### How to Multiply Larger Numbers in Your Head

I was in a meeting last week when the presenters had a 32 by 9 matrix of items on the board. He was trying to calculate the number of total cells in this matrix. I blurted out 288. He looked at the board again, and after a few seconds, agreed that there was 288 items in the matrix. Later on, he was calculating the totoal of some small scale costs figures on the board, \$9.25/hour for 12 hours. Again, I blurted out \$111.00. He looked back at me with an expression that I have never seen before.

After the meeting, he asked me how I was able to multiply so quickly. I told him that I don’t multiply like other people. In back in school, I learned that they teach elementary multiplication inefficiently. As some of you may remember, here’s elementary schools teach you how to multiply:

Problem: 32 X 9

Step 1: Rearrange the numbers into a vertical format.

```
32
X     9
--------```

Step 2: Multiply 2 X 9. The answer, 18, is more than 9, so the second digit is placed under the line, and carry the other digit. So, the 8 is placed in the answer and the 1 is carried.

```     1
32
X     9
--------
8```

Step 3: Multiply 3 X 9, then add the carry. Again, the answer, 28 is more than 9. Like before, The 8 is placed in the next spot in the answer and the 2 is carried.

```    21
32
X     9
--------
88```

Step 4: Now, there is no more digits in the first number, so the carry (2) drops down in the next place in the answer

```    21
|32
X   | 9
--------
288```

Now, you have your answer, 288. But, I definitely did not do this in my head, because it is a lot of work. What did I do? I took advantage of a multiplication property called distributivity. What I did was break down the problem into pieces and distribute the pieces over simple addition.

So, a multiplication of larger numbers can break down into one or more simpler representations. For example, 32 X 9 can be broken down into simple pieces:

```32 X 9 = (30 + 2) X 9
= (30 X 9) + (2 X 9)
= 270 + 18
= 288```

Now, for me, multiplying 30 X 9 is trivial, but for some, further (or even a different) breakdown is needed. So, now:

```32 X 9 = (30 + 2) X 9
= (30 X 9) + (2 X 9)
= (3 X 10 X 9) + (2 X 9)
= (3 X 90) + (2 X 9)
= 288```

Now, addition is not the only way. If it can work for you, subtraction is a good choice, too. Subtraction is just another form of addition. So, I can round the 9 up to 10, then compensate using subtraction:

```32 X 9 = 32 X (10 - 1)
= (32 X 10) - (32 X 1)
= 320 - 32
= 288```

In general, I like to break down problems into multples of 10 (10, 20, 30…) because it’s real trivial to multiply by 10. Add a zero at the end of your operand!

For more complex problems, the method is the same, but more addition is needed. For the money problem:

```\$9.25 X 12 = (\$9.00 + \$0.25) X (10 + 2)
= (\$9.00 X 10) + (\$0.25 X 10) + (\$9.00 X 2) + (\$0.25 X 2)
= \$90.00 + \$2.50 + \$18.00 + \$0.50
= \$111.00```

