# Tutorial on Decimals

Decimals represent fractions. For example 1/2 = 0.5, 3/4 = 0.75

We can write 0.75 as just .75 also. The zero on the left hand side of the decimal is optional.

To convert 0.75 into a fraction.

Step 1
Write 0.75 as 0.75/1.

Step 2
Multiply numerator and denominator by 100

(0.75 x 100)/(1 x 100) = 75/100 = 3/4

If we have a decimal with three digits on the right side, for example 0.725 we simply multiply the numerator and denominator by 1000.

(0.725 x 100) / (1 x 1000) = 725/1000 = 29/40

So if we have two digits on the right side of the decimal we multiply by 100 if there are three digits we mutiply by 1000 and so on.

Also 0.75 = 0.750 = 0.7500

Adding zeros on the right hand side does not change the value of the decimal.

We know that 75/100 = 0.75 and 725/1000 = 0.725

Similarly 75/1000 = 0.075 and 75/10000 = 0.0075

Each time we add a zero to the denominator we add a zero between 75 and the decimal.

We will add 3 and 2.75

Step 1
Write the numbers as follows:

3.00
2.75

Notice we place two zeros on the right side of the decimal after 3. This is done to match the number of digits in the two numbers.

Step 2
Add the two numbers just as you would add two whole numbers

300
275
575

Step 3
Now where should the decimal point be placed in the result? Well, it is placed just beneath the position of the decimal points in the added numbers

3.00
2
.75
5
.75

Subtracting Decimals
This is similar to adding decimals. Say we want to solve 3 - 2.75

Step 1

3.00
-2.75

Step 2

300
-275
025

Step 3

3.00
-2
.75
0
.25

Multiplying Decimals
Let us multiply 2.32 with 0.5

Step 1
Write the numbers without the decimal and multiply them

232 x 5 = 1160

Step 2
Now we have to place the decimal point in the result. For doing this count the number of digits after the decimal. This is also called counting the number of decimal places.

Number of decimal places in 2.32 are two and 0.5 has one decimal place.

The total decimal places in 2.32 and 0.5 are two plus one which equals three.

To place the decimal point in the result, count these many places and put the decimal. So the result becomes 1.160

Dividing Decimals
Say we want to calculate 120.086/0.02 or in other words to divide 0.02 into 120.086. To do this follow the following steps:

Step 1
Move the decimal as many places to the right in the division so as to make it a whole number. In 0.02 this will mean two places to the right giving us 2.

Step 2
As we have changed the divisor from 0.
02 to 2 we must change the dividend also exactly in the same manner. To do this we move the decimal to the right two steps in 120.086 also. This gives us 12008.6

Step 3
Now divide 12008.6 by 2 as you would divide a whole number. In the result leave the decimal place where it is. The answer will be 6004.3

Multiplication and Division of Decimals by 10
Study the following examples

21.3267 x 10 = 213.267 (decimal moves one step right)
213.267 x 10 = 213
2.67 (decimal moves one step right)
2132.67 x 10 = 2132
6.7 (decimal moves one step right)

Each time we multiply by 10 the decimal jumps to the right by one step. Multiplying by 10 is as simple as that!

For division study the following:

213267 / 10 = 21326.7 (decimal moves to the left by one step)
21326.7 / 10 = 2132.
67 (decimal moves to the left by one step)
2132.67 / 10 = 213.
267 (decimal moves to the left by one step)

In each division by 10 the decimal jumps to the left by one step.

Multiplication and Division of Decimals by 10
Study the following multiplication examples:

21.3267 x 100 = 2132.67 (decimal moves right by two steps)
2132.67 x 100 = 2132
67 (decimal moves right by two steps)

Study the following division examples:

213267 / 10 = 21326.7 (decimal moves left by one step)
21326.7 / 10 = 2132.
67 (decimal moves left by one step)

Multiplication and division of decimals by 100 is similar to that by 10. The only difference being the number of steps the decimal jumps each time. In the case of 100 it is two steps and in 10 it is one step.

How much would the decimal jump for multiplication and division by 1000? The answer is three steps.

Fractions and Decimals
1/2 is a fraction and is equal to 0.5

2/5 is a fraction and is equal to 0.4

If we take the fraction 1/3 and try to covert it to a decimal we get:

0.333333-------

This is a repeating decimal.

In calculations it is better to use the fraction in place of a repeating decimal.