Classification of Fractions: In many real-life situations, every quantity to be measured cannot be an absolute whole number. In other words, we may be dealing with portions of a whole or parts of a whole. Fractions play an essential role in this. A fraction represents the portion or part of the whole thing.

A fraction has two parts, the numerator and the denominator. The number on the top is called the numerator, and the number on the bottom is called the denominator. Any whole quantity can be divided into parts, and each part is known as a fraction. Let us learn about the different types of fractions, like proper, improper, unit, mixed, equivalent, like, and unlike fractions.

Fraction

A fraction refers to the numerical quantity that is a part of something. We have a large rectangular piece of paper that we cut into nine equal pieces, and colour four parts dark pink and five parts light pink. Then, each part of the paper is the only \(\frac{1}{9}^{\text {th }}\) of the total paper. The dark pink coloured paper can be written as the \(\frac{4}{9}^{\text {th }}\) part of the paper and the light pink coloured paper can be written as the \(\frac{5}{9}^{\text {th }}\) part of the paper. The quantities representing the parts of the paper like \(\frac{1}{9}, \frac{3}{9}\), and \(\frac{5}{9}\) are fractions.

Parts of a Fraction

A fraction contains two parts, the numerator, and the denominator. The numerator is one of the parts placed on the top of the fraction bar, while another part placed on the bottom is called the denominator. 

The numerator shows the number of parts being considered, while the denominator shows the total number of parts in the whole.

In the above example, for \(\frac{1}{9}, 1\) is known as the numerator, and \(9\) is known as the denominator, for \(\frac{4}{9}, 4\) is known as the numerator, and \(9\) is known as the denominator. Similarly, for \(\frac{5}{9}, 5\) is known as the numerator, and \(9\) is known as the denominator.

We don’t always deal with whole objects in our regular life. Let us take another example of a whole number. It can give you a little bit more precise concept. Now, have a look at the figure.

Number \(7\) is a whole number, and it is divided into two parts that are \(4\) and \(3\). So, number \(7\) is a whole, and \(4\) and \(3\) are the parts of number \(7\). It means one of the parts represents \(3\) out of \(7\), and the other represents \(4\) out of \(7\). To express each part in fraction form, we need to write it as \(\frac{3}{7}\) and \(\frac{4}{7}\) respectively.

So, the numbers of the form \(\frac{x}{y}\) Where \(x\) and \(y\) are whole numbers, and \(y \neq 0\) are called fractions.

Here, \(x \rightarrow\) Numerator and \(y \rightarrow\) Denominator.

Classification of Fractions

We deal with parts or portions of whole objects very often. To quantify them, we need fractions. Although there are several types of fractions, the three most important types of fractions that are identified based on the numerator and the denominator are:

1. Proper Fraction
2. Improper Fraction
3. Mixed Fraction

Proper Fractions 

Proper fractions are those in which the numerator is always smaller than the denominator.

The condition of the proper fraction is \(\text {numerator}< \text {denominator}\).
For example, \(\frac{1}{2}, \frac{3}{4}, \frac{5}{6}, \frac{11}{20}, \frac{21}{25}\) are the proper fractions.

The fraction value of a proper fraction is always less than one.

For example, \(\frac{1}{2}=0.5, \frac{3}{4}=0.75, \frac{5}{6}=0.833, \frac{11}{20}=0.55, \quad \frac{21}{25}=0.84\)

We can see that for all these fractions, the value of the fraction is less than \(1\).

Improper Fractions

In an improper fraction, the numerator is always greater than or equal to the denominator. The condition of the improper fraction is the \(\text {numerator} \geq \text {denominator}\).
For example, \(\frac{3}{2}, \frac{7}{4}, \frac{6}{5}, \frac{15}{11}, \frac{10}{10}\) are the improper fractions.

According to the definition, it is improper if the fraction value is greater than or equal to one.

