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

Question

Anything to speed up or simplify calculations.

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

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

mental-calculations tips tricks shortcut cheats time-saver

Answer 1

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

Answer 2

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.

Answer 3

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)

Answer 4

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.

Answer 5

For two digit numbers, if the first digit is the same, the sum of two second digits is 10. (This includes Moor Xu's case as a special case), then

xa*xb = x*(x+1) append a*b

e.g

46*44 = 4*5 append 4*6 = 2024

93*97 = 9*10 append 7*3 = 9021

Following this pattern you can generalize 3 digits multiplications (with complication), and two digits multiplication with different leading digit.