Math Tricks - 2

4. Multiplying together two numbers that differ by a small even number

This trick only works if you’ve memorized or can quickly calculate the squares of numbers. If you’re able to memorize some squares and use the tricks described later for some kinds of numbers you’ll be able to quickly multiply together many pairs of numbers that differ by 2, or 4, or 6.
Let’s say you want to calculate 12×14.
When two numbers differ by two their product is always the square of the number in between them minus 1.
12×14 = (13×13)-1 = 168.
16×18 = (17×17)-1 = 288.
99×101 = (100×100)-1 = 10000-1 = 9999
If two numbers differ by 4 then their product is the square of the number in the middle (the average of the two numbers) minus 4.
11×15 = (13×13)-4 = 169-4 = 165.
13×17 = (15×15)-4 = 225-4 = 221.
If the two numbers differ by 6 then their product is the square of their average minus 9.
12×18 = (15×15)-9 = 216.
17×23 = (20×20)-9 = 391.


5. Squaring 2-digit numbers that end in 5

If a number ends in 5 then its square always ends in 25. To get the rest of the product take the left digit and multiply it by one more than itself.
35×35 ends in 25. We get the rest of the product by multiplying 3 by one more than 3. So, 3×4 = 12 and that’s the rest of the product. Thus, 35×35 = 1225.
To calculate 65×65, notice that 6×7 = 42 and write down 4225 as the answer.
85×85: Calculate 8×9 = 72 and write down 7225.


6. Multiplying together 2-digit numbers where the first digits are the same and the last digits sum to 10

Let’s say you want to multiply 42 by 48. You notice that the first digit is 4 in both cases. You also notice that the other digits, 2 and 8, sum to 10. You can then use this trick: multiply the first digit by one more than itself to get the first part of the answer and multiply the last digits together to get the second (right) part of the answer.
An illustration is in order:
To calculate 42×48: Multiply 4 by 4+1. So, 4×5 = 20. Write down 20.
Multiply together the last digits: 2×8 = 16. Write down 16.
The product of 42 and 48 is thus 2016.
Notice that for this particular example you could also have noticed that 42 and 48 differ by 6 and have applied technique number 4.
Another example: 64×66. 6×7 = 42. 4×6 = 24. The product is 4224.
A final example: 86×84. 8×9 = 72. 6×4 = 24. The product is 7224


7. Squaring other 2-digit numbers

Let’s say you want to square 58. Square each digit and write a partial answer. 5×5 = 25. 8×8 = 64. Write down 2564 to start. Then, multiply the two digits of the number you’re squaring together, 5×8=40.
Double this product: 40×2=80, then add a 0 to it, getting 800.
Add 800 to 2564 to get 3364.
This is pretty complicated so let’s do more examples.
32×32. The first part of the answer comes from squaring 3 and 2.
3×3=9. 2×2 = 4. Write down 0904. Notice the extra zeros. It’s important that every square in the partial product have two digits.
Multiply the digits, 2 and 3, together and double the whole thing. 2×3x2 = 12.
Add a zero to get 120. Add 120 to the partial product, 0904, and we get 1024.
56×56. The partial product comes from 5×5 and 6×6. Write down 2536.
5×6x2 = 60. Add a zero to get 600.
56×56 = 2536+600 = 3136.
One more example: 67×67. Write down 3649 as the partial product.
6×7x2 = 42×2 = 84. Add a zero to get 840.
67×67=3649+840 = 4489.


8. Multiplying by doubling and halving

There are cases when you’re multiplying two numbers together and one of the numbers is even. In this case you can divide that number by two and multiply the other number by 2. You can do this over and over until you get to multiplication this is easy for you to do.
Let’s say you want to multiply 14 by 16. You can do this:
14×16 = 28×8 = 56×4 = 112×2 = 224.
Another example: 12×15 = 6×30 = 6×3 with a 0 at the end so it’s 180.
48×17 = 24×34 = 12×68 = 6×136 = 3×272 = 816. (Being able to calculate that 3×27 = 81 in your head is very helpful for this problem.)


9. Multiplying by a power of 2

To multiply a number by 2, 4, 8, 16, 32, or some other power of 2 just keep doubling the product as many times as necessary. If you want to multiply by 16 then double the number 4 times since 16 = 2×2x2×2.
15×16: 15×2 = 30. 30×2 = 60. 60×2 = 120. 120×2 = 240.
23×8: 23×2 = 46. 46×2 = 92. 92×2 = 184.
54×8: 54×2 = 108. 108×2 = 216. 216×2 = 432.
Practice these tricks and you’ll get good at solving many different kinds of arithmetic problems in your head, or at least quickly on paper. Half the fun is identifying which trick to use. Sometimes more than one trick will apply and you’ll get to choose which one is easiest for a particular problem.