Decimals and Fractions

A decimal fraction is a fraction where the denominator is a power of 10, such as 10, 100, or 1000. The bottom number of a fraction, known as the denominator, is a power of 10, such as 10, 100, 1000, and so on. In a decimal fraction, we write the fraction without the denominator by using a decimal point. Using decimals instead of fractions makes arithmetic operations like addition and multiplication easier. For example, 1/10 is a fraction, and we can write it as 0.1. Likewise, 4/100 can be written as 0.04. 
 

Using a decimal point (.), we separate whole numbers from the fractional part of a decimal number. The numbers to the right of the decimal point are called decimal places, and the numbers to the left are whole numbers or integers. The place values of whole numbers start from ones, then move to tens, hundreds, thousands, and so on. Each place value is ten times more than the previous number. However, the place values of decimal numbers start from tenths, hundredths, and thousandths. A decimal number can be read using the term “point.” For example, 68.54 is read as sixty-eight point five four. The given table represents the place values of decimal numbers. 
 

A portion of a whole is represented by a fraction, which is in the form a/b, where a is the numerator and b is the denominator. The numerator shows how many parts we have, while the denominator shows how many equal parts the whole is divided into. For example, 1/5, 3/8, and 7/9 are a few examples of fractions. 

Decimals and Fractions

We can represent every decimal number in the form of a fraction. To express a decimal number as a fraction, follow these steps: 

Step 1: Remove the decimal point and make the number a whole number.  

Step 2: Based on the decimal place value, write the number using the power of 10. 
For example, 0.46 is a decimal number, and the place value of the decimal places is in hundredths. So we can write this number as a fraction: 
0.46 = 46/100. 
If possible, we can simplify the fraction to its lowest form by finding the greatest common divisor of both numerator and denominator. Thus, 46/100 can be simplified to 23/50.
46/100 = 23/50 
We can also convert a recurring decimal to a fraction. For instance, we can write 0.444….as 4/9. 
Similarly, we can convert a fraction to a decimal by using the long division method or multiplication method. In the long division method, we divide the numerator by the denominator. However, in the multiplication method, we multiply both the numerator and denominator by a number that makes the denominator a power of 10. Then, the result will be written as a decimal.

For example, we can convert 3/5 into a decimal: 

 Step 1: To make the denominator a power of 10, multiply the numerator and denominator by 2. 
   (3 ×2 ) / (5 × 2) = 6/10 

Step 2: 6 divided by 10 gives 0.6. Hence, write the answer in the decimal form.
Thus, 3/5 = 0.6

Additionally, we can convert a fraction to a decimal by direct division. For example, convert 3/5 into a decimal:
  3 ÷ 5 = 0.6 

Decimals and Percentages 

Just like fractions, we can represent every decimal as a percentage. To compare two numbers easily, we  can convert the decimals to percentages. To convert a decimal to a percentage, follow these steps:

Step 1: Multiply the decimal number by 100.

Step 2: Place the % symbol on the answer. 
For instance, we can convert 0.67 to a percentage. 
      0.67 × 100 = 67%

We can also convert a percentage into a decimal by following these steps: 

Step 1: Divide the given percentage by 100. 

Step 2: Remove the % symbol.   
For example, we can convert 56% to a decimal. 
      56 ÷ 100 = 0.56 

To read a decimal, start with the numbers on the left of the decimal point. If there is no number or it is zero, we read the decimal part separately. For example, we read 235.87 as two hundred thirty-five point eight seven. After the decimal point, instead of combining the values of the digits, we read each digit individually after the decimal point. Similarly, we read 0.45 as zero point four five.   

How to Represent Decimal Fraction? 

In a decimal fraction, we write the fraction without its denominator by using a decimal point. The denominator of a decimal fraction is a power of 10. We can represent a decimal fraction as a fraction and the denominator will be a power of 10, such as 10, 100, 1000, etc. For example, 

0.7 = 7/10 
Furthermore, a decimal point can be used to represent a decimal fraction. For instance, 
30/100 = 0.30.
A decimal fraction can also be expressed in words. For instance, 0.56 is read as “fifty-six hundredths” because it represents 56/100. 

What are the Operations on Decimal Fractions?

We can perform the four basic mathematical operations: addition, subtraction, multiplication, and division, on decimals. 

Addition of decimal fraction: When we add two fractions together, we must first convert them into a decimal form. 
For example, add 35/100 and 35/1000.
35/100 = 0.35 
35/1000 = 0.035
Now we can easily add the decimals. 
    0.35 + 0.035 = 0.385 

Subtraction of decimal fractions: If we need to subtract fractions, we must convert them into decimal forms. 
For instance, subtract 35/1000 from 35/100. 
35/1000 = 0.035
35/100 = 0.35
Now, we can easily subtract the two decimals. 
      0.35 - 0.035 = 0.315 

Multiplication of decimal fraction: Depending on how many zeros there are in the power of 10, move the decimal point to the right when multiplying a decimal fraction.
For example, multiply 4.4321 × 100 = 443.21
Here, the power of 10 (100) has two zeros, so we move the decimal two places to the right.

Division of decimal fractions: Depending on how many zeros there are in the power, move the decimal point to the left when dividing a decimal fraction by a power of 10.
For instance, 4333.4 ÷ 100 = 43.334
Here, we shift the decimal two places to the left.  

What are the Types of Decimal Fractions?

The two main types of decimals are: 
Terminating decimal 
Non-terminating decimal 

a) Repeating decimal 
b) Non-repeating decimal 

Now, let us understand each of them in detail. 

Terminating decimal: The digits after the decimal point of a terminating decimal are limited and do not continue indefinitely. For example, 1.4, 2.54, 4.984, etc. 

Non-terminating decimal: The non-terminating decimals have an infinite number of digits after the decimal point. It goes infinitely and may or may not repeat again and again. For instance, 1.8666…, 4.782118…etc

Repeating decimals: The numbers after the decimal point of a repeating decimal follow a pattern in which the digits repeat endlessly. For example, 1. 323232…, 1.151515….etc.

Non-repeating decimals: Non-repeating decimals continue endlessly but should not follow a specific pattern. For example, 1.252552555…., 1.141441444 ... .etc. 
 

In our daily lives, we use decimals and fractions in various situations, from counting to distributing resources fairly. Here are some real-world applications of decimals and fractions: 

• In finance and banking, decimals and fractions are used to represent money accurately, such as $45.50 cents. Additionally, bankers use fractions and decimals to calculate interest rates, loan amounts, and savings balances for account holders. For example, if a bank offers 2.5% annual interest on a savings account, it means customers can earn $2.50 for every $100 saved each year.  

• When we cook, we can measure ingredients precisely using fractions and decimals. For example, baking a cake requires 1/5 cup of flour and 1.5 liters of milk, and measuring them in decimals and fractions ensures an exact measurement. 

• Companies and stores calculate the price of items with discounts, offers, and taxes, such as 25% off or a $45 discount. Fractions are used to divide items fairly, such as dividing a 10-pack of items into 1/2 and 1/4 portions. 

• In sports, coaches and players can track performance using decimal values that show accurate timing and scores. For example, an athlete completes a 100 m sprint in 8.39 seconds helps coaches and players to figure out areas for improvement.