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.

_____________________

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.

The tricks in this article all involve multiplication.

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:

Comments (271) Trackbacks (22)
  1. thanks good site.to multiplie 142 by 11.
    take 142 start from righthand side first nunber is 2 add it to 4=6 add the 4 to 1= 5 remaining nunder 1.
    answer1562 .this works better if you start on the left.

  2. 93×97=

    93-7.
    97-3.
    multiply7x3=21 the last two numbersin answer.
    cross subtract to get 90 the first two numbers.
    ans=9021.you can do this in your head.

  3. 99×11=1089.
    from the left put down the 9.
    next add 9+9 18.
    finallyputdown the last9 9.
    ——-
    1089

  4. thank you!

  5. Basicly, you need to understand the exact rule to correct the mistake

    rule says that the numbers on the left (8 and 8) – checks
    but another condition must be met…that is the right numbers MUST add up to be 10 for the rule to hold true! which is not the case with these 2 numbers
    good luck

    engineer abbas aldelaimy

    the

  6. You can prove to your friend that 10+10=10, and 10-10=20 by performing this trick:
    1. Show all ten digits of your hands.
    2. Show all ten digits of a pair of gloves.
    3. Put on the gloves and show your hands.(10+10=10)
    4. Remove the gloves and show both hands and gloves. (10-10=20)

  7. very helpful indeed

  8. If you want to have fun doing mental math or practice you can use this:
    http://www.nixroot.com/icompute/
    A really a fun way to stay sharp! 🙂

  9. thanks tom that is cool

  10. NICE TRICK!!!

  11. I love school.I love math and i has hoping to find ways to solve mentally more than the ones i found at home. 5 days! i am only 9 years and got 99 in math and 94 in average.The tricks stink.

  12. These techniques are great. I did so poorly on last year’s mental math at ARML. I hope this’ll help me for this year. =)

  13. A trick for multiplying a 9 digit number by 142857143. For example 586362931*142857143.
    Write the 9 digit number twice: 586362931586362931 then divide by 7. you get 83766133083766133.

    This works because 7*142857143=1000000001 then multiplying by the 9 digit number on both sides causes the 9 digit number to be repeated twice on the right. Dividing both sides by 7 yields the desired result.

    This is very impressive when done on a blackboard because you write down your answer from left to right. (Especially if you do the division in your head. It’s not hard–it just takes practice.)

  14. ja, your right about not being critical to others… im asian and IM not saying any thing so lay off 🙁

    nice article 🙂

  15. suppose take a number between 10-100 then reverse the such as if you take a number 12 then make it as 21 add 11 to the obtained number 21+11=33 it is divisible by 11 for example take 45 then reverse it then it would become 54 add 11 to it then it is 66

  16. Here is an alternative way to multiply a no. by 9:We have 53X9.
    Step 1: Take only the rightmost digit of the number 53 (i.e., 3)Find 10 – 3 = 7.
    53X9 = …7
    Step 2:Subtract the number (i.e., 53 ) by one more than the rest of the digits to its left ., in this case .,5:One more than 5 is(5 + 1 ) = 6
    5 3 X 9 = (5 3 – 6 ) / 7 = 47 / 7 i.e., 477

    Another example: 234 X 9
    Step 1: Take only the rightmost digit of the number 234 (i.e., 4) Find 10 -4, this is 6.
    234X9 = …6
    Step 2: Subtract the number (i.e., 234) by one more than the rest of the digits to its left ., in this case ., 23. One more than 23 is (23 + 1 ) = 24
    234 X 9 = (234 – 24 ) / 6
    = 210 / 6
    = 2106

  17. pls tell me how to get percentage base and rate
    because that is my only weakness in math……….. but when it comes to social studies, i can memorize a whole book…. hehehe pls. answer my comment…

  18. thank u so much for these tricks….hey i really want to know how to multiply big numbers(RANDOM) like 57 * 89..so if anyone could help me…. actually i know one of the tricks…but i want to know a more easy one…… i wil post the trick in my next comment…..so i will be grateful if anyone helps me..and thank u for the above techniques…

  19. I have a new trick.
    How we find sum of a number series like.
    1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = ?

