# What calculation shortcuts exist to help or speed-up mental (or paper) calculations?

Anything to speed up or simplify calculations.

A simple example might be to get a multiple of $19$, for instance, $38 \cdot 19 = 38 \cdot 20 - 38$.

(This is hard to tag with so few tags in play!)

mental-calculations tips tricks shortcut cheats time-saver

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I have started a meta thread, meta.math.stackexchange.com/questions/358/… – Akhil Mathew Jul 29 '10 at 23:07
Please make this community wiki – Casebash Jul 29 '10 at 23:11
the New Math worked pretty well for me. – Tom Stephens Jul 30 '10 at 0:48

To square a number ending in 5:

Remove the ending 5. Let the resulting number be n, and compute n(n+1). Append 25 to the end of n(n+1) and that's your answer.

Example: 852. Here, we drop the last digit to get 8, compute 8*9 = 72, so 852 = 7225. Similarly, we can compute 1152. Here, we drop the last digit to get 11, compute 11*12 = 132, so 1152 = 13225.

How does this work?: Note that (10n + 5)2 = 100n2 + 100n + 25 = 100 * n(n+1) + 25.

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When squaring a number, break the calculation into three smaller calculations and add them using the FOIL method.

Example:

302^2 = 300^2 + 2*(300*2) + 2^2.

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The term "FOIL" should be banned. It makes it seem there is something in it... – Mariano Suárez-Alvarez Oct 29 '10 at 16:26
@MarianoSuárez-Alvarez I'd hope that as a joke. FOIL seems to consist of a pretty good mnemonic for some people, and also acronyms like this don't have to get pronounced as words, but instead just spelled out, like "F O I L". – Doug Spoonwood Jan 15 '12 at 13:23

One simple one that you're probably familiar with already: When you multiply a one-digit number, n, by 9, the result has n-1 in the 10s place, and then the ones digit is such that the sum of the digits is 9.

Example: 9*6 = 54 because 5 is 6-1, and then 5+4 is 9.

What's really cool is that you can use this trick, plus your fingers, to get the answer instantly. Hold out your hands with all 10 fingers up, then put down the n'th finger (which might be on either hand. Then the number is just (how many fingers there are to the left of the finger you put down)*10 + (how many fingers there are to the right of the finger you put down)

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Art Benjamin is your man! He has many tricks to speed up mental calculation and other fun mathemagical tricks. He also wrote two books on the subject!

Here is a video of him in action: http://www.youtube.com/watch?v=M4vqr3_ROIk

Here is his new book: http://www.amazon.com/Secrets-Mental-Math-Mathemagicians-Calculation/dp/0307338401

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Not sure if you are an iOS user, but...

https://itunes.apple.com/us/app/mathemagics-mental-math-tricks/id306586847?mt=8

That has a plethora of mental math tricks with practice problems...

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Multiplying becomes especially easy if the numbers are of the same parity, through $$(a-b)(a+b)=a^2-b^2$$

$$27\times33=(30-3)(30+3)=30^2-3^2=891$$

You can memorize a fairly large number of squares pretty quickly; then you just need to get fast at taking averages (which is why even$\times$odd can be trickier) and subtracting.

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