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.

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.

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Comments (271) Trackbacks (22)
  1. Hey there!
    Just wanted to stop by and say thanks for a great post and read – math doesn’t have to be boring! :)

    Have a nice day,
    Alex

  2. Alex,

    You’re quite welcome. Glad you liked it.

    Sol

  3. Hey Sol i don’t expect to be using them to impress anybody, but some of those techniques will come in very handy. Thanks a lot. I’ll give it a Stumble so others can learn them too.

  4. Karen,

    Thanks for the kind words and the stumbling.

    Sol

  5. WOW!!!

    I wish I knew this while I was in school!!!

    That is neat…

  6. Very cool – I plan to use this as often as possible. By the way there is a minor typo…
    “So, 46×99 = 4600x(100-1) = 4600-46 = 4554.” should read “So, 46×99 = 46x(100-1) = 4600-46 = 4554.”

    that’s ok though – this page is so cool i think i’ll let this one slide.

  7. Alan,

    I’m glad you like the page. I’m not seeing the error, though. Your correction looks to me the same as what I wrote. Please elaborate.

    Thanks.

  8. mental math tricks helps to perform arithmetic calculations quickly
    Mental math

  9. Multiplying by 11 is easier when adjusting the rule for multiplying by 9. Just think of 11 as (10+1)

    So 436 × 11 = 4360 + 436 = 4796…that’s the simpler way of explaining why the digits add up to each other like you wrote:

    4360
    + 436
    —–
    4796

  10. Excellent Stumble. Thumb Up.

  11. @DJ, BPM — Thanks for your kind comments.

    @IB – yes, your way of showing why this “adding the pair” approach for multiplying by 11 is right on.

  12. You are saying 46×99 = 4600x(100-1) when it should be 46 instead of 4600.

  13. Fred, Alan:

    Thanks for the correction. Now I see it!
    Article is now fixed.

  14. These are just retarded, any idiot could think them up…

  15. omg the trick for multiplying squares is awesome.

    -Rob

  16. Rob,

    Glad you liked it.

  17. Even primary school students know these simple tricks in China…..

  18. Shao Han,

    Can you recommend any books in English where I could learn about what Math Chinese students learn?

  19. Hi, I has got a problem with multipling:
    If i multiply some numbers in that metod some things go wrong:
    E.G. (agree):
    87 * 81
    88*80 + 7 = 7047

    38*20
    38*20 + 0 = 760

    E.G.(doesnt agree):
    56 * 17
    63*10 + 42 =’ 672
    Real: 952

    75 * 88
    93*70 + 40 =’ 6550
    Real:6000

    85 * 26
    91*20+30 =’ 1850
    Real: 2210

    Please tell me what mistake was doing!

  20. Hi BlueS,

    I assume you’re trying to use technique #4 in the article. That technique only works when the two numbers differ by a small even amount and when you can easily calculate the square of the number in the middle of the two numbers (i.e. the average).

    In your example of 56×17 I see what you’re trying to do but it’s different than this trick.

    Let’s look at your example:

    Let a=56
    Let b=17

    You want to calculate a*b, right?

    I see that you added 7 to 56 and subtracted 7 from 17 so that you could multiply by 10. That’s a good idea.

    So, you were computing (a+7)x(b-7).
    (a+7)x(b-7) =
    (axb)-(7xa)+(7xb)-49 =
    (axb)-7x(a-b)-49

    So, (axb) = (a+7)x(b-7) + 7x(a-b)+49

    Or, 56×17 = 63×10 + 7x(39)+49 = 630 + 273 + 49
    = 952

    This approach is not easy for these two numbers.

    What you could do with what you’ve noticed is to say that 56×17 = 56x(10+7) = 56×10 + 56×7
    = 560+392 = 952.

    Does this help?

  21. everyone should already know this in my opinion. it’s basic basic math.

  22. I agree that everyone should know things like how to multiply by 9 or 11. However, the method used to achieve the answer may be quite different. I was taught multiplication and agree that it is basic math, however I was never taught “tricks” such as this; basically easy ways to remember how to multiply certain numbers. I am horrible with math so ordinarily I cannot do multiplication in my head. However, with these tips, I may get better at it.

  23. @Alex: Knowing these tricks is largely about having a relationship with numbers. I’m glad you have it but not everyone does.

    @Amanda: Do report back on how these techniques help you if they do. The Vedic Math approach allows people to do multiplication without knowing more than up to 5×5 in their multiplication tables. I’ll post some Vedic Math techniques in the future.

  24. good tricks mate. you make it easier. nice one.

    dont listen to them < >.

  25. Your annotation for the multiplying by 9’s is wrong. You have:

    9×9 is just 9x(10-1) which is 9×10-9 which is 90-9 or 81

    If you follow the acronym PEMDAS you would do what is in the parenthesis first and then multiply which would give you 9×9. You should have stated you need to use the distributive method. Which would mean it would read (9×10)-(9×1)= 90-9= 81.

