Proper and Improper Fractions: A fraction is a number representing a part of a whole. The whole may be a single object or a group of objects. The fraction representing four parts out of \(9\) equal parts in which the whole is divided can be denoted by \(\frac{4}{9}\).

The types of fractions are proper fractions, improper fractions, and mixed fractions. This article will study the definitions of proper fractions and improper fractions, examples of proper fractions and improper fractions, the difference between proper fractions and improper fractions, and solve some example problems on proper fractions and improper fractions.

What is a Fraction?

Sangliana had heard about fractions in his previous classes so he would try to use fractions whenever possible. One occasion was when he forgot his tiffin box at home. His friend Shankar invited him to share his lunch. He had five idlies in his lunch box. So, Sangliana and Shankar took two idlies each. Then, Shankar made two halves of the fifth idly and gave one half to Sangliana and took the other half himself. Thus, both Sangliana and Shankar had \(2\) full idlies and one-half idly.

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Sangliana knew that one-half is written as \(\frac{1}{2}\) and remembered that a fraction is a number representing part of a whole. The whole may be a single object or a group of objects. A fraction means a part of a group or a region. Example, \(\frac{3}{11}\) is a fraction. We read it as three-eleventh. Here, \(3\) is called the numerator, and \(11\) is called the denominator.

Example: We know that there are \(12\) hours in the daytime. If we need to know, what fraction of a day is \(7\) hours, we can write it as \(7\) hours \(= \frac{7}{12}\) of a daytime.

Fractions on Number Line

A number line is a straight line on which numbers are marked at equal intervals to illustrate simple numerical operations. To represent whole numbers on a number line, we draw a straight line and mark points \(0, 1, 2, 3, 4,…\) on the line at equal distances to the right of \(0\).

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Similarly, we can show fractions on a number line. To represent \(\frac{1}{2}\) on a number line, first, mark a point \(A\) to represent \(1\). Now, divide the gap between \(O\) and \(A\) into two equal parts. Let \(P\) the point of division. Then, point \(P\) represents \(\frac{1}{2}\).
To represent \(\frac{1}{3}\) on a number line, we split the gap between \(O\) and \(A\) into \(3\) equal parts. Let \(P\) and \(Q\) be the points of division. Then, \(P\) represents \(\frac{1}{3}\), and \(Q\) represents \(\frac{2}{3}\), because \(\frac{2}{3}\) means \(2\) sections out of \(3\) equal parts.

Proper Fractions

Fractions whose numerators are less than the denominators are called proper fractions.
Example: \(\frac{3}{4},\,\frac{1}{2},\,\frac{9}{{10}},\,\frac{0}{3},\,\frac{7}{8}\)

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If we represent all the proper fractions on a number line, it will lie to the left of \(1\). So, the value of the proper fraction is always less than \(1\).
A proper fraction is a number constituting part of a whole. In a proper fraction, the denominator represents the number of parts into which the whole is divided, and the numerator shows the number of parts that are considered. Therefore, in a proper fraction, the numerator is always less than the denominator.

Improper Fractions

Fractions with the numerator either equal to or greater than the denominator are called improper fractions.
Improper fractions are usually written in mixed number form, as mixed fractions are easier to understand.
Example: \(\frac{8}{5},\,\frac{5}{2},\,\frac{9}{2},\,\frac{3}{3},\) etc.
If we represent all the above fractions on a number line, it will lie on \(1\) or to the right of \(1\). So, the value of the improper fraction is always greater than \(1\).

Mixed Fractions

The sum of a whole number and a fraction is known as a mixed fraction. In other words, we can write a mixed fraction as
\({\text{Quotient}} + \frac{{{\text{Remainder}}}}{{{\text{Denominator}}}}\) or \({\text{Quotient}} \frac{{{\text{Remainder}}}}{{{\text{Denominator}}}}\)
Example: \(6 \frac{1}{8}\) is a mixed fraction, in which \(6\) is a whole number, \(1\) is a remainder and \(8\) will be the denominator.
We can convert the given mixed fraction into an improper fraction and vice versa. But we cannot convert a mixed fraction into a proper fraction.

