## Fast Math Tricks

_{Fun with number 5}

There are many quick arithmetic tricks that can be used when we deal with number 5. Below we are going to review with examples few of these quick techniques. If you want to quickly calculate in an exam and calculator is not handy, give these some practice. This list is only concentrated on the number 5 and its unique properties in mathematics.

### 1. Multiply or Divide any number with 5

When we multiply any number with 5, a quicker way of performing multiplication is multiplying it with 10 instead and then dividing by 2. Multiplying with 10 is always simple as we only need to add a zero in the end. Similarly dividing by 2 is quicker for most people as well. By increasing the number of steps to smaller bite sizes we increase the calculation time.

**Examples :**

**234**= (234 x 10 ) /2 = 2340 /2_{X}5**= 1170****819**= (819 x 10 ) /2 = 8190 / 2_{X}5**= 4095****6532**= 65320 / 2_{X}5**= 32660**

On the flip side, when dividing by 5, just divide by 10 and multiply by 2.

**Examples :**

**430 / 5**= (430 / 10) * 2 = 43 * 2**= 86****589 / 5**= (589 / 10) * 2 = 58.9 * 2**= 117.8****4374 / 5**= 437.4 * 2**= 874.8**

### 2. Multiply or Divide by 25

Similar to above methods when we multiply by 25, we multiply by 100 (instead of 10 as in case of 5) and then divide by 4 or divide by 2 , two times

**Examples:**

**256**(256 x 100) / 4 = (25600 /2) /2 = 12800 /2 =_{x}25 =**6400****713 x 25**= (713 x 100) / 4 = (71300 / 2) /2 = 35650 /2 =**17825****523 x 25 = (**52300 / 2 ) /2 = 26150 /2 =**13075**

Multiplication is reverse. Divide by 100 and then multiply by 2 twice.

**Examples :**

**2300 / 25**= (2300**/**100 ) * 2**x**2 = (23) * 2**x**2 = 46 x 2 =**92****1250 / 25**= (1250**/**100 )*****2**x**2 = (12.5 )* 2**x**2 = 25 x 2 =**50****6523 / 25**= (65.23 ) * 2**x**2 = (130.46 ) 2 =**260.92**

### 3. Squaring any 2 digit number which begins with 5

This is a medium level trick. To find the square of any 2 digit number which begins with 5 simply add 25 to the second digit and append square of the last digit. Answer will always be a 4 digit number

54^{2} = 54 x 54

- Add 4 ( the second digit) to 25
- 25+4 = 29
- Find square of second digit
- 4 x 4 = 16
- Append to first part to get answer
- 54
^{2 }= 2916

**Examples :**

**53**= 25+3 && 3 x 3 = 28 && 09 =^{2 }**2809****59**= 25+9 && 9 x 9 = 34 && 81 =^{2}**3481****51**= 25+1 && 1 x 1 = 26 && 01 =^{2}**2601**

### 4. Squaring any number which ends in a 5

This is a bit advanced trick. Any number (N) that ends in a 5 has following format

N = c x 10 + 5

Where "c" is an integer.

For example , the number 5 can be written as follows, the constant "c" being a zero.

5 = **0** x 10 + 5

Square of any given number which ends in a 5 is equal to constant "c" multiplied with constant "c"+1 and number 25 appended to the end.

**Example :**

**5 = 0 **x 10 +5

c = 0

5^{2 }= **0** x (**0**+1 ) && 25 (where && means append)

5^{2} = **0** x 1 && 25

5^{2} = **0** && 25 = 25

Now lets take a little complex number lets say 225. When we write 225 in our constant plus number 5 format we get the following "c"

225 = 22 x 10 + 5

c = 22

Now We compute 225^{2}

225^{2} = 22 x (22+1 ) && 25

225^{2} = 22 x 23 && 25

At this point we need to know how to fast multiply two digit numbers. There are many ways this can be done, below is one of the methods. If you prefer any other quick multiplication, feel free to do that at this point in time.

- Multiply first digits form left to get first digit(s) of answer ( 2 x 2 = 4)
- Multiply last digits to get end digit of answer (2 x 3 = 6)
- Right now our number is something like this

- 22 x 23 = 4 _ 6

- We cross multiply and add
- = 2*3 + 2* 2
- = 6 + 4
- = 10

- = 4 10 6
- = 4
^{+1 }0 6 - = 506

Another method that is gaining popularity is the Japanese lines method,also called magic math lines. If you search you tube you will find videos explaining the method. Use which ever method you find easier.

So back to our quick square

225^{2} = 22 * 23 && 25

225^{2} = 506 && 25

225^{2}= 50625

**More Examples :**

- 65
^{2}= 6*7 &&25 = 42 &&25 = 4225 (c = 6) - 105
^{2}= 10 x 11 &&25 = 110 &&25 = 11025 (c = 10) - 115
^{2}= 11 x 12 &&25 =132 &&25 = 13225 (c = 11) **3352 = 33 x 34 &&25 =1122 &&25 = 112225 (c = 33)**

### 5- Squaring any number which has repeating 3 and ending with 5.

If a number only consists of 3 and end with a 5, the square is equal to

- Number of 1 equal to number of 3
- Append number of 2 equal to number of 3
- Append with number 25

Examples

- 35 = 1 && 2 && 25 = 1225
- 335 = 11 && 22 && 25 = 112225
- 3335 = 111 && 222 && 25 = 11122225
- 333333335 = 11111111 && 22222222 && 25 = 111111112222222225

### 6- Squaring any number which has repeating 6 and ending with 5.

If a number only consists of 6 and end with a 5, the square is equal to

- Number of 4 equal to repeating 6
- Append number of 2 equal to repeating 6
- Append with number 25

Examples

- 65 = 4 && 2 && 25 = 4225
- 665 = 44 && 22 && 25 = 442225
- 6665 = 444 && 222 && 25 = 44422225
- 333333335 = 44444444 && 22222222 && 25 = 444444442222222225

### 5- Squaring any number which has repeating 9 and ending with 5.

If a number only consists of 9 and end with a 5, the square is equal to

- Number of 9 equal to nines (9)
- Append number of 0 equal to nines (9)
- Append with number 25

Examples

- 95 = 9 && 0 && 25 = 9025
- 995 = 99 && 00 && 25 = 990025
- 9995 = 999 && 000 && 25 = 99900025
- 999999995 = 99999999 && 00000000 && 25 = 999999990000000025

This is all from number 5. Have fun applying these cool tricks to speed up your calculation times.

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