  26. This is a great system developed a long time ago by Jakow Trachtenberg whilst in a Nazi camp. More info here; http://en.wikipedia.org/wiki/Trachtenberg_system

  27. there is an easier way to multiply by 9.

    this is the way I learned when i was in school.

    this works all the way up to 9 x 9, but if you’re in elementary school, it can come in handy.

    take the number you are multiplying 9 by, and subtract one. then figure out what number plus that number equals nine.

    put the first and second answer beside each other and you get tour answer.

    it’s simpler than it sounds…..
    example:
    9 x 7 = ?
    7 – 1 = 6
    6 + 3 = 9
    the answer is 63!

    9 x 3 = ?
    3 – 1 = 2
    2 + 7 = 9
    the answer is 27!

    this was the easiest way for me.

  28. Well let me put the technique #4. Generically it uses the fact that (a+b)*(a-b) = a^2 – b^2. Here b is half of the difference between the numbers, hence a is the average. In the example where difference is 6, the minus 9 comes coz (6/2)^2 = 9. Similary it can be done for large differences also, but it depends how comfortable is one is with squaring numbers.

  29. I love this!
    There is one problem with technique #6 though:
    Take 81*89 where the first digits are equal, and where the last digits’ sum is 10.
    By applying your method the answer should be: (8*9)(9*1) which gives 729, when the correct answer is 7209. This happens in all cases where the product of the last digits is less than 10 (ie with 9 and 1), so a quick fix would be to always make sure you have two digits in the product, and if not then add a zero in front.

  30. Oh!
    I just dicovered that the problem I mentioned above also applies to technique #7!
    If the square of the second number is less than 10, you also have to add a zero before it in order to get the right answer:
    Wrong:
    92^2
    => 180 =>360
    814
    360
    1174
    Right:
    92^2
    => 180 => 360
    8104
    360
    8464

  31. This is great for “impressing” the somewhat math smart girls. I’m sure that the math wizzes already know this.

    Bookmarked

    Adam

    ===
    http://www.BecomingAPUA.com – V is the #1 Pick Up Artist
    ===

  32. I invited #8 a long time ago… at least, I thought I did. LOL

  33. very nice math trick.. wish i had known when i was in school.. if you ask me, i would suggest to put them in elementary school’s curriculum :)

  34. Great work, very useful.

  35. A fantastic work! I didn’t just like it, I loved it. I have so much respect and appreciation for people like you who spend their time doing something productive on the net.

    Thanks man!

  36. have never been good with numbers, but these things do help quite a bit. how come we were never taught this at school??? :(

  37. Anyone who has taken CAT exam in India know all these techniques and more.

  38. :%s/math/maths/g ;-)

  39. Great post! Good tricks for life.

  40. heres a real trick

    (a+b)^2 = 2a^2 + 2ab + b^2, so

    86^2 = (80+6)^2 = 80^2 + 2*80*6 + 6^2

    so 80^2 is easy 8^2+10^=6400
    2*80*6=2*10*6*8=2*480=960
    6^2 = 36
    add

    =7396

    works for any number 0-99 pretty easily

  41. @Varun: True.. very true!

  42. @ EPIc

    my little brother when he was 7 taught me (age 21) the easiest way to multiply by 9 (only works up to 9×9):

    Hold your ten fingers out in front of you. Now let’s say we multiply 9×3.

    Starting from your left pinkie, count three fingers (end up at your left middle finger). Fold it down. Now read your fingers. 2 (fold) 7. 27.

    Try 9×6. Count from left pinkie, end up at right thumb. Fold down. Read… five fingers, fold, four fingers. 54.

    EASY!

    And to all of you from China, India, etc, it’s great that your primary schools taught math tricks. But many (most?) in America do not, and your comments are not constructive, almost hurtful. I go to MIT yet I still can’t do basic arithmitic in my head. Don’t belittle people because they want to learn- it’s never too late to learn!

  43. @ naomi

    You’re not quite right about that finger trick- I use it too, and it can be easily adapted to 9×10-9×20 (it works higher than this too but takes some playing around, and not all numbers work out perfectly- I’ll let you try that!)

    ie 9×13

    Do the same thing for 9×3 as naomi says. Except now your leftmost pinkie is 100. Read left to right- 1 finger (100), 1 finger (10), fold, seven fingers. Answer = 117.

    9×16 therefore would be left pinkie (100), four, fold, four. 144.

  44. Boy this was cool one. Simply splendid.After reading this my maths seem to have improved

  45. Hi My wife and I would like to thank you all for this web site. Hours of pleasure and all

  46. BlueS, Sol –

    For 56×17, the 2x rule seems easier here. 2x2x2x7 = 56, so do 7×17, then x2 three times.

    7×17 = 119 (7×10 + 7×7)
    x2 = 238
    x2 = 476
    x2 = 952

    Great article, Sol. Lots of fun.

  47. Regarding #2. A similar trick works for multiplications with 111, 1111, … you just need to make the “pipeline” longer.

    Example:
    24253 * 111 = (2) (2+4) (2+4+2) (4+2+5) (2+5+3) (5+3) (3) = (2) (6) (8) (11) (10) (8) (3) = 2692083

    It works for 101, 1001 too, where you need to “skip” a position or two when adding. Example:
    24253 * 101 = (2) (4) (2+2) (4+5) (2+3) (5) (3) = 2449553


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