Proper Fraction

A fraction is called a proper fraction if its numerator is less than its denominator. The value of a proper fraction is always less than 1. For example, Sam got a chocolate bar and he divided it into four equal parts. He took one part and gave three parts to his sister Rachel. We represent Sam's portion as 1/4 and Rachel's portion as 3/4. Both these fractions are proper fractions because here the numerator is less than the denominator.

A fraction in which the numerator value is always less than the denominator value is known as a proper fraction. For example, 18/25, 19/45, 62/78, 1/6, 1/9, are proper fractions. A fraction consists of two parts, the numerator, and the denominator, and various types of fractions are identified on the basis of these two values. The two primary fractions which are distinguished on this criterion are proper and improper fractions.

Now, let us learn how to perform basic arithmetic operations on proper fractions.

Adding Proper Fractions

In order to add two proper fractions, we add the numerators if the given fractions are like fractions, i.e., if the denominators are the same. For example, 2/8 + 3/8 can be easily added by just adding the numerators since the denominators are the same. The sum of 2/8 + 3/8 will be 5/8.

However, to add proper fractions that are unlike, in which the denominators are different, we take the LCM (Least Common Multiple) of the denominators and rewrite the fractions as equivalent fractions using the LCM as the common denominator. Now, when all the denominators become the same, we add the numerators and write the result as the final numerator on top of the common denominator. For example, to add 2/3 + 4/5, we take the LCM of the denominators. The LCM of 3 and 5 is 15. Now, we multiply both the fractions with such a number (in this case, 5 and 3 respectively) so that the denominators become equal. This results in (10 + 12)/15 = 22/15.

Subtracting Proper Fractions

The subtraction of proper fractions is similar to addition. If we need to find the difference of proper fractions that are like fractions, we simply find the difference of the numerators, keeping the same denominator. For example, the difference of 6/9 - 4/9 will be 2/9.

Now, if we want to subtract proper fractions that are unlike, in which the denominators are different, we take the LCM (Least Common Multiple) of the denominators and rewrite the fractions as equivalent fractions using the LCM as the common denominator. When all the denominators become the same, we subtract the numerators and write the result on the common denominator. For example, to subtract 8/9 - 3/4, we take the LCM of the denominators. The LCM of 9 and 4 is 36. Now, we multiply both the fractions with such a number (in this case, 4 and 9 respectively) so that the denominators become equal. This results in (32 - 27)/36 = 5/36.

Multiplying Proper Fractions

Unlike addition and subtraction, the multiplication and division of proper fractions are easier in a way. We simply multiply the given numerators, then multiply the denominators, and finally simplify or reduce the resultant fraction. For example, to multiply 2/6 × 5/4, we multiply the numerators 2 and 5, to get 10, and we multiply the denominators 6 and 4 to get 24. The product is written as 10/24 which can be further reduced to 5/12.

Dividing Proper Fractions

The division of proper fractions is similar to multiplication. The only difference is that we change the division sign to the multiplication sign and then we multiply the first fraction with the inverse (reciprocal) of the second fraction. For example, let us divide: 4/9 ÷ 2/3. This will become 4/9 × 3/2 = 12/18. This can be further reduced to 2/3.

The opposite of a proper fraction is the improper fraction. When the numerator is less than the denominator, it results in a proper fraction and when it is equal to or greater than the denominator, we get an improper fraction. So, we can decide which one is a proper fraction and which is an improper fraction by comparing the values of numerator and denominator. An example of a proper fraction is 5/7, and when we take its reciprocal, i.e. 7/5, we get an improper fraction. Now let us learn the tabular difference between a proper fraction and an improper fraction.

Difference between Proper Fraction and Improper Fraction

Look at the table given below showing proper fraction vs. improper fraction.

Mathematically, an improper fraction can be converted to a mixed fraction, which is the combination of a whole number and a proper fraction. For example, let us convert an improper fraction 13/5 to a mixed fraction by using the following steps:

• Divide the numerator by the denominator. Here, on dividing 13 ÷ 5, we get 2 as the quotient and 3 as the remainder.
• Write the obtained quotient as the whole number and the remainder as the numerator on the same denominator. Here, the quotient (2) will be the whole number, the remainder (3) will be the new numerator and the denominator will be the same.
• This converts the given improper fraction to a proper fraction which has a whole number. Therefore, 13/5 can be written as \(2\dfrac{3}{5}\).

To add a mixed fraction to a proper fraction we simply convert the mixed fraction to an improper fraction and then we add the two fractions in the usual way of the addition of fractions. After finding the answers we again convert the result into a mixed fraction. For example, let us add \(2\dfrac{1}{5}\) + \(\dfrac{1}{4}\).

Convert the mixed fraction into an improper fraction that is \(2\dfrac{1}{5}\) = 11/5.

11/5 + 1/4 = (44 + 5)/20 = 49/20

Now, convert 49/20 into a mixed fraction which will be = \(2\dfrac{9}{20}\). This is how we add a mixed number to a proper fraction.

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Check these interesting topics related to proper fractions.

• Unit Fraction
• Decimals and Fractions
• Like Fractions and Unlike Fractions
• Comparing Fractions

What is a Proper Fraction with Example?

The fraction in which the numerator is less than the denominator is called a proper fraction. For example, 7/12, 7/8 are proper fractions.

What is the Difference between Proper Fraction and Improper Fraction?

The fraction in which the denominator is greater than the numerator is called a proper fraction. For example, 3/4, 27/39 are proper fractions. On the other hand, a fraction in which the numerator is greater than or equal to the denominator is called an improper fraction. For example, 18/5, 49/23, 46/46 are improper fractions.

How to Convert Improper Fraction to Proper Fraction?

Improper fractions can be converted to mixed fractions which is a combination of a whole number and a proper fraction. This is the only way we can convert an improper fraction to a proper fraction. To do this, we divide the numerator by denominator and write the quotient as a whole number, the remainder as the numerator, and the denominator will remain the same.

How to Add Proper Fractions with Different Denominators?

To add proper fractions with different denominators, we take the LCM of the denominators and convert the given fractions to their equivalents to make the denominator the same. Then, we write the sum of numerators over the common denominator.

How to Multiply Proper Fractions?

In order to multiply proper fractions, we multiply the given numerators, then multiply the denominators, and then reduce the fraction to its lowest terms. For example, to multiply 4/9 × 3/6, we multiply the numerators 4 and 3 to get 12, and we multiply the denominators 9 and 6 to get 54. The product is 12/54 which can be further reduced to 2/9.

How to Divide Proper Fractions?

We divide proper fractions in the same way as we multiply them. The only difference is that we multiply the first fraction with the reciprocal (inverse) of the second fraction. For example, let us divide: 2/3 ÷ 4/5. We will write the reciprocal of the second fraction and then multiply the fractions. This will give us 2/3 × 5/4 = 10/12. This can be further reduced to 5/6.

Is 12/8 a Proper Fraction?

No, 12/8 is not a proper fraction because the numerator (12) is greater than the denominator (8). It is an improper fraction.

Is 7/7 a Proper Fraction?

No, 7/7 is not a proper fraction because the numerator is equal to the denominator. It is considered as an improper fraction.

Is 5/8 a Proper Fraction?

Yes, 5/8 is a proper fraction because the numerator (5) is less than the denominator (8). Thus, 5/8 is a proper fraction.