The Fastest and most Efficient Methods of Mental Math.

Mental Calculation, Mathematics, and Puzzles
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I just finished reading Secrets of Mental Math by Arthur Benjamin and I was amazed by the simplicity of most of his methods and how easy it was to learn them, beauty lies in simplicity. I was wondering if there are more efficient and faster methods to do mental calculations than what this book provides? So in your opinion guys what are the best, faster most efficient methods for mental addition, subtraction, multiplication, division, cube root, squaring etc? I’ll appreciate if you explain why you think x method is the best and provide an example of it.

I think Benjamin methods of multiplication work great, he solves multiplications left to right and uses this methods:

Addition method:
Example-46x42= 46x42(40+2)
46x40=1840
2x46=92
1840+92=1932
46x42=1932

Factoring Method:
Example-46x42= 46x(7x6)
46x7=322
322x6=1932
46x42=1932

Multiplying left to right:
Example: 856x8= 8x8=64 and you add two zeros=6400
8x5=40 and you add a zero=400
8x6=48
6400+400+48=6848

If you have trouble remembering the numbers in the larger 3x3 or 5x5 multiplications just use the PAO mnemonic method.

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I have written extensively about other methods, here on this forum.
One example is the difference of squares.

Example-46x42= 44X44 - 4
44x44=16 X 121 = 1600 +320 +16 = 1936
1936 - 4 =1932

Just look up my old posts here on this subforum.

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You ask ‘which methods are best’.
IMHO, there is no best. There is only good in a certain situation.

There are general methods. Methods that will work on any multiplication. Arthur Benjamins way of multiplication is one.
They are good because they can be used in all situations.

And then there are methods that work in special cases.
They can speed up a calculation but only work well in certain limited cases.

Here is one. First an example:

43 X 47 =
40X40 + 40 X (3+7) +3X7 =
1600 + 400 + 21 =
2021

The speed advantage in this calculation is in recognizing that if both numbers are of the same amount of tens, then we can add the singles together and multiply that number with the tens.
In this case we can add the 3 and the 7 to get 10, and then multiply this 10 by the 40 to get 400.

If we recognize that 42 means ‘40 plus 2’ then 38 can be seen as '40 minus 2’.
The same algorithm can then be utilized in this calculation:

38 X 43 =
40X40 + 40X(-2+3) + (-2X3) =
1600 + 40 + -6 =
1634

I other words, if we can write 42 as 4|2, where the first number are the tens and the second number are the singles, then we can teach ourselves to see 38 simultaneously as 3|8 and 4|-2.

If you do this then 43 +38 becomes 4|3 + 4|-2 = 8|1.
This might not seem as a speed advantage, until you realize that if you calculate 3+8 you have to deal with the carry, where in 3-2 there is no carry.

Avoiding the carry is thus another way of speeding calculations up.

And in this way there are numerous small changes you can make to a calculation in order to get a speed increase.

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I sometimes do a different calculation and later take care of the changes.

An example is this:
23 X 56

I would first do 25X56=100X14=1400.
Then realize I need to subtract 2X56=112.

1400 - 112 = 1288
Done.

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That are some great methods Kinma, any advice on faster division methods? I don’t think Benjamin’s division methods offer that great of advantage. I am also studying the trachtenberg system, really useful rules to multiply very large numbers, what do you think of it?

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Sure. I have written about fast and easy division here:

Let me know if there is a special kind of division calculation that you want to speed up. There almost always is a speedier way.

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