Decimal Fraction

A fraction comprises a numerator and a denominator. In a decimal fraction, the fraction has a denominator that is a power of 10. When converting a decimal to a fraction, first express the denominator as a power of 10. Also, the number of zeros in the power of 10 should be equal to the number of decimal places in the given number. In short, a decimal fraction will have a denominator of 10 or its powers, like 100, 1000, 10000, and so on.

For example, 
 

    \({0.5} = {5\over10}\)
 

    \({0.25} = {25\over100}\)
 

    \({0.75} = {75\over100}\)

These fractions all have denominators that are powers of 10 (10, 100, etc.).

Decimal fractions can be read in two ways: by naming each digit separately after the decimal point or by using place values. Understanding how to read them correctly helps in math and everyday life. 

Step 1: Read the whole number (if any) before the decimal point. For example, in 3.125, the whole number is 3.

Step 2: Say the word “point” when you reach the decimal (.).

Step 3: Read each digit, one by one, after the decimal point separately. For example, 3.125 can be read as “three point one two five”.  
 

Step 4: You can also read the decimal as a fraction by using place values. For example, 3.125 as “three and one hundred twenty-five thousandths”.

Basic mathematical operations are applicable on decimal fractions. The following are the detailed operations on decimal fractions:

Addition of Decimal Fractions: The addition of decimal fractions can be done in two ways. 

Convert decimal fractions to decimal form before adding.
 

Step 1: First, convert them into decimal form. 
 

For example, if we need to add \(45\over100 \) and \(65\over1000\), 
 

• \({45\over100} = 0.45\)
   
• \({65\over1000} = 0.065\)
  
   

Step 2: Now, adding them together:

 \(0.45 + 0.065 = 0.515\) 

 

Step 3: To make it back to a fraction, look for how many decimal places are there. 

 

Here, there are three digits after the decimal point. So we use 1000 as the denominator. To convert 0.515 into a fraction, we need to multiply and divide it by 1000. 
 

\({{0.515 \over 1000} \times 1000} = {515\over 1000}\)
 

So the answer is \(515\over1000\)

Convert the given decimal fractions to like fractions before adding. 

Step 1: First, make both the fractions into like fractions. 
 

For example, \(45\over100 \)and \(65\over 1000\)
 

For that, find the LCM of both denominators, 100 and 1000.
 

The LCM is 1000

    
Step 2: Convert each fraction

\({45\over100}  = {{45 \times 10} \over{100 \times 10}} = {450\over1000}\)
 

\(65\over1000\) already has a denominator of 1000, so it remains the same. 

    
Step 3: Add them together 
 

\({450\over1000} + {65\over1000} = {515\over1000}\)
 

 So the answer is \({515\over1000}\)

Subtraction of Decimal Fractions: Similarly, to subtract decimal fractions, convert them into decimal form first. For example, if we subtract \(65\over1000 \)from \(45\over100\).
 

\({45\over100} = 0.45\)
 

\({65\over1000} = 0.065\)
 

Now, subtracting
 

\( {0.45} - {0.065} = {0.385}\)

Convert back into fraction, 

\({0.385\over1000} × 1000 = {385\over1000}\)
 

So the answer is \(385\over1000\).

Multiplication of Decimal Fractions: While multiplying a decimal fraction by a power of 10, it is all about calculating the place of the decimal point as per the number of zeros in the power of 10.

For example, \(​​10^2\), here, the power of 10 is 10². We multiply it by 63.457, we count the number of zeros and shift the decimal point two places accordingly.
 

Division of Decimal Fractions: When we divide a decimal fraction by 10 or any power of 10, we move the decimal point to the left. We change the number of places by counting how many zeros there are in the power of 10 we divide. 

For example, \(63.457 \div 100 = 0.63457 \)

(since the denominator 100 (\(10^2\)) has two zeros, we move the decimal two places to the left. 

