Multiplying two numbers that differ from a number by the same small number

Follow the following examples to understand the simple idea behind it.

Example 1:

99 * 101 = (100 - 1) * (100 + 1) = (100 * 100) - 1 = 10000 - 1 = 9999

Rem:

99 * 101 = (100 * 100) - (1 * 1)

99 * 101 = 100^2 - 1^2

Example 2:

997 * 1003 = (1000 - 3) * (1000 + 3) = (1000 * 1000) - (3 * 3) = 1000000 - 9 = 999991

Rem:
997 * 1003 = (1000 * 1000) - (3 * 3)

999 * 1003 = 1000^2 - 3^2

Example 3:

47 * 53 = (50 - 3) * (50 + 3) = 2500 - 9 = 2491

Rem:
47 * 53 = (50 * 50) - (3 * 3)

47 * 53 = 50^2 - 3^2 

Now, do the following exercises:

1. 195 * 205 (Rem: 195 * 205 = 200^2 - 5^2)

2. 36 * 44

3. 77 * 83

4. 148 * 152

Answers:

1. 39975

2. 1584

3. 6391

4. 22496

Squaring Numbers Ending in Nine

To square a number ending in nine; e.g. 29, 59, 129, ..., first square the next number that is ending in zero, and then subtract both the number and the next number from it.


Example 1:

To square 29 (29^2):

29: the number
30: the next number

29^2 = 30^2 - (29 + 30) = 900 - 59 = 841


Example 2:

To square 69 (69^2):

69: the number
70: the next number

69^2 = 70^2 - (69 + 70) = 4900 - 139 = 4761


Example 3:

To square 129 (129^2):

129: the number
130: the next number

129^2 = 130^2 - (129 + 130) = 16900 - 259 = 16641

Squaring Numbers Ending in One

To square a number ending in one; e.g. 21, 51, 121, ..., first square the prior number that is ending in zero, and then add both the prior number and the number to it.


Example 1:

To square 21 (21^2):

21: the number
20: the prior number

21^2 = 20^2 + 20 + 21 = 441


Example 2:

To square 61 (61^2):

61: the number
60: the prior number

61^2 = 60^2 + 60 + 61 = 3721


Example 3:

To square 121 (121^2):

121: the number
120: the  prior number

121^2 = 120^2 + 120 + 121 = 14641

Squaring 14, 24, 34, ..., 94

To square a two digit number ending in 4; i.e. 14, 24, 34, ..., 94, first square the next number that is ending in 5 (as explained in "squaring a two digit number ending in 5"), and then subtract the number and the next number from it.


Example 1:

To square 14 (14^2):

14: the number
15: the next number
Based on "squaring a two digit number ending in 5", we know that 15^2 = 225. So:

14^2 = 225 - 14 - 15 = 196


Example 2:

To square 44 (44^2):


44: the number
45: the next number
Based on "squaring a two digit number ending in 5", we know that 45^2 = 2025. So:

44^2 = 2025 - 44 - 45 = 1936


Example 3:

To square 94 (94^2):


94: the number
95: the next number
Based on "squaring a two digit number ending in 5", we know that 95^2 = 9025. So:

94^2 = 9025 - 94 - 95 = 8836.

Squaring 16, 26, 36, ..., 96

To square a two digit number ending in 6; i.e. 16, 26, 36, ..., 96, first square the prior number that is ending in 5 (as explained in the previous post), and then add the number and the prior number to it.


Example 1:

To square 16 (16^2):

16: the number
15: the prior number
Based on the previous post, we know that 15^2 = 225. So:

16^2 = 225 + 16 + 15 = 256


Example 2:

To square 46 (46^2):


46: the number
45: the prior number
Based on the previous post, we know that 45^2 = 2025. So:

46^2 = 2025 + 46 + 45 = 2116



Example 3:


To square 96 (96^2):


96: the number
95: the prior number
Based on the previous post, we know that 95^2 = 9025. So:

96^2 = 9025 + 96 + 95 = 9216.

Squaring 15, 25, 35, ..., 95

To square a two digit number ending in 5; i.e. 15, 25, 35, ..., 95, first multiply the tens digit by the next whole number and then affix 25 to it.

Example 1:

To square 15 (15^2):

The tens digit of 15 is 1 and the next whole number of 1 is 2. So:
1 * 2 = 2
Now, we affix 25 to 2. The answer is 225.


Example 2:

To square 35 (35^2):

The tens digit of 35 is 3 and the next whole number of 3 is 4. So:
3 * 4 = 12
Now, we affix 25 to 12. The answer is 1225.


