Wild About Math!Making Math fun and accessible

11Nov/07271

Impress your friends with mental Math tricks

See Math tricks on video at the Wild About Math! mathcasts page.

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Being able to perform arithmetic quickly and mentally can greatly boost your self-esteem, especially if you don't consider yourself to be very good at Math. And, getting comfortable with arithmetic might just motivate you to dive deeper into other things mathematical.

This article presents nine ideas that will hopefully get you to look at arithmetic as a game, one in which you can see patterns among numbers and pick then apply the right trick to quickly doing the calculation.

Don't be discouraged if the tricks seem difficult at first. Learn one trick at a time. Read the description, explanation, and examples several times for each technique you're learning. Then make up some of your own examples and practice the technique.

As you learn and practice the tricks make sure you check your results by doing multiplication the way you're used to, until the tricks start to become second nature. Checking your results is critically important: the last thing you want to do is learn the tricks incorrectly.

1. Multiplying by 9, or 99, or 999

Multiplying by 9 is really multiplying by 10-1.

So, 9x9 is just 9x(10-1) which is 9x10-9 which is 90-9 or 81.

Let's try a harder example: 46x9 = 46x10-46 = 460-46 = 414.

One more example: 68x9 = 680-68 = 612.

To multiply by 99, you multiply by 100-1.

So, 46x99 = 46x(100-1) = 4600-46 = 4554.

Multiplying by 999 is similar to multiplying by 9 and by 99.

38x999 = 38x(1000-1) = 38000-38 = 37962.

2. Multiplying by 11

To multiply a number by 11 you add pairs of numbers next to each other, except for the numbers on the edges.

Let me illustrate:

To multiply 436 by 11 go from right to left.

First write down the 6 then add 6 to its neighbor on the left, 3, to get 9.

Write down 9 to the left of 6.

Then add 4 to 3 to get 7. Write down 7.

Then, write down the leftmost digit, 4.

So, 436x11 = is 4796.

Let's do another example: 3254x11.

The answer comes from these sums and edge numbers: (3)(3+2)(2+5)(5+4)(4) = 35794.

One more example, this one involving carrying: 4657x11.

Write down the sums and edge numbers: (4)(4+6)(6+5)(5+7)(7).

Going from right to left we write down 7.

Then we notice that 5+7=12.

So we write down 2 and carry the 1.

6+5 = 11, plus the 1 we carried = 12.

So, we write down the 2 and carry the 1.

4+6 = 10, plus the 1 we carried = 11.

So, we write down the 1 and carry the 1.

To the leftmost digit, 4, we add the 1 we carried.

So, 4657x11 = 51227 .

3. Multiplying by 5, 25, or 125

Multiplying by 5 is just multiplying by 10 and then dividing by 2. Note: To multiply by 10 just add a 0 to the end of the number.

12x5 = (12x10)/2 = 120/2 = 60.

Another example: 64x5 = 640/2 = 320.

And, 4286x5 = 42860/2 = 21430.

To multiply by 25 you multiply by 100 (just add two 0's to the end of the number) then divide by 4, since 100 = 25x4. Note: to divide by 4 your can just divide by 2 twice, since 2x2 = 4.

64x25 = 6400/4 = 3200/2 = 1600.

58x25 = 5800/4 = 2900/2 = 1450.

To multiply by 125, you multipy by 1000 then divide by 8 since 8x125 = 1000. Notice that 8 = 2x2x2. So, to divide by 1000 add three 0's to the number and divide by 2 three times.

32x125 = 32000/8 = 16000/4 = 8000/2 = 4000.

48x125 = 48000/8 = 24000/4 = 12000/2 = 6000.

4. Multiplying together two numbers that differ by a small even number

This trick only works if you've memorized or can quickly calculate the squares of numbers. If you're able to memorize some squares and use the tricks described later for some kinds of numbers you'll be able to quickly multiply together many pairs of numbers that differ by 2, or 4, or 6.

Let's say you want to calculate 12x14.

When two numbers differ by two their product is always the square of the number in between them minus 1.

12x14 = (13x13)-1 = 168.

16x18 = (17x17)-1 = 288.

99x101 = (100x100)-1 = 10000-1 = 9999

If two numbers differ by 4 then their product is the square of the number in the middle (the average of the two numbers) minus 4.

11x15 = (13x13)-4 = 169-4 = 165.

13x17 = (15x15)-4 = 225-4 = 221.

If the two numbers differ by 6 then their product is the square of their average minus 9.

12x18 = (15x15)-9 = 216.

17x23 = (20x20)-9 = 391.

5. Squaring 2-digit numbers that end in 5

If a number ends in 5 then its square always ends in 25. To get the rest of the product take the left digit and multiply it by one more than itself.