A number can be expressed as a decimal with either a finite or infinite number of decimal places. Decimals can be classified into three types:
 

• Terminating Decimals
   
• Non-terminating Repeating Decimals
   
• Non-terminating Non-repeating Decimals
   

A decimal fraction can be represented as a decimal with a finite number of decimal places. The number of zeros in the power of 10 in the denominator determines the number of decimal places. Hence, decimal fractions can be considered as terminating decimals, but not all terminating decimals are decimal fractions unless their denominator can be expressed as a power of 10. 
 

To convert numbers into decimal fractions, different methods are used depending on whether the number is a fraction or a decimal or a mixed fraction. The methods are discussed below:

 

• Converting Fractions to Decimal Fractions
  
  Let’s take the example of the fraction \(5\over4\).
  
  Step 1: First, find a number that, when multiplied by the denominator, results in 10 or a multiple of 10. 
  
  Step 2: In this case, multiplying 4 by 25 gives 100.
  
  Step 3: Multiply with the same number to the numerator and denominator.
   

        \({(5 \times 25) \over (4 \times 25)}  = {125\over100}\)
 

Thus, the decimal fraction of \(5\over4 \) is \(125\over100\) or 1.25 


 

• Converting Mixed Numbers to Decimal Fractions
  
  Let’s take an example of \({2{1\over 5}}\)
  
  Step 1: First, we need to convert the mixed number, \({2{1\over 5}}\) into an improper fraction.
  
  \({2{1\over 5}} = {11\over 5}\)
  
  Step 2: Next, to get a denominator of 10, we must multiply both the numerator and denominator by 2.
  
  \({{11 \times 2}\over {5 \times 2}} = {22\over10} \)
  
  Therefore, the decimal fraction of \(2{1\over5}\) is \(22\over 10\).  

• Converting Decimal Numbers to Decimal Fractions
  
  Let’s take an example of 6.2
  
  Step 1: Write the decimal number as a fraction with 1 in the denominator.
  
  \(6.2\over1\)
  
  Step 2: Shift the decimal one place to the right, changing the denominator from 1 to 10.
  
  \({6.2\over1 } = {6.2\over10}\)
  
  Step 3: Now that the numerator is a whole number, we get the decimal fraction.
  
  \(6.2 = {62\over10}\)
  
  Thus, 6.2 as a decimal fraction is \(62\over10\).

Decimal fractions are an important concept in mathematics and everyday life. Mastering them helps in accurate calculations, precise measurements, and a better understanding of money, science, and technology. In this section, we will learn some tips and tricks to master decimal fractions. 

• Understand the place value: In a decimal fraction, it is important to understand the place value, that the first digit after the decimal point is tenths, the second is hundredths, the third is thousandths, and so on.
   
• Practice shifting the decimal point: To move the decimal point, we multiply or divide the number by 10, 100, or 1000. For example, \(3.45 × 10 = 34.4\). 
   
• Rounding the decimal: Always round the number to the nearest whole number or nearest tenth for the quick answer. 
   
• Align decimals when calculating: When performing calculations with decimals, always write the numbers vertically with decimal points aligned when adding or subtracting. 
   
• Use visual aids: Decimals can be represented using a number line, charts, and grids, which make it easier to understand the concept.  

Decimal fractions simplify calculations, and shifting the decimal point allows for precise and accurate results. They are commonly used in daily life whenever exact measurements or values are required. Here are a few applications of decimal fractions.

• In finance, decimals are involved in prices, bills, and bank transactions. So, decimal fraction is used to calculate discounts, interest rates, and taxes for accuracy.
   
• Decimal fractions are used to measure length, weight, and volume. For example, 2.5 kg, 1.75 liters, 23.65 meters, etc.
   
• In science and medicine, medication doses and scientific measurements like temperature, speed, or mass are represented using decimals. For example, 2.25 mg, 36.7 °C, 25 N, 9.8 m/s².
   
• In sports and fitness, decimals are used to perform statistics, running time, distances, body measurements, weight tracking, and calorie counting also involve decimal precision. For example, 6.75 meters, 12.45 seconds, 30 kg.
   
• In technology and engineering, digital systems, electronics, and engineering projects often use decimal values for accuracy, for example, 3.6 volts, 2.25 mm components.