysterious” Multiplication Problem
So what’s the answer to the mysterious and fascinating mouthful of a multiplication problem:
111,111,111 x 111,111,111 = ?
It turns out the answer is: 12 quadrillion, 345 trillion, 678 billion, 987 million, 654 thousand, 321. But this number itself in quadrillions, trillions, billions, and so on isn’t really what has me so interested. No, I’m actually interested because of the pattern of the numbers. Namely, that the answer starts with the number 1 and then continues digit-by-digit counting up to the number 9, and finally back down again to the number 1. In other words, the answer is the number:
Now that’s pretty amazing, right? Especially since we got it simply by multiplying a bunch of ones together. Yes, it is amazing, but it turns out that it’s not a mystery—it’s completely understandable in terms of the distributive property.
How to Multiply Two-Digit Numbers by Eleven
But before we’re ready to explain all of this, we need to talk about a trick for easily multiplying any 2-digit number by 11. The quick and dirty tip for solving this type of problem is to add the digits of the 2-digit number and then put this sum between the two original digits—that will be the answer!
For example, in the problem 25 x 11, start by adding the two original digits—2 + 5 = 7, and then stick that number 7 between the 2 and 5 from 25 to get the answer 275—that’s a 2 from 25, a 7 from the sum of the digits 2+5, and a 5 from 25. That’s it!
Well, actually, there’s one special case to worry about: What if the sum of the 2 numbers is greater than 9? For example, in the problem 78 x 11, the sum of 7 + 8 = 15. How do we stick 15 between the 7 and 8? Is the answer just 7-15-8…7,158? No, that’s not the right answer. Instead, when the sum of the 2 digits is 10 or more, you need to carry the 1 and add it to the first number. For example, in 78 x 11 we see that 7 + 8 = 15. So we need to carry the 1 from 15 and add it to the 7 in front…which gives us a final answer of 78 x 11 = 858—that’s an 8 from 7 + 1 (the 7 is from 78, and we carried the 1 from 15), then a 5 from the non-carried piece of 15, and finally an 8 from 78.
How to Multiply any Number by Eleven
Okay, it’s great to have that trick up our sleeves for multiplying a 2-digit number by 11, but wouldn’t it be nice to be able to multiply any number by 11? Well, it turns out there’s a related trick to help us do exactly that. The quick and dirty tip is to add up the various pairs of neighboring numbers in your 3 or more digit problem, and stick the results between the 2 outside digits of the original number while removing the inside number(s). This one is a little harder to understand in words, but an example should help clarify things.
How about the problem 351 x 11? According to our tip, all we have to do is add the “3” and “5” from 351 to get 8, and the “5” and “1” from 351 to get 6, and then stick these two digits between the outside digits of 351 while removing the middle 5. So the final answer must be 3-8-6-1, or 3,861. Or for the problem 242 x 11, the answer has to be 2,662, right? Also, the same caveat about carrying the 1 when numbers add to 10 or more applies here just like it did before.
Relationship to the Distributive Property
[[AdMiddle]But why do these tricks work? Well, it’s all thanks to the distributive property and the fact that we can write 11 = 10 + 1. Just to make sure we’re all on the same page, let’s quickly recap last week’s tip on using the distributive property to perform lightning fast multiplication. There are two basic things to recall:
The distributive property says that a x ( b + c ) = a x b + a x c.
We can use this distributive property to make a multiplication problem easier. For example, in the problem we saw earlier: 25 x 11 = 25 x (10 + 1). And according to the distributive property, that just equals 25 x 10 + 25 x 1 = 250 + 25.
Now, if you look at the tens column of this last addition problem, 250 + 25, you’ll see that we’re left adding up the two digits from our original number—2 and 5—just as the trick prescribes. And it’s all nicely explained as a result of the distributive property.
Demystifying the “Mysterious” Multiplication Problem
But what about the “mystery” multiplication problem we started with:
111,111,111 x 111,111,111 = ?
Are we ready to understand that? Yes, we are…but we’re also out of time for today. So as not to leave you completely hanging, I’ll leave you with this suggestion: Try solving the following problems and thinking about the pattern you get:
11 x 11 = ?
111 x 111 = ?
1111 x 1111 = ?
If you do that, you’ll be well on your way to understanding the answer to our “mystery” problem. And then be sure to check out next week’s article to get my full explanation!
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Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!