Math Tricks - 1

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, 9×9 is just 9x(10-1) which is 9×10-9 which is 90-9 or 81.
Let’s try a harder example: 46×9 = 46×10-46 = 460-46 = 414.
One more example: 68×9 = 680-68 = 612.
To multiply by 99, you multiply by 100-1.
So, 46×99 = 46x(100-1) = 4600-46 = 4554.
Multiplying by 999 is similar to multiplying by 9 and by 99.
38×999 = 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, 436×11 = is 4796.
Let’s do another example: 3254×11.
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: 4657×11.
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, 4657×11 = 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.
12×5 = (12×10)/2 = 120/2 = 60.
Another example: 64×5 = 640/2 = 320.
And, 4286×5 = 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 = 25×4. Note: to divide by 4 your can just divide by 2 twice, since 2×2 = 4.
64×25 = 6400/4 = 3200/2 = 1600.
58×25 = 5800/4 = 2900/2 = 1450.
To multiply by 125, you multipy by 1000 then divide by 8 since 8×125 = 1000. Notice that 8 = 2×2x2. So, to divide by 1000 add three 0’s to the number and divide by 2 three times.
32×125 = 32000/8 = 16000/4 = 8000/2 = 4000.
48×125 = 48000/8 = 24000/4 = 12000/2 = 6000.