35x35 ends in 25. We get the rest of the product by multiplying 3 by one more than 3. So, 3x4 = 12 and that's the rest of the product. Thus, 35x35 = 1225.

To calculate 65x65, notice that 6x7 = 42 and write down 4225 as the answer.

85x85: Calculate 8x9 = 72 and write down 7225.

6. Multiplying together 2-digit numbers where the first digits are the same and the last digits sum to 10

Let's say you want to multiply 42 by 48. You notice that the first digit is 4 in both cases. You also notice that the other digits, 2 and 8, sum to 10. You can then use this trick: multiply the first digit by one more than itself to get the first part of the answer and multiply the last digits together to get the second (right) part of the answer.

An illustration is in order:

To calculate 42x48: Multiply 4 by 4+1. So, 4x5 = 20. Write down 20.

Multiply together the last digits: 2x8 = 16. Write down 16.

The product of 42 and 48 is thus 2016.

Notice that for this particular example you could also have noticed that 42 and 48 differ by 6 and have applied technique number 4.

Another example: 64x66. 6x7 = 42. 4x6 = 24. The product is 4224.

A final example: 86x84. 8x9 = 72. 6x4 = 24. The product is 7224

7. Squaring other 2-digit numbers

Let's say you want to square 58. Square each digit and write a partial answer. 5x5 = 25. 8x8 = 64. Write down 2564 to start. Then, multiply the two digits of the number you're squaring together, 5x8=40.

Double this product: 40x2=80, then add a 0 to it, getting 800.

Add 800 to 2564 to get 3364.

This is pretty complicated so let's do more examples.

32x32. The first part of the answer comes from squaring 3 and 2.

3x3=9. 2x2 = 4. Write down 0904. Notice the extra zeros. It's important that every square in the partial product have two digits.

Multiply the digits, 2 and 3, together and double the whole thing. 2x3x2 = 12.

Add a zero to get 120. Add 120 to the partial product, 0904, and we get 1024.

56x56. The partial product comes from 5x5 and 6x6. Write down 2536.

5x6x2 = 60. Add a zero to get 600.

56x56 = 2536+600 = 3136.

One more example: 67x67. Write down 3649 as the partial product.

6x7x2 = 42x2 = 84. Add a zero to get 840.

67x67=3649+840 = 4489.

8. Multiplying by doubling and halving

There are cases when you're multiplying two numbers together and one of the numbers is even. In this case you can divide that number by two and multiply the other number by 2. You can do this over and over until you get to multiplication this is easy for you to do.

Let's say you want to multiply 14 by 16. You can do this:

14x16 = 28x8 = 56x4 = 112x2 = 224.

Another example: 12x15 = 6x30 = 6x3 with a 0 at the end so it's 180.

48x17 = 24x34 = 12x68 = 6x136 = 3x272 = 816. (Being able to calculate that 3x27 = 81 in your head is very helpful for this problem.)

9. Multiplying by a power of 2

To multiply a number by 2, 4, 8, 16, 32, or some other power of 2 just keep doubling the product as many times as necessary. If you want to multiply by 16 then double the number 4 times since 16 = 2x2x2x2.

15x16: 15x2 = 30. 30x2 = 60. 60x2 = 120. 120x2 = 240.
23x8: 23x2 = 46. 46x2 = 92. 92x2 = 184.
54x8: 54x2 = 108. 108x2 = 216. 216x2 = 432.

Practice these tricks and you'll get good at solving many different kinds of arithmetic problems in your head, or at least quickly on paper. Half the fun is identifying which trick to use. Sometimes more than one trick will apply and you'll get to choose which one is easiest for a particular problem.

Multiplication can be a great sport! Enjoy.

See Math tricks on video at the Wild About Math! mathcasts page.

Check out these related articles:

1. thank you for having this trick!!!!!

2. very good excellent tricks………………

3. i like this maths

4. Thanks for great trick……

5. I’ve got study some of the content articles on your website now, and I enjoy your style of blogging. I included it to my favorites web-site list and will be looking at back soon.

6. THERE IS VERY NICE STEP FOR SOLVING MATHS I LIKE IT

7. This is really nice….Now, i am using this trick. It is very kind for me to use in my daily math solving.

8. Hi friends!
Everyone are just talking about tricky proof. Here I am sharing some tricks like 2=3, 2+2=5 or 2+2=6. OK, It is as follows.

