Decimal fractions are those in which the denominators are powers of 10.

$\frac{1}{10}$ , $\frac{1}{100}$, $\frac{1}{1000}$ are respectively the tenth, the hundredth and the thousandth part of 1. $\frac{8}{10}$ is 8 tenths written in decimals as 0.8, $\frac{14}{100}$ is 14 hundredths written in decimals as 0.14, $\frac{6}{1000}$ is 6 thousandths written in decimals as 0.006 and so on.

## Expressing Common Fraction as Decimal Fraction

Let us first understand what a common fraction is. A fraction as a term would mean a type of division which is considered as a part or piece of a whole or that which indicates the division of a whole into equal units or parts. A common fraction would therefore definitely refer to an expression where the denominator would be the same for all fractions like $\frac{1}{5}$, $\frac{2}{5}$, etc.

The expression if and when expressed in decimals would qualify as a decimal fraction so for each of the common fractions like $\frac{1}{5}$, $\frac{2}{5}$ or $\frac{3}{5}$ can be expressed in decimal terms as well.

$\frac{1}{5}$ = 0.2

$\frac{2}{5}$ = 0.4

$\frac{3}{5}$ = 0.6

Common fraction is in fact another way of indicating division and is nothing but getting the numerator divided by the denominator into decimals. To change a common fraction into its decimal fraction we need to follow the steps mentioned below.

Step 1: The numerator is divided by the denominator.
Step 2: Carry the division beyond the needed number of decimal places by one digit.
Step 3: Round off the quotient wherever required to the number of decimal places.

### Examples on Expressing a Common Fraction as a Decimal Fraction

Here are some examples based on common fraction as a decimal fraction:

Example 1:

Find the decimal fractions of common fractions $\frac{2}{12}$ , $\frac{5}{6}$ and $\frac{7}{6}$

Solution : Given $\frac{2}{12}$ , $\frac{5}{6}$ and $\frac{7}{6}$

Step 1: Convert the fractions into its lowest equivalents ($\frac{2}{12}$ = $\frac{2\times1}{2\times6}$ = $\frac{1}{6}$ )

Step 2: Convert into common fractions ( $\frac{1}{6}$ , $\frac{5}{6}$ and $\frac{7}{6}$ )

Step 3: Convert into decimals (0.1667) , (0.8333) and (1.1667)

Example 2:

Find the decimal fractions of common fractions $\frac{2}{7}$, $\frac{6}{14}$, and $\frac{4}{7}$

Solution:

Basic idea: Convert the fractions into common fractions

Step 1: Convert the fractions into its lowest equivalents ( $\frac{6}{14}$ = ($\frac{2\times3}{2\times7}$) = $\frac{3}{7}$ )

Step 2: Convert into common fractions ( $\frac{2}{7}$, $\frac{3}{7}$, $\frac{4}{7}$ )

Step 3: Convert into decimals (0.2857) (0.42857) (0.57142857)

## Expressing Measuring Units in Decimal Fraction

When we think of units we always relate these to early measuring ideas. Whenever we measure a certain amount of substance with different sized measuring cups or glass, the number of measures changes with the size of the measuring unit. Every fraction depends on some unit and when we convert these into decimals then we have measuring units expressed as decimals.

With the passage of time and corresponding improvements in measuring technology, greater accuracy became possible. Smaller parts of inches or feet were converted into smaller units by dividing the units into 1000 parts and each of these parts referred as “thousandths of an inch” and depicted as 0.001

 One inch 1 One thousandth of an inch 0.001 One tenth thousandth 0.0001 One millionth 1e-06

Let us take the millimetre as a unit of measure.

The millimetre (mm) is slightly smaller than $\frac{1}{16}$ of an inch.