Conversion of Improper Fractions into Mixed Fractions

Below are the steps used to convert an improper fraction to a mixed fraction.
1. Divide the numerator by the denominator.
2. Find the remainder from the above step.
3. Present the numbers in the following way, quotient followed by \(\frac{m}{n}\) Where \(m\) is the remainder and \(n\) is the denominator of the improper fraction.
Example: \(\frac{31}{9} = 3 \frac {4}{9}\)

Conversion of Mixed Fractions into Improper Fractions

Below are the steps used to convert a mixed fraction to an improper fraction.
1. Obtain the mixed fraction.
2. Identify the whole number part and the numerator and denominator of the fraction.
3. Multiply the whole number or the quotient part by the denominator of the fraction and add the result to the fraction’s numerator.
4. Write the fraction having a numerator equal to the number obtained in the above step and the denominator same as the denominator of the fraction given in the fraction.
So, improper fraction \( = \frac{{{\text{(Whole number }} \times {\text{ Denominator)}} + {\text{Numerator}}}}{{{\text{Denominator}}}}\)
Example: \(4 \frac {5}{3} = \frac {17}{3}\)

Difference between Proper Fraction and Improper Fraction

In proper fractions, the numerator is always less than the denominator, while in improper fractions, the numerator is always greater than the denominator.
We can convert the improper fractions to mixed fractions and vice versa, while we cannot convert the proper fractions into mixed fractions.

Solved Examples – Proper and Improper Fractions

Q.1. Express the below-given mixed fractions as improper fractions.
a. \(3 \frac{2}{7}\)
b. \(4 \frac{5}{9}\)
c. \(3 \frac{2}{5}\)
Ans:
a. Given: \(3 \frac{2}{7}\)
To convert a given mixed fraction to an improper fraction, we have,
Improper fraction \( = \frac{{{\text{(Whole number }} \times {\text{ Denominator)}} + {\text{Numerator}}}}{{{\text{Denominator}}}}\)
Here, whole number \(= 3\), numerator \(= 2\), denominator \(= 7\)
So, \(3 \frac{2}{7}\) can be written as \(\frac{{(3 \times 7) + 2}}{7} = \frac{{21 + 2}}{2} = \frac{{23}}{2}\)
Therefore, \(3 \frac{2}{7} = \frac{23}{2}\)
b. Given: \(4 \frac{5}{9}\)
To convert a given mixed fraction to an improper fraction, we have,
Improper fraction \( = \frac{{{\text{(Whole number }} \times {\text{ Denominator)}} + {\text{Numerator}}}}{{{\text{Denominator}}}}\)
Here, whole number \(= 4\), numerator \(= 5\), denominator \(= 9\)
So, \(4 \frac{5}{9}\) can be written as \(\frac{{(4 \times 9) + 5}}{9} = \frac{{36 + 5}}{9} = \frac{{41}}{9}\)
Therefore, \(4 \frac{5}{9} = \frac{{41}}{9}\)
c. Given:\(3 \frac{2}{5}\)
To convert a given mixed fraction to an improper fraction, we have,
Improper fraction \( = \frac{{{\text{(Whole number }} \times {\text{ Denominator)}} + {\text{Numerator}}}}{{{\text{Denominator}}}}\)
Here, whole number \(= 3\), numerator \(= 2\), denominator \(= 5\)
So, \(3 \frac{2}{5}\) can be written as \(\frac{{(3 \times 5) + 2}}{5} = \frac{{15 + 2}}{5} = \frac{{17}}{5}\)
Therefore, \(3 \frac{2}{5} = \frac{{17}}{5}\)

Q.2. Write the below improper fractions as mixed fractions:
a. \(\frac{17}{4}\)
b. \(\frac{13}{5}\)
c. \(\frac{28}{5}\)
Ans:
a. Given: \(\frac{17}{4}\)
If we divide \(17 ÷ 4\), we get