(1-3/2)^2=(3/2-1)^2 …………..(i)

(-1/2)^2=(1/2)^2

1/4=1/4

2=2 (Here 2=2 proofs that equation (i) is correct)

Again taking equation (i)

(1-3/2)^2 = (3/2-1)^2

(1-3/2)=(3/2-1) (canceled squares from both side)

1+1=3/2+3/2

2=3 (proved)

Now, 2+2

2+3=5 (replacing 2 from 3)

3+3=6 (replacing both 2 with 3 as we have proved 2=3)

9. Guys, all these are part of Vedic Mathematics. These were the ancient methods of calculation used by Hindu saints. Buy one Vedic Mathematics book written by Puri Sankaracharya, all are available in it. Best of luck.

10. Here’s one 3*9 (this works for all of 9 times table)

take away one from 3! =2
find out how far 3 is from 10!=7

11. maths is never wrong .we can never prove that any two different gives same value……in comment of Zaid also he hasn’t taken +- sign while taking root ………maths always right

12. Very nice article-I like point 8 particularly where you halve and double numbers to arrive at answers quickly. Mental math can also be extended to stuff like graphing hyperbolas, parabolas etc without having to resort to any explicit paper calculations; I teach my students these skills to help them become more efficient when solving graphing related problems.

13. This is really appreciated website to learn about tricks for Mathematics.

14. how to multiply 69321×11

15. thanks It’s awesome

16. @umang—->>> use a calculator = 762531

17. 69321×11= 762531. to solve this mentally, 69321 has 5 digits, add 1 so that the answer will be a 6-digit no. (1) copy the last digit. (2) add the last digit and the second last digit to get 3. (3) Then add 3 and 2 to get 5. (4) to get 2, add 3 by its neighbor digit 9. since 3+9 is 12 , a two digit no, regroup 1 and remain 2. (5) to get 6, 9+6 = 15 + 1 (regroup from 12). Since 15 +1=16 , again regroup 1 from 16. then, to get 7 , the digit at the highest place value is 6 , add 1 ( regroup from 16. technique : just add the digit from the right to the digit neighbor from left.

18. Dear Sir,

I am Mahesh, from Madras, India.

I have a website for the puzzle game based on the magic square puzzles.

The URL of the website is http://www.magicsquarepuzzles.com

The magic square puzzles were discovered by me. These puzzles are Arithmetic puzzles based on the concepts of the magic squares.

I have conducted classes about these puzzles in few schools around my place of residence.

I find the response from the student community very encouraging and enthusiastic.

The students slowly shed their math phobia when they play this puzzle game.

They also tend to do all the basic arithmetic operations in their mind only, coming out of the influence of the calculators and computers to do simple problems.

I want this good message to spread.

I have also given a link to your website from my site.

Thanks.

Yours Turuly,

T.N. Mahesh

19. i m very ipressed.thanks for such useful site.

20. This is a very useful website with a good collection of math tricks. I recently started blogging.

21. thax yaar for such typs of site…its vry useful n vry interesting…i like it.

22. thnxxxxx for such an awsum tricks………….

23. thanks for this nice tricks
this will help me lot

24. thanks a lot………….great work
now i found maths to b intresting
this will help me lot

25. nyc tricks buudy thankssssssss

26. nice thanks

27. the tricks are awesome…………….and very fascinating,thanks.

28. U hv done a greatful job. Really superb & more interesting. many of them should visit tz website. u’l add some more tricks like multiplying 3 digits, 4 digits and so on. Thnks a lot.

29. really intresting , Thank you.

30. I love these trick! They helped with my math project. Merci Mille fois!

31. It’s really intresting.

32. There’s another way of solving no. 8

14×16 = 28×8 = 56×4 = 112×2 = 224
(this is quite long and a kid would find it difficult)

MY way is fast and easy. This can be apply if you are dealing numbers from 11 to 19, multiply by any numbers from 11 to 19.

Here’s how to do it!

14 x 16 = (14 + 6, equals 20), (4×6, equals 24)

so, 20
+ 24
———–
224

Another one,

13 x 18 = (13 + 8, equals 21), (3×8, equals 24)

so, 21
+ 24
———–
234

33. nice,very nice. you did a very good job. i also impressed.

34. cool.. like it much! 🙂

35. good tips

n n
)
(—-)

36. In no. 7, instead of saying double the product of the digits and add zero at the end, the easier way to remember this step would be to say mulitply by 20.

37. If you have to square numbers ending in 5 (e.g 25*25, 35*35), then add 1 to the first digit and multiply with the second digit and then place 25 at the end
25*25 = (2+1)*2=6, add 25 at the end = 625
35*35 = (3+1)*3 = 12, add 25 at the end = 1225
45*45 = (4+1)*4 = 20, add 25 at the end = 2025
125*125 = (12+1)*12 = 156, add 25 at the end = 15625

38. this is my way of multipliying number that have a one:

13 x 12 easy drop the one, add 3 plus 2 equals 5, 3times 2 equals 6.