Millimeter: $\frac{1}{1000}$ of meter = 0.001 m

Millimeter: $\frac{1}{10}$ of centimetre = 0.1 cm

Liquid data for measuring units:

1 gallon of water weighs 8.34 pounds
1 gallon of milk weighs 8.6 pounds
1 cubic foot of water = 7.48 gallons

### Examples on Measuring Units in Decimal Fraction

Here are some examples on measuring units in decimal fraction.

Example 1:

Express 2($\frac{1}{4}$) meters into millimeters in decimal fraction.

Solution:

Step 1: Convert the measuring unit of meter into an improper fraction ($\frac{9}{4}$) m
Step 2: Convert the improper fraction into a decimal fraction (2.250 m)
Step 3: Convert the decimal fraction into a millimeter (2.250 x 1000) mm

2($\frac{1}{4}$) meters = $\frac{9}{4}$ meters

$\frac{9}{4}$ meters = 2.25 meters

1 meter = 1000 m

Therefore, 2.25 m x 1000 = 2250 mm

Example 2:

Express 1($\frac{1}{4}$) meters into millimetres in decimal fraction.

Solution:

Step 1: Convert the measuring unit of meter into improper fraction ($\frac{5}{4}$) m
Step 2: Convert the improper fraction into decimal fraction (1.250 m)
Step 3: Convert the decimal fraction into millimetre (1.250 x 1000) mm.

1($\frac{1}{4}$) meters = $\frac{5}{4}$ meters

$\frac{5}{4}$ meters = 1.25 meters

1 meter = 1000 mm

Therefore, 1.25 m x 1000 = 1250 mm

## Simplified Method of Reading Decimal Fraction

A decimal fraction is a fraction whose denominator has the power of ten. The decimal fraction originates from dividing a unit into 10 equal parts and then each of these parts into other equal parts and so on indefinitely. The fractional units are then found to have tenths, hundredths, thousandths or in any order of parts.

Decimal fractions are more often expressed by writing the numerator only with the decimal point coming before it.

$\frac{4}{10}$ could be written as ‘0.4’ and may be read as 4 tenths

$\frac{11}{100}$ may be written as ‘0.11’ and may be read as 11 hundredths.

$\frac{9}{1000}$ could be written as 0.009 and read as 9 thousandths.

$\frac{115}{10000}$ may be written as 0.0115 and may be read as 115 ten-thousandths.

## Rounding Decimal Fraction

When we need to round a number to a particular decimal place, we have to observe one place to the right of that number.

• If the digit is 5 or higher, we have to add one to the digit in the desired place and drop all the following digits thereafter.
• If the digit is 4 or lower, leave the digit in the desired place as it is and drop all the following digits.

Rules for rounding off numbers

If the last digit before the number we are supposed to round off is greater than equal to 5, we have to add one to the last digit, or else we have to retain that digit.

### Examples for Rounding Decimal Fraction

Here are some examples on rounding a decimal fraction

Example 1:

Convert the decimal fraction 1.367 using rounding off rules to one decimal place.

Solution:

Given 1.367

Step 1: As we can see that 7 is greater than 5 so 6 is rounded off and the new number is 1.37
Step 2: Again we can see that the rounded off digit, in this case 7 is greater than 5 so 3 is rounded off to 4 and we have a new answer as 1.4

Example 2:

Convert 1.563 using significant figures rules and round off to two decimal places.

Solution:

Given 1.563

Step 1: As we can see 3 is smaller than 5 so we cannot use the rounding off rule for this digit
Step 2: Again, the second digit is 6 and so we leave the answer as 1.56

When we add decimal fractions, we might use the same rule as adding whole numbers but with the decimal difference. Adding decimal fraction with another decimal fraction would be easy as compared to decimal fraction with a whole number. To add a decimal fraction into a whole number, we have to re-write the whole number in the decimal form with a zero after decimal and follow up the normal decimal fraction addition process.

Let us add 11 and 3.4

Step 1: Re-write the whole number in a decimal form (11.0)
Step 2: Add a decimal fraction with the whole number keeping the decimal right below one another.
Step 3: Follow the normal addition procedure (11.0 + 3.4 = 14.4).