For example, \(\frac{3}{2}=1.5, \frac{7}{4}=1.75, \frac{6}{5}=1.2, \frac{15}{11} 1.36, \frac{10}{10}=1\)

We can see that for all these fractions, the value of the fraction is either \(1\) or greater than \(1\).

Mixed Fractions

A mixed fraction is a fraction that is a combination of both whole and a proper fraction in the same fraction. A mixed fraction has a value that is always greater than one.

For example, \(1 \frac{1}{2}, 2 \frac{3}{4}, 5 \frac{5}{6}\) are the mixed fractions.

Every mixed fraction can be converted to an improper fraction, and every improper fraction can be converted to a mixed fraction.

Note: To convert improper fractions to mixed fractions, we need to divide the numerator by the denominator. Then, we write it in the mixed number form by placing the quotient as the whole number, the remainder as the numerator, and the divisor as the denominator.

Other Classifications of Fractions

Depending on the denominator, we classify two or more fractions into like fractions and unlike fractions.

Like Fractions

The group of two or more fractions with the same denominators or identical denominators are called like fractions.

For examples, \(\frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5}, \frac{7}{5}\), etc., are all like fractions.

Fraction like \(\frac{3}{5}, \frac{3}{7}, \frac{3}{11}, \frac{3}{17}\) are not like fractions. Here the numerators are the same, but the denominators are different.

Unlike Fractions

The group of two or more fractions that have different denominators is unlike fractions. For examples, \(\frac{1}{6}, \frac{2}{4}, \frac{2}{5}, \frac{4}{7}, \frac{5}{8}\), etc., are all unlike fractions.

Fraction like \(\frac{3}{5}, \frac{3}{7}, \frac{3}{11}, \frac{3}{17}\) are unlike fractions as the denominators are different, even though the numerators are the same.

Unit Fractions

A unit fraction is any fraction with \(1\) as its numerator and a non-zero whole number as the denominator.

For example, \(\frac{1}{9}, \frac{1}{4}, \frac{1}{5}, \frac{1}{3}, \frac{1}{8}\), etc., are all unit fractions.

Decimal Fractions

A decimal fraction is a fraction whose denominator is a positive exponent of \(10\), such as \(10^{1}, 10^{2}, 10^{3}\) etc or a multiple of \(10\) like \(100,1,000,10,000\), etc. For example, \(\frac{2}{10}, \frac{15}{100}, \frac{3}{10}, \frac{67}{1000}\) are all decimal fractions.

Equivalent Fraction

Equivalent fractions are those fractions equal to the same value irrespective of their numerators and denominators.
Equivalent fractions are the fractions that have different numerators and different denominators but are equal to the same value when simplified or reduced.

For example, \(\frac{2}{4}, \frac{3}{6}, \frac{4}{8}\) are all equal to \(\frac{1}{2}\). So, these fractions are equivalent.

Let us multiply \(\frac{1}{2}\) by \(2,3,4,5\), respectively.

\(\frac{1}{2}=\frac{1 \times 2}{2 \times 2}=\frac{1 \times 3}{2 \times 3}=\frac{1 \times 4}{2 \times 4}=\frac{1 \times 5}{2 \times 5}\)

It shows that multiplying the numerator and the denominator of a fraction by the same nonzero number does not change the value of the fraction.

Similarly,\(\frac{2 \div 2}{4 \div 2}=\frac{3 \div 3}{6 \div 3}=\frac{4 \div 4}{8 \div 4}=\frac{5 \div 5}{10 \div 5}=\frac{1}{2}\)

It shows that dividing the numerator and the denominator of a fraction by the same nonzero number does not change the value of the fraction.

Solved Examples – Classification of Fractions

Q.1. 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 \(\frac{13}{5}, \frac{19}{8}, \frac{8}{5}\), and \(\frac{16}{3}\).

Q.2. 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 a mixed fraction.
So, the mixed fraction is \(4 \frac{3}{4}\).

