The decimal numbers can be written in a fractional notation. A fraction or a mixed number whose denominator is a power of 10 is known as a decimal fraction. Some of the examples of decimal fractions are $\frac {2}{10} = 0.2, \frac {7}{10} = 0.7, \frac {67}{100} = 0.67, \frac {534}{1000} = 0.534$ etc.
Let’s find out more about decimal fractions, the method of converting decimal to fraction, and vice-versa and their arithmetic operations.
What is Decimal Fraction?
As you know that a fraction is a relation between a part(numerator) and a whole(denominator). In the case of a decimal fraction, a part can be any whole number such as $7$, $24$, or $439$, but the whole is always a power of $10$.
Note: The numbers $10$, $100$, $1000$, $10000$, etc. are called the power of $10$s, as all these numbers can be represented in the powers of $10$. $\left( 10 = 10^{1}, 100 = 10^{2}, 1000 = 10^{3}, 10000 = 10^{4}, … \right)$
For example, $\frac {7}{10}$ means $7$ parts out of a total of $10$ parts. Similarly, $\frac {24}{100}$ means $24$ parts out of a total of $100$ parts. And in the same way, $\frac {439}{1000}$ refers to $439$ parts out of $1000$ parts.
In the same way, a decimal fraction $1.6$ can be written as $1\frac {6}{10}$, which means $1$ whole parts and $6$ parts out of $10$ parts, where $1$ whole is made up of $10$ parts.
Note: The fractions $\frac {3}{7}$, $\frac {1}{9}$, $\frac {21}{75}$, $\frac {129}{389}$ are not decimal fractions, since in all these fractions denominators are not powers of $10$.
Conversion of Decimal Fractions
As seen above the decimal fractions are another way of representing a decimal as a fraction with a denominator as a power of $10$.
Now, let’s see how you can convert a decimal to a fraction and vice versa.
Conversion of Decimal to Fraction
The following steps are used to convert a decimal to a decimal fraction.
Step 1: Write the decimal fraction as a numerator but without a decimal point. E.g., for $0.7$ the numerator will be $7$, and similarly, for $0.834$ the numerator will be $834$.
Step 2: Count the number of digits in decimal places and write the denominator as a power of $10$, with the number of decimal places as power. For example, in the case of $0.7$, the number of digits after the decimal point is $1$, therefore denominator will be $10^{1} = 10$. And similarly, in the case of $0.834$, the number of digits after the decimal point is $3$, therefore denominator will be $10^{3} = 1000$.
Step 3: Reduce the fraction, if required.
Examples
Ex 1: Convert $0.6$ to a fraction.
The numerator will be $6$ (Decimal number without decimal point).
Number of digits after decimal places = $1$.
Therefore, denominator will be $10^{1} = 10$.
And fraction $= \frac {6}{10}$.
Reduce it to the lowest/simplest form. $\frac {6}{10} = \frac {3}{5}$.
Note: $\frac {6}{10} $ and $\frac {3}{5}$ are equivalent fractions.
Ex 2: Convert $0.953$ to fraction.
The numerator is $953$ (Decimal number without decimal point).
Number of digits after decimal places = $3$.
Therefore, denominator is $10^{3} = 1000$.
And fraction $= \frac {953}{1000}$.
Ex 3: Convert $2.26$ to fraction.
The numerator is $226$ (Decimal number without decimal point).
Number of digits after decimal places = $2$.
Therefore, denominator is $10^{2} = 100$.
And fraction $= \frac {226}{100}$.
Note: It’s an improper fraction and hence can be converted to a mixed fraction.
$\frac {226}{100} = 2 \frac {26}{100}$
Convert it to lowest form ${26}{100} = {13}{50}$.
Therefore, $2.26 = 2 \frac {13}{50}$.
Conversion of Fraction to Decimal
The steps used to convert a decimal fraction to a decimal number are.
Step 1: Note down the numerator of a fraction
Step 2: Count the number of zeroes in the denominator
Step 3: Put a decimal point after a number of places equal to the number of zeroes.
Examples
Ex 1: Convert $\frac {2|{10}$ into decimal
Numerator is $2$.
The number of $0$s in the denominator is $1$.
So, the decimal number is $0.2$.
Ex 2: Convert $\frac {543|{1000}$ into decimal
Numerator is $543$.
The number of $0$s in the denominator is $3$.
So, the decimal number is $0.543$.
Operations With Decimal Fractions
Like other types of numbers, with decimal fractions also, you can perform any of the four arithmetic operations – addition, subtraction, multiplication, and division.
There are two ways of performing the arithmetic operations on decimal fractions
- using fraction operations
- by converting them to decimals
Operations With Decimal Fractions Using Fractions
Since decimal fractions are a type of fractions, you can use the process of addition & subtraction or multiplication & division of fractions.
Addition of Decimal Fractions Using Fractions
Let’s consider a few examples to add decimal fractions.
Ex 1: Add $0.3$ and $0.4$.
$0.3 = \frac {3}{10}$ and $0.4 = \frac {4}{10}$.
Here $\frac {3}{10}$ and $\frac {4}{10}$ are like fractions as the denominators are same. So, to add these two fractions, just add the numerators and the denominator will remain the same.
