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Multiplying or dividing by powers of 10
If you want to multiply or divide some number by a power of ten, e.g.
13,3 * 100,
the fastest way it can be done is the method of "shifting the decimal point". Each power of ten means a shift of 1 to the right, e.g.
13.34 * 10 = 133.4
or
13.34 * 1000 = 13340.0 = 13340.

Divisions are the inverse, so dividing by a power of ten is a shift of 1 to the left, e.g.
13.34 / 10 = 1.334
or
13.34 / 1000 = 0.001334.

Note that we added 0s to the left, because the shift was greater than the amount of decimal places to the left.
This strategy can be combined with other strategies, e.g.
x * 50
is the same as
(x * 5) * 10,
so first calculate
x*5,
and then perform a decimal shift.
Multiplying a two digit number by 11
This trick is very useful and yet simple to learn. Let's say you want to multiply 63 by 11. One way to do this is to split the calculation into
(63*10) + (63*1) = 630 + 63 = 693.
While this works fine, it takes way to much time. A neat trick for this is to simply add the two digits of 63 and squeeze them in between. In our case we have
6 + 3 = 9.
and thus
63 * 11 = 6 9 3.
One more example. Let's say you want to calculate 81*11. First we do
8 + 1 = 9
and then
81 * 11 = 8 9 1.
Mint, but what if the two digits sum up to something greater than 9? Don't worry, this will also be very simple. Let's take a look at 87 * 11. First we
8 + 7 = 15.
Squeezing 15 in between 8 and 7 doesn't seem correct. So first we add 1 to the first digit, i.e.
1+8 = 9.
Now we squeeze 5 in between and get
87 * 11 = 957.
Try some numbers by yourself to convince yourself. There is also a similar trick in order to divide numbers by 11, I am sure you will be able to figure this one out for yourself.
Everyone, and I mean literally everyone, learned to add numbers from right to left. The reasons for this are unknown to mankind. So I urge you to forget about that and start adding numbers from left to right. You read numbers from left to right, so why would you do anything different when calculating the number in question? You will become a lot quicker, trust me. Let's do some examples. First we look at 68 + 25. We start with
68 + 20 = 88,
and now simply add 5 to it to get
88 + 5 = 93.
Now 873 + 138. Again
873 + 100 = 973,
then
973 + 30 = 1003,
finally
1003 + 8 = 1011.
This might feel slow when you start practicing, but trust me you will become way faster than with the right to left technique. You should use the same style when subtracting two numbers. One final remark. Sometimes you might find it useful to switch the numbers. For example 582 + 300 seems to be quicker for me than 300 + 582, but this is totally up to ones personal preference.
Squarring two digit numbers.
First of all everyone should know squares up to 25 (or at least 20) by hard. This won't take you more than 5 minutes to master, no worries. However, here is a neat to tip to square two digit numbers. Let's look at 15 * 15. You should know that this is equal to 225, but let's say you did forget it (shame on you). What are numbers in the near distance of 15, which are easy to multiply? Clearly, 20 and 10 are really nice. Even better, they have the same 'distance' from 15 (namely 5). So what to do now? Simple:
15 * 15 = 20 * 10 + 5 * 5 = 225.
It is essential to pick two numbers with the same distance to the number you want to square. You might be able to proof why this trick works even if you never enjoyed any serious mathematical training. Lets try 14*14. Here we get
14 * 14 = 18 * 10 + 4 * 4 = 196.
We will now turn our heads towards more 'serious' examples. Let's investigate 42*42.
42*42 = 44 * 40 + 2*2 = 1760 + 4 = 1764.
Mint, isn't it? It is essential to find suitable neighbors, this will take less practice that you might think. OK, one more example. 57*57. This might already cause some heart attacks, but don't you worry.
57 * 57 = 60 * 54 + 3 * 3 = 3000 + 240 + 9 = 3249.
This trick can also be used in some inverse way. Look at 16 * 14. Easy:
16 * 14 = 15 * 15 - 1 * 1 = 224.
Notice that here you subtract the distance, I am sure you will find out yourself why. One more:
19 * 15 = 17 * 17 - 2 * 2 = 285.