Or the subtraction method (somthing simpler):

```\$9.25 X 12 = (\$10.00 X 12 ) - (\$0.75 X 12)
= \$120.00 - \$9.00
= \$111.00```

So, with this powerful method of multiplication, you can impress your friends and peers, too!

#### 26 Responses to “How to Multiply Larger Numbers in Your Head”

1. 1 alex July 4, 2008 at 1:47 pm

Well this is pretty much what I do naturally as I suck at maths. Never learnt the name for what I was doing tho. Only problem I have is I forget what my previous calculation was before adding something else, so have to start again.

• 2 Shah Atherr Alam (@AtherrAlam) April 19, 2012 at 9:32 am

you can use the placement method, place the number in a special corner in ur mind. Its fun. More like you throwing something in the corner of your room!

2. 3 alex July 4, 2008 at 1:51 pm

Well this is pretty much what I do naturally as I suck at maths. Never learnt the name for what I was doing tho. Only problem I have is I forget what my previous calculation was before adding something else, so have to start again.

Try
23.12 x
14.23

3. 4 Ralph Allan Rice July 5, 2008 at 4:06 am

@Alex,

Ahh, two decimal numbers. The technique still works, but I would make it simpler.

To solve 23.12 x 14.23 in my head, I would multiply each operand by a multiple of 10, in this case, 100. This way, the problem becomes multiplication of two integers. However, when you are done, you need to put your decimal places back. In this example, I would need to divide by 10000 (100 x 100).

4. 5 John L February 21, 2009 at 10:54 pm

My father used to talk about how my grandmother would multiply four digit numbers in her head as the kids would work them out on paper. He said her responses where almost instantaneous. She had an 8th grade education. Is there a trick to multiplying four digit numbers or was she some kind of savant?

5. 6 Ralph Allan Rice February 22, 2009 at 1:18 am

@John L

Your grandmother probably used one of many algorithms used for fast multiplication. For example, the algorithm that I described in the post is called Booth’s Algorithm. I did not know at the time I wrote the post, I did not know it had a name or even if it existed. I just used it naturally because it made sense. Your grandmother probably did the same.

6. 7 ashlie September 19, 2009 at 3:28 am

that’s cool!!!
🙂
this is the coolest meathod i have ever seen!!!!!!!

7. 8 jordan October 17, 2009 at 2:42 pm

if you are having trouble remembering the numbers from the previous calculations then i found a trick for you!!
i am very good @ seeing in 3d in my head so i make block numbers in my head of the numbers so that they become an image which is easier to remember
if you can’t do that well try this try to write the numbers of something in the room not literally but in you mind take for instance 7*182 = 1374 so 7*100=700 and pretend to write than on the ceiling then 7*80= 560 so pretend to write that on the floor
then 2*7=14 so add 14 to whats on the floor then add that sum to whats on the ceiling!!!
it does take a good pit of practice but when you can do it your friends will be in awe when you try to explain this to them.

• 9 Paul Pinson November 9, 2010 at 2:06 am

@jordan

7*182 is 1274

8. 10 Edimouddy January 2, 2010 at 4:05 am

9. 11 munch May 27, 2010 at 7:20 pm

I do this but I can’t do it superfast.

10. 12 Ray July 16, 2010 at 8:10 pm

I’m going to keep trying this until I can master up to 4digit numbers. oh wont this be sweet to use at school?

• 13 Ralph Allan Rice July 16, 2010 at 8:56 pm

@Ray

I don’t see a reason why you shouldn’t use this in school. However, this technique, at least in the USA, is not taught as the primary method of solving multiplication problems. In fact, while I was in school, I had points deducted on tests because I did not show the work “as it was taught”. Silly really, but I understood.

• 14 Zaceij B February 15, 2012 at 11:35 am

I, too, always had points deducted for not “showing my work.” I knew they thought I was cheating somehow because I wouldn’t write anything save the answer. Because I thought the notion was silly, I would take the lost points on the chin everytime. I refused to use scratch paper, in fact I counted it as a personal failure if I ever had to resort to it, all the way up until 5th grade pre-calc. School was for the most part boring otherwise, especially in those years. I didn’t find it interesting at all to work out a complex problem on paper, especially when I could have more fun and finish my exam FASTER juggling all the numbers in my head. It would agitate me sometimes (“what’s the difference how I get there if I arrived at the correct result everytime?”) After a while most teachers I’d had would give up. It usually took a few weeks but sooner or later they would realize they were fighting a losing battle. I can’t really say that I understood then, or that I do now. Every mind has a different method of absorption, a different mechanism that coordinates effort and attentiveness with problem-solving and execution newly learned principles. As far as I am concerned, if you don’t accept that, then you have no business being a teacher. You will spend your career butting your head into brick walls, all the while letting down the young minds you are responsible for shaping.

Interesting to read this one’s known as Booth’s Algorithm. Like you’ve written, our minds resort to these methods out of reflex; it just makes sense. It’s always a bit amusing that something so seemingly instinctual in so many of us even has a mathematician’s name attached to it. Haven’t they been doing this since the beginning? 🙂

11. 15 Dean Fleury February 29, 2012 at 10:11 am