From the above division, we have quotient \(= 4\), remainder \(= 1\), denominator \(= 4\)
Therefore, \(\frac{17}{4} = 4 \frac{1}{4}\)
b. Given: \(\frac{13}{5}\)
If we divide \(13 ÷ 5\), we get
From the above division, we have quotient \(= 2\), remainder \(= 3\), denominator \(= 5\)
Therefore, \(\frac{13}{5} = 2 \frac{3}{5}\)
c. Given: \(\frac{28}{5}\)
If we divide \(28 ÷ 5\), we get
From the above division, we have quotient \(= 5\), remainder \(= 3\), denominator \(= 5\)
Therefore, \(\frac{28}{5} = 5 \frac{3}{5}\)

Q.3. Identify the proper fractions from below.
\(\frac{{13}}{5},\,\frac{9}{{16}},\,5\frac{3}{4},\,\frac{{19}}{8},\,\frac{8}{5},\,\frac{{16}}{3},\,\frac{3}{{28}}\)
Ans: We know that fractions whose numerators are less than the denominators are called proper fractions.
Therefore, the proper fractions are \(\frac{{9}}{16},\,\frac{3}{28}\).

Q.4. Identify the improper fractions from below.
\(\frac{5}{{28}},\,\frac{{13}}{5},\,\frac{9}{{16}},\,\frac{{19}}{8},\,\frac{8}{5},\,\frac{{16}}{3},\,7\frac{3}{4}\)
Ans: We know that fractions with the numerator equal to or greater than the denominator are called improper fractions.
Therefore, the improper fractions are \(\frac{{13}}{5},\,\frac{19}{8},\,\frac{{8}}{5},\,\frac{16}{3}\).

Q.5. Identify the mixed fractions from below.
\(\frac{{19}}{5},\,\frac{7}{{16}},\,4\frac{3}{4},\,\frac{{17}}{8},\,\frac{3}{5},\,\frac{{17}}{3},\,\frac{6}{{28}}\)
Ans: We know that the sum of a whole number and a fraction is known as a mixed fraction.
So, the mixed fraction is \(4 \frac{3}{4}\).

Summary

In the above article, we learned about the definition of a proper fraction, improper fraction, and mixed fraction. Also, we have learned the conversion between mixed fractions to improper fractions and vice versa. Also, we have studied the difference between proper and improper fractions and solved some example problems on the same.

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Frequently Asked Questions – Proper and Improper Fractions

Q.1. What is a proper and improper fraction?
Ans: The fractions whose numerators are less than the denominators are called proper fractions.
The fractions with the numerator either equal to or greater than the denominator are called improper fractions.

Q.2. How to identify proper and improper fractions?
Ans: In the given fraction, if the numerator is less than the denominator, then it is a proper fraction.
In the given fraction, if the numerator is greater than the denominator, then it is a proper fraction.

Q.3. What is the difference between a proper fraction and an improper fraction?
Ans: In proper fractions, the numerator is always less than the denominator, while in improper fractions, the numerator is always larger than the denominator.
We can convert the improper fractions into mixed fractions and vice versa, while we cannot convert proper fractions into mixed fractions.

Q.4. What are mixed fractions?
Ans: The sum of a whole number and a fraction is known as a mixed fraction.
Example: \(9 \frac{3}{2}\).

Q.5. How to convert mixed fractions to improper fractions?
Ans: Below given are the steps used to convert a mixed fraction to an improper fraction.
1. Obtain the mixed fraction.
2. Identify the whole number part and the numerator and denominator of the fraction.
3. Multiply the whole number or quotient part with the fraction’s denominator and add the result to the numerator of the fraction.
4. Write the fraction having a numerator equal to the number obtained in the above step and the denominator same as the denominator of the fraction given in the fraction.
So, improper fraction \( = \frac{{{\text{(Whole number }} \times {\text{ Denominator)}} + {\text{Numerator}}}}{{{\text{Denominator}}}}\)

Related Concepts:

Fraction in Simplest Form	Division of Fractions
Multiplication of Fractions	Conversion of Fractions
Addition and Subtraction of Fractions	Equivalent Fractions