Q.3. Identify the proper fractions from below.
\(\frac{5}{28}, \frac{13}{5}, \frac{9}{16}, \frac{19}{20}, \frac{8}{15}, \frac{16}{3}, 7 \frac{3}{4}\)
Ans: We know that fractions with the numerator less than the denominator are called improper fractions.
Therefore, the proper fractions are \(\frac{5}{28}, \frac{9}{16}, \frac{19}{20}, \frac{8}{15}\).

Q.4. Identify the like fractions from below.
\(\frac{5}{2}, \frac{3}{4}, \frac{9}{6}, \frac{6}{4}, \frac{8}{5}, \frac{16}{3}, 6 \frac{3}{4}\)
Ans: The group of two or more fractions that have the same denominators is like fractions.
Therefore, the like fractions are \(\frac{3}{4}, \frac{6}{4}, 6 \frac{3}{4}\).

Q.5. Express the following mixed fractions into improper fractions
(i) \(5 \frac{3}{4}\)
(ii) \(7 \frac{1}{7}\)
Ans: Given mixed fractions are \(5 \frac{3}{4}\) and \(7 \frac{1}{7}\).
To convert the mixed fractions to an improper fraction, multiply the denominator with the whole and add that to the numerator keeping the same denominator.
Mixed fractions are converted to an improper fraction as follows:
\(\frac{\text { (whole×denominator) + numerator) }}{\text { denominator }}\)
\( \Rightarrow \left( i\right)\,5\frac{3}{4} = \frac{{(5 \times 4) + 3}}{4} = \frac{{20 + 3}}{4} = \frac{{23}}{4}\)
\( \Rightarrow (ii)\,7\frac{1}{7} = \frac{{(7 \times 7) + 1}}{7} = \frac{{49 + 1}}{7} = \frac{{50}}{7}\)

Summary

In the above article, we learnt about the definition of a fraction, the classification of the fractions such as proper fractions, improper fractions, and mixed fractions. Then we saw like and unlike fractions, unit fractions, decimal fractions, and equivalent fractions. We learnt to identify the fractions based on their classifications.

Frequently Asked Questions (FAQ) – Classification of Fractions

Q.1. What do you understand by classification of fractions?
Ans: The classification of fractions means dividing the fractions into different groups based on different parameters.
Although there are several types of fractions, the fractions are mainly classified into three based on the numerator and the denominator. They are:
1. Proper fraction
2. Improper fraction
3. Mixed fraction
The other classification of fractions are
1. Like fractions
2. Unlike fractions
3. Unit fractions
4. Decimal fractions
5. Equivalent fractions

Q.2. What is an equivalent fraction?
Ans: Two or more fractions representing the same part of a whole or the fractions that provide the same value at their simplest forms are equivalent fractions.
\(\frac{1}{2}=\frac{2}{4}=\frac{3}{6}=\frac{4}{8}=\frac{5}{10}\) these fractions are called equivalent fractions.

Q.3. What are fractions? Explain with an example.
Ans: A fraction is a quantity that is used to represent a part of the whole. When we divide a whole into parts, then each part obtained is called a fraction.
Example: \(\frac{2}{3}\)

Q.4. What is like and unlike fraction?
Ans: Fractions with the same denominators are called the like fractions.
For example, \(\frac{3}{7}, \frac{5}{7}, \frac{6}{7}\) and \(\frac{8}{7}\) are like fractions.
Fractions with different denominators are called, unlike fractions.
For example, \(\frac{4}{11}, \frac{7}{15}, \frac{9}{17}\) and \(\frac{11}{28}\) are unlike fractions.

Q.5. What is a mixed fraction?
Ans: A mixed fraction is a fraction that is a combination of both whole and a proper fraction in the same fraction. A mixed fraction has a value that is always greater than one.
For example, \(1 \frac{1}{2}, 2 \frac{3}{4}, 5 \frac{5}{6}\) are the mixed fractions.

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