I’ve been searching the internet for ways to multiply numbers in your head quickly, and I may add that this site is the closest way that I have seen. I am not as punctual with my responses as I may have been in the past, but I think the way that I do math “in my head” is more reliant. If you know, or good at remembering numbers, this is the easiest way to produce answers more efficiently.

Note: You need to be able to “store” numbers in your mind to administer later.

Firstly, let’s take a simple multiplication: 75 x 33 = ?

Now the 1st question you ask yourself is, “where do I want to start?”
Well this is where your brilliance should shine through.
This is what I suggest that you do:

You look at the numbers involved ( 75,33 ), then you find a “lazy way” to break down ONE ( I repeat ONE ) of the numbers; the easiest. I’ll go with, hmmm 75?
Now, let’s look at the number 75 ( don’t forget about 33 )
Well, how do you want to break down 75 ? –> 25×3? 15×5 ? which one?
Let’s go with 25 x 3…k.
Let’s not forget about your base numbers 75, and especially 33, since your are manipulating 75 and 33 is left on the shelf.
Now, think about this, are you comfortable multiplying 33 x 25, or are you more comfortable with 33 x 3? I’d go with 33 x 3 eh., which is 99. Yaaaa man.
So now we’ve actually taken care of 33, right? This has given us 99…woot
We have 99, what’s up? This is where you need to remember your initial question: 75 x 33.
Since we eliminated 33, we can hopefully remember what we did with 75?
We broke down 75, 3 times, 3 x 33 =99, now the 25.
This leaves us with 99 x 25 right ?

How bout break down 25 even further ( if you can remember 99 ), and go with
99 x 5 = 495, then 495 x 5 = ?…nahhh, seems a bit much eh.

or, you can change 99 to 100, then multiply 100 x 25 = 2500
then minus one 25, which equals 99 25’s, from 2500 which equals 2475.

75 x 33 = 2475

I may also add that I would have changed 33 to –>11 x 3, then multiplied
11 x 75 = 825 , then 825 x 3 = 2475 ( but that’s me ) . All in my head, most of the time!

I will be writing a book to help people overcome numbers, and what revering to certain numbers can make you more efficient to produce products more readily.

• 16 Dean Fleury February 29, 2012 at 10:25 am

I will be writing a book to help people overcome numbers, and what revering

Don’t you mean “reverting” not “revering?”

Ya I do, boo hoo

• 17 Ralph Allan Rice May 16, 2012 at 2:55 pm

Sorry for not replying to this comment earlier. If I was presented with 75 x 33, I would choose to break up the 33 instead of 75, making the equation (75 x 3) + (75 x 30).

Why? Here is my reasoning:
* Multiplying by a number ending in a 5 is easy.
* Multiplying by a number ending in a 0 is easier.
* Multiplying by a single digit is easy.
* I would to multiply 75 x 3 once, but actually use it twice after applying a simple breakdown.

Back to the problem at hand: 75 x 3 = 225, 75 x 30 = 75 x 3 x 10 = 2250, and 225 + 2250 = 2475.

My point: Strategy is very important, but I would happily admit everyone’s thinking is different. What works for me may not work for anyone else. I like to multiply by factors of 5 and 10 efficiently, but someone else may be able to multiply by factor of 3 and 7 better.

Good luck with the book!

• 18 Cherie May 23, 2012 at 7:46 am

An easier way to break this down is 75 x 3 add a zero ‘0’ = 2250 + 75×3 (225) = 2475. Basically what you did – but for some it becomes complicated when you say, 75 x 3 x ’10’ – because we were taught weird in school, we try to ‘visualize’ the actual multiplication problem as though it were written down – that’s when we start losing track of what goes where and get confused. This is a great method that I’ve been learning for dealing Pot Limit Poker games. I’ve got a lot of practice to do still; and I tend to ‘freeze’ when I’m at the actual table dealing. I’m hoping to get better and gain more confidence for these games.

12. 19 Jack March 10, 2012 at 1:13 pm

This is just the distributive property and understanding that a number has different units (ones, tens, hundreds etc.). It’s on the grade 6 curriculum at my kid’s school and in my opinion, a must.. I ask her random questions all the time like 6 x 135 and 7 x 47.
Calculations in your head become more difficult (that is, you need to remember more intermediate numbers AND you must be able to add well) for 3 digit by 3 digit or 4 digit by 3 digit questions.

13. 20 Mathematical Man May 15, 2012 at 5:45 pm

This technique and others can be found in “Secrets to Mental Math”

14. 21 Mathematical Man May 15, 2012 at 5:47 pm

Basically, the gist of Secrets to Mental Math is to work left to right (the way we read anyway) rather than right to left, which is how we were all taught to solve on paper.

• 22 Ralph Allan Rice May 15, 2012 at 5:51 pm

@Mathematical Man Sounds very interesting. I have to check this book out. Thanks!

15. 23 lateef August 5, 2012 at 4:11 am

Well,these are good tricks but yet the memory power is important,so first increase your memory then intimate these tricks.

16. 24 Darien October 5, 2012 at 6:54 pm

lol, This is probably irrelevant but it seems like some people cant even do something like 56×8 in there head because there not very good at basic multiplication(me) so what I learned to do was something like this:

5×8=40 6×8=48
40 48
\ /
448
Not sure how popular this is, but if you learn it, you can figure out how to do it on very large numbers.

17. 25 vivek December 19, 2012 at 6:28 am

the all above techniques are time consuming,

18. 26 Jj April 25, 2013 at 5:33 am

You’re great! Knowledge is meant to be shared. Ignorance is the cause of most if not all human evil. Keep sharing brother. Kiera on Lexical Illusion Jj on How to Multiply Larger Numbers… hannesh on MySQL Query Browser in Ubuntu… Chris38B on MySQL Query Browser in Ubuntu… Ralph Allan Rice on MySQL Query Browser in Ubuntu…