Example 3:

To square 95 (95^2):

The tens digit of 95 is 9 and the next whole number of 9 is 10. So:
9 * 10 = 90
Now, we affix 25 to 90. The answer is 9025.

Multiplying by 20, 25, and 30

Multiplying By Twenty

Multiplying by 20 is done in two stages, by 2 and then by 10.


Multiplying By Twenty Five

To multiplying by 25:

Divide the number by 4. Then:
If you have a remainder of 0 then add 00 at the end.
If you have a remainder of 1 then add 25 at the end.
If you have a remainder of 2 then add 50 at the end.
If you have a remainder of 3 then add 75 at the end.

Example: 143 * 25

143 / 4 = 35 remainder: 3

so
143 * 25 = 35 & 75 (for a remainder of 3 add 75 at the end)
143 * 25 = 3575


Multiplying By Thirty

Multiplying by 30 is done in two stages, by 3 and then by 10.

Multiplying by 5, 10, and 15

Multiplying By Five

Multiplying by 5 is just multiplying by 10 and then dividing by 2. Example:

15 * 5 = (15 * 10) / 2 = 150 / 2 = 75


Multiplying By Ten

Everybody knows that to multiply by 10, a zero should be added to the end of the number. Example:

26 * 10 = 260


Multiplying By Fifteen

To multiply by 15, you can use one of these two methods:

1) Multiply by 5 then then add it up with the product of the number and 10. Example:

43 * 15 = (43 * 5) + (43 * 10) = (43 * 10 / 2) + 430 = 215 + 430 = 645

In other words, add half of the product of the number and 10 to the product of the number and 10; i.e.:

43 * 15 = 430 + (1/2 * 430) = 645

2) Multiply by 30 and then divide by 2 (in the next post, a method is presented for "Multiply by 30"). Example:

57 * 15 = (57 * 30) / 2 = 1710 / 2 = 855

Multiply 11*11 to 19*19

Use this algorithm in your head:

Take 16 * 18 for an example:

1) Add 16 + 8 = 24
2) Multiply it by 10: 24 * 10 = 240
3) Add up the product of 6 and 8: 240 + (6 * 8) = 288
16 * 18 = 288

Another example:

14 * 17 = ?

1) Add 14 + 7 = 21
2) Multiply it by 10: 21 * 10 = 210
3) Add up the product of 4 and 7: 210 + (4 * 7) = 238
14 * 17 = 238

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Multiplying by 9, 99, 999, ...

Multiplying by 9 is as multiplying by 10-1. So, 9 * 8 is as (10 - 1) * 8 that is:

9 * 8 = (10 - 1) * 8 = 80 - 8 = 72

Another example:

26 * 9 = 26 * (10 - 1) = 260 - 26 = 234

To multiply by 99, one must multiply by 100-1:

53 * 99 = 53 * (100 - 1) = 5300 - 53 = 5200 + 100 - 53 = 5200 + 47 = 5247

Similarly, to multiply by 999, one must multiply by 1000-1:

87 * 999 = 87 * (1000 - 1) = 87000 - 87 = 86000 + 1000 - 87 = 86000 + 913 = 86913

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The Magic of 1001

Attach a 3-digit number to itself (e.g. 547 -> 547547). Divide it by 7.
The remainder will be zero!

Example: Take 826. Then 826826 / 7 = 118118

Now, divide the quotient by 11. The remainder will be zero!

Example:  826826 / 7 =  118118   -->   118118 / 11 = 10738

Again, divide the recent quotient by 13. The remainder will be zero!

Example:  826826 / 7 =  118118    -->   118118 / 11 = 10738  -->   10738 / 13 = 826

Where 826 is the selected number!

Another example:

Take 743!
Concatenate it with itself: 743743
Divide it by 7: 743743 / 7 = 106249
Divide the quotient by 11: 106249 / 11 = 9659
Divide the recent quotient by 13: 9659 / 13 = 743

Why?

7 * 11 * 13 = 1001
Multiplying 1001 by any 3-digit number will be as concatenating it with itself:

100 * 1001 = 100100

945 * 1001 = 945945

So, any number that is obtained by concatenating a 3-digit number with itself can be divided by 1001 and as 1001 is equal to 7 * 11 * 13, it can be divided by 7, 11, and 13.

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Rapid Math Tricks & Tips: 30 Days to Number Power
Numbers to Symbols: Strategies for Learning and Teaching Mathematics
50 Activities for Interpersonal Skills Training