    So, First take next digit of last number that is 11.
    and then take mid digit of series that is 5.
    11 * 5 == 55

    this is the also the sum of=
    1 + 2 + 3 + 4 + 5+ 6 + 7 + 8 + 9 + 10 = 55

  20. Also Middle number series like
    21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 = ?

    Simple, The tenth value of this series is 2.
    then value is 2 55 = 255

  21. A few years ago I observed an activity involving the numbers 1 to 25 arranged in a 5 x 5 grid. The numbers were each a color ( I think there were 5 colors.) The teacher would leave the room and the kids would agree on a number from the board. When the teacher returned he would ask two or three questions and be able to tell the kids the number they picked.

    Have you heard of this? I’d like to do it with my math class.

    Steve

  22. 11x rule under 10

    eg. 9*7

    = the number before the 7 is 6
    now think in your head what plus 6 =9?
    its 3! so 9*7=63

    eg2. 9*5 = 45 (5-1=4)(9-4=5)

  23. Thanks All .This Website is very intersting
    11*11=121
    111*111=12321
    1111111*1111111=1234567654321((Fist asending order and desending order)
    This is same rule apply many times same digit but just one different.The squre is multiplyed.
    ex
    22*22=484(2*2=4)
    step1
    22*22=121*4=484
    2222*2222=1234321*4=4937284
    This rule apply multiple same digit(1 to 9)
    Try it very intersting
    999*999=12321*81=ans

  24. hi. these tricks are really worthy n amazing. i am an engineer bt i did not know it. these are really awesome.thanks 4 them

  25. Hi Sol,
    great post…

    Also Middle number series like
    21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 = ?

  26. I believe that mathematics is the most sweet between the other lessons

  27. I have the proof of 2*2=5 ??. How should I send it to you? and also the proof of 1+1=0. So if you think that you are genius so send me its proofs.

  28. this are all taking from methametic vedic book.

  29. i like your tricks

    but i need help with 12s

  30. These methods are good.
    I have derived a method by which ypu can write any multiplication table without memorizing it. only tables up to 9 are to be memorized. by this method you can write tables of any number of digits e.g. tables of 29,49,99,42,67,119 etc.

    Pravin Somayya

  31. incredible and very interesting…..good work

  32. It is sad that not everybody knows this. I’m lucky that I was in the mood to increase my math skills today, or I probably wouldn’t have stumbled upon these for months or years. Thanks for posting these. I’ll be using these for years to come.

  33. Can you please post this method on this site?

  34. Last comment by me was @Pravin.

  35. Dear Sir

    kindly send me the proof of the 1+1 = 0 & 2*2 = 5.

    I will be always greatful to you, as i want to know it eagerly.

  36. I love math.math is my life.This tricks very helpful.my mom dad and friends r impressed they call me “master of math”

  37. Thanks for cool tricks. I have a simple trick for squaring two digit number. If we have to square !ab! then the answer wil be |a*a|2*a*b|b*b|. For example if we have to square 54 then answer wil be |5*5|2*5*4|4*4| 4*4=16=6, 1 carry.
    2*5*4=40+1carry=41=1, 4 carry.
    5*5=25+4carry=29. So the square of 54 is 2916. This trick i learn frm vedik maths.

  38. verry good math tricks!
    it just one simple thing i think could be easier.
    you can take ig: 12*5= 1*5=50 + 2*5= 10.
    and then the total will be : 50+10=60
    and thats is a easy brain training to answer things quiq.

  39. I have a prooof of -1 = 1…

  40. your’s site is very useful for all competative students

  41. nice tricks….thanks…..its realy necessary for all students

  42. cananyone tell me about multiplication by 75

  43. Thanks a lot,this tricks helps each & every students to improve their Math,
    Dear sir,plz sent me the proof of
    1+1=0
    &
    2*2=5,
    I’m egar to know about these proof

  44. this site is very,very,very good to the children LIKE ME.

  45. Hey frndz,what’s up?
    I’m a Civil Engineer,
    I’ve the proof of
    1+1=0
    &
    2+2=0
    &
    2*2=0,
    if you wanted to know about this proof & lots of tricky proof like:
    1=2,
    &
    2=3,
    & Funny things of Physics & Chemistry sø contact me,
    bye,

  46. thank u so much for this…it really helps a lot especially when mentally multiplying 2 digit numbers by another 2 digit number…I know the answer within 3 sec…THANK YOU this will help me in my MATH CLUB!

  47. Thanks, this was helpful

  48. what is the result if we multiply 64*63?

  49. THIS WEBSITE IS A GOOD WEBSITE. I HAD LEARNT MANY TRICKS FROM THIS AND THEY WERE ALL SUPER. I LIKE THAT THE MULTIPLY BY 5.


Leave a comment