In some cases, we have decimal fractions added to fractions. When we have to add a group of fractions with decimal fraction (s), we have to make sure that the denominators are equal for the fractions. To get the same denominator we have to make use of a common denominator by multiplication. Once we have the common denominator, convert each of the fractions to a common fraction and convert these into decimals. Once we have the decimal fractions we can add these decimal fractions.

Let us add $\frac{2}{3}$ + $\frac{3}{5}$ + 1.4

Step 1: Find the common denominator of the fractions and add the fractions ($\frac{2}{3}$ + $\frac{3}{5}$) = $\frac{(10+9)}{15}$ = $\frac{19}{15}$

Step 2: Convert the fraction into a decimal fraction (1.266)
Step 3: Add the decimal fractions (1.266 + 1.4) = 2.666

### Example on Adding Decimal Fractions

Solution:

Given 22.46 and 8.12

Step 1: Add the decimal fraction with the whole number keeping the decimal right below one another.
Step 2: Follow the normal addition procedure ( 22.46 + 8.12 = 30.58)

## Subtracting Decimal Fractions

The operation of subtraction of decimal fractions is same as addition and we have to follow the same pattern of converting the fractions (if any) into a decimal fraction and then subtract them together keeping in mind the number operations.
We have to line up the decimal points and fill in as many zeros as placeholders so that all the decimal fractions have the same number of digits to the right of the decimal point.

Let us subtract 4.31 from 13.1

Step 1: Line up the decimal fractions and fill in with zeros wherever necessary as place holders.
Step 2: Subtract 13.10 - 4.31
Step 3: The final answer for the decimal fraction is found to be 8.79

### Example on Subtracting Decimal Fractions

Subtract 7.31 from 21.82

Solution:

Given 21.82 - 7.31

Step 1: Line up the decimal fractions and fill in with zeros wherever necessary as place holders.
Step 2: Subtract 21.82 - 7.31
Step 3: The final answer for the decimal fraction is found to be 14.51

## Multiplying Decimal Fractions

While multiplying decimal fractions, we have to keep in mind that the decimal fractions should be considered as whole numbers and then we have to place as many places from the first number on the right as the sum of decimal places in the multiplier and multiplicand.

Let us multiply 2.345 and 8.11

Step 1: Consider the decimal fraction as a whole number and follow the normal operations for multiplication (2345 x 811)
Step 2: Once the product is obtained apply the rule of the decimal (1901795) = (19.01795)

### Examples on Multiplying Decimal Fractions

Multiply 3.561 x 6.42

Solution:

Given 3.561 x 6.42

Step 1: Consider the decimal fraction as a whole number and follow the normal operation for multiplication ( 3561 x 642 )
Step 2: Once the product is obtained apply the rule of the decimal ( 2286162 ) = ( 22.86162 )

## Dividing Decimal Fractions

When we have to divide a decimal fraction, we have to move the decimal point to the right and both the divisor and the dividend to as many places it is to the left to form a whole number in the divisor and then proceed with it as a whole number.

The decimal point in the quotient should be immediately placed above the decimal point in the dividend. The quotient will therefore, contain as many places as the dividend and less than the numbers in the divisor.

Let us divide 1.38483 by 60.21

There are five decimal places in the dividend and two in the divisor and so there must be three places in the quotient.

Step 1: Keep the decimals directly below one another.
Step 2: Divide by normal division rules.
Step 3: Add prefix of ‘0’ zero to quotient (0.023)

### Example on Dividing Decimal Fractions

Divide 4.57682 by 45.32

Solution:

Given 4.57682 ÷ 45.32

There are five decimal places in the dividend and two in the divisor and so there must be three places in the quotient.

Step 1: Keep the decimals directly below one another
Step 2: Divide by normal division rules
Step 3: Add prefix of '0' zero to quotient (0.100)

### Numerator Denominator

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