The word fraction is derived from the Latin word ‘*fractio*’ meaning ‘to break’ and it refers to a part of a whole. The earliest civilization to use fractions was the Egyptians. Although fractions have been a part of our history for a long time, fractions were not considered numbers.

The way fractions are represented – two numbers written one above the other separated by a horizontal line was introduced in India, but without a line. Later on, the line was added by the Arabs.

## What is Fraction?

In Mathematics, fractions are represented as a numerical value, which defines a part of a whole. This **whole** can be either of the two:

- a thing
- a collection of objects

### Fraction as a Part of a Thing

When the whole is divided into equal parts, the number of parts we take makes up a fraction.

If a cake is divided into eight equal pieces and one piece of the cake is placed on a plate, then each plate is said to have $\frac {1}{8}$ of the cake. It is read as ‘one-eighth’ or ‘$1$ by $8$’.

### Fraction of a Collection of Objects

Let’s consider a group of $7$ children among which $3$ are boys and $4$ are girls.

It means $4$ out of $7$ are girls. So, the fraction of girls is four-sevenths $\left( \frac {4}{7} \right)$.

Similarly, $3$ out of $7$ are boys. So, the fraction of boys is three-sevenths $\left( \frac {3}{7} \right)$.

All these numbers $\frac {2}{8}$, $\frac {3}{4}$, $\frac {4}{5}$, $\frac {3}{11}$, and $\frac {17}{99}$ represent fractions.

## Examples of Fractions in Everyday Life

You may not even notice, but fractions are all around us! Some examples of everyday fractions include:

- Splitting a bill at a restaurant into halves, thirds, or quarters
- Working out price comparisons in the supermarket when something is half price
- Figuring out amounts in the kitchen – half teaspoon of sugar, a quarter cup of flour, etc.
- Adding up monetary amounts
- Looking at time! Half an hour and a quarter past are both common things to hear where time is concerned.

## Representation of Fraction – Parts of a Fraction

All fractions consist of a numerator and a denominator and they are separated by a horizontal bar known as the fractional bar.

**Numerator:** The numerator indicates how many sections of the fraction are represented or selected. It is placed in the upper part of the fraction above the fractional bar.**Denominator:** The denominator indicates the number of parts into which the whole has been divided. It is placed in the lower part of the fraction below the fractional bar.

## Types of Fractions

The numerator and denominator are the primary parts of any fraction. Based on these, different types of fractions can be defined. Let us look at some common types of fractions.

### Unit Fractions

The fractions in which the numerator is 1 are called unit fractions.

Examples of unit fractions are $\frac {1}{2}$ (one-half), $\frac {1}{3}$ (one-third), $\frac {1}{4}$ (one-fourth), $\frac {1}{8}$ (one-eighth).

### Proper Fractions

The fractions in which the numerator is less than the denominator are called proper fractions.

Examples of proper fractions are $\frac {12}{19}$, $\frac {2}{3}$, $\frac {5}{9}$, and $\frac {13}{79}$.

**Note:** All unit fractions are proper fractions.

### Improper Fractions

The fractions in which the numerator is greater than the denominator are called improper fractions. Improper fractions can be converted to mixed fractions.

Examples of improper fractions are $\frac {17}{7}$, $\frac {5}{2}$, $\frac {21}{9}$, and $\frac {99}{32}$.

**Note:** All improper fractions are greater than a whole.

### Mixed Fractions

The fractions which are a mixture of a whole number and a proper fraction are called mixed fractions. Mixed fractions can be converted to improper fractions.

Examples of mixed fractions are $5\frac {2}{3}$, $9\frac {5}{7}$, $25\frac {2}{3}$, and $16\frac {5}{9}$.

### Like Fractions

Two or more fractions having the same denominator are called like fractions.

For example in a group of fractions $\frac {1}{2}$, $\frac {2}{3}$, $\frac {4}{5}$, $\frac {5}{2}$, $\frac {2}{7}$, $\frac {5}{9}$, $\frac {2}{9}$, $\frac {3}{7}$, $\frac {1}{5}$

$\frac {1}{2}$ and $\frac {5}{2}$ are like fractions.

$\frac {4}{5}$ and $\frac {1}{5}$ are like fractions.

$\frac {2}{7}$ and $\frac {3}{7}$ are like fractions.

$\frac {5}{9}$ and $\frac {2}{9}$ are like fractions.

### Unlike Fractions

Two or more fractions having different denominators are called unlike fractions.

For example in a group of fractions $\frac {1}{2}$, $\frac {2}{3}$, $\frac {4}{5}$, $\frac {5}{2}$, $\frac {2}{7}$, $\frac {5}{9}$, $\frac {2}{9}$, $\frac {3}{7}$, $\frac {1}{5}$

$\frac {1}{2}$, $\frac {2}{3}$, $\frac {4}{5}$, $\frac {2}{7}$, $\frac {5}{9}$ are unlike fractions.

### Equivalent Fractions

Fractions that represent the same value after they are simplified are called equivalent fractions. To get equivalent fractions of any given fraction:

- we can multiply both the numerator and the denominator of the given fraction by the same number.
- we can divide both the numerator and the denominator of the given fraction by the same number.

For example, $\frac {1}{2}$, $\frac {2}{4}$, $\frac {3}{6}$, $\frac {12}{24}$, $\frac {50}{100}$ are all equivalent fractions. When simplified each of these reduces to $\frac {1}{2}$.

## Fractions on a Number Line

A number line is used to represent numbers – natural numbers, whole numbers, or integers. The number line can also be used to represent the fractions.

The fractions on a number line can be represented by making equal parts of a whole, i.e., from $0$ to $1$. The denominator of the fraction would represent the number of equal parts in which the number line will be divided and marked.

For example, if we want to represent $\frac {1}{4}$ on the number line, we divide the space between $0$ and $1$ into $4$ equal parts. Then, the first interval represents $\frac {1}{4}$. Similarly, the next interval represents $\frac {2}{4}$, the next one represents $\frac {3}{4}$, and the last one $\frac {4}{4}$ which is equal to $1$.

Again from $1$ to $2$, we divide the space into $4$ equal parts. The first interval after $1$ represents $\frac {5}{4}$. The second $\frac {6}{4}$, next $\frac {7}{4}$ and the last one $\frac {8}{4}$, which is equal $2$.

Similarly, we can divide the space between other numbers.

**Note**: The last interval representing $\frac {4}{4}$ means $1$. Any fraction of the form $\frac {a}{a}$ means $1$.

For example $\frac {2}{2}$, $\frac {7}{7}$, $\frac {19}{19}$ all represent $1$ (**the whole**).

## Conclusion

Fractions are numbers that represent a part of a whole. Depending on the relation between their numerator and denominator the fractions can be proper, improper, or mixed. When in a group the fractions can be classified as like, unlike, or equivalent.

## Practice Problems

- Which of the following are proper fractions? And which of them are improper?
- $\frac{7}{9}$
- $\frac{3}{7}$
- $\frac{5}{3}$
- $\frac{15}{37}$
- $\frac{19}{5}$

- Which of the following are represented correctly as mixed fractions?
- $3\frac{5}{9}$
- $1\frac{12}{7