$\frac {3}{10} + \frac {4}{10} = \frac {3 + 4}{10} = \frac {7}{10} = 0.7$
Ex 2: Add $0.5$ and $0.23$.
$0.5 = \frac {5}{10}$ and $0.23 = \frac {23}{100}$
In this case the fractions $\frac {5}{10}$ and $\frac {23}{100}$ are unlike fractions as the denominators are different. So, the first step will be to take the LCM of denominators i.e., $10$ and $100$, and then convert numerators accordingly.
$\frac {5}{10} = \frac {5\times 10}{10 \times 10} = \frac {50}{100}$
Therefore, $\frac {5}{10} + \frac {23}{100} = \frac {50}{100} + \frac {23}{100}$
= $\frac {50 + 23}{100} = \frac {73}{100} = 0.73$.
Subtraction of Decimal Fractions Using Fractions
Let’s consider a few examples to subtract decimal fractions.
Ex 1: Subtract $0.2$ from $0.8$.
$0.2 = \frac {2}{10}$ and $0.8 = \frac {8}{10}$.
Here $\frac {2}{10}$ and $\frac {8}{10}$ are like fractions as the denominators are same. So, to subtract these two fractions, just subtract the numerators and the denominator will remain the same.
$\frac {8}{10} – \frac {2}{10} = \frac {8 – 2}{10} = \frac {6}{10} = 0.6$
Ex 2: Subtract $0.4$ from $0.59$.
$0.4 = \frac {4}{10}$ and $0.59 = \frac {59}{100}$
In this case the fractions $\frac {4}{10}$ and $\frac {59}{100}$ are unlike fractions as the denominators are different. So, the first step will be to take the LCM of denominators i.e., $10$ and $100$, and then convert numerators accordingly.
$\frac {4}{10} = \frac {4\times 10}{10 \times 10} = \frac {40}{100}$
Therefore, $\frac {59}{100} – \frac {4}{10} = \frac {59}{100} – \frac {40}{100}$
= $\frac {59 – 40}{100} = \frac {19}{100} = 0.19$
Multiplication of Decimal Fractions Using Fractions
For multiplication also, we’ll use the process of multiplication of fractions. Let’s consider a few examples to multiply decimal fractions.
Ex 1: Multiply $0.8$ and $0.6$
$0.8 = \frac {8}{10}$ and $0.6 = \frac {6}{10}$
Therefore, $0.8 \times 0.6 = \frac {8}{10} \times \frac {6}{10}$
$=\frac {8 \times 6}{10 \times 10} = \frac {48}{100} = 0.48$
Ex 2: Multiply $0.2$ and $0.67$
$0.2 = \frac {2}{10}$ and $0.67 = \frac {67}{100}$
Therefore, $0.2 \times 0.67 = \frac {2}{10} \times \frac {67}{100}$
$= \frac {2 \times 67}{10 \times 100} = \frac {134}{1000} = 0.134$
Note: You can directly multiply two or more, unlike fractions without converting them to like fractions.
Division of Decimal Fractions Using Fractions
You know that division is nothing but multiplying the reciprocal of the second fraction with the first fraction, hence the process of division is the same as that of multiplication with one additional step!
Ex 1: Divide $0.7$ by $0.5$
$0.7 = \frac {7}{10}$ and $0.5 = \frac {5}{10}$
Therefore, $0.7 \div 0.5 = \frac {7}{10} \div \frac {5}{10} = \frac {7}{10} \times \frac {10}{5}$$ = \frac {7 \times 10}{10 \times 5} = \frac {70}{50} = \frac {7}{5} = 1\frac {2}{5}$ (Converted to mixed fraction).
Real-Life Application of Decimal Fractions
Decimal fractions are used for understanding precise quantities instead of whole numbers. You will also use them for expressing percentages. For instance, 97% can be written as 97/100 for ease of calculation.
Here are some of the situations where you might encounter decimal fractions:
- Coins (They are a fraction of Rupees)
- Weighing products
- Measuring ingredients while cooking
Conclusion
Decimal fractions are fractions with denominators as a power of $10$. These fractions help us to express precise quantities and help us understand weights like $2.8$ kg and distances like $2.54$ km.
Practice Problems
- Which of the following are decimal fractions?
- $\frac {5}{10}$
- $\frac {15}{2}$
- $\frac {17}{100}$
- $\frac {25}{1000}$
- $\frac {7}{93}$
- Convert the following decimals to fractions
- $0.005$
- $0.234$
- $1.56$
- $0.208$
- $3.2335$
- Convert the following fractions to decimals
- $\frac {3}{10}$
- $\frac {56}{10}$
- $\frac {78}{100}$
- $\frac {432}{1000}$
- $\frac {203}{10000}$
- Perform the following operations
- $0.8 + 0.1$
- $1.2 + 0.5$
- $0.6 + 0.3$
- $0.7 – 0.2$
- $0.9 – 0.6$
- $1.5 – 0.2$
- $0.2 \times 0.1$
- $0.3 \times 0.28$
- $0.74 \times 0.22$
- $0.7 \div 0.2$
- $0.4 \div 0.9$
- $0.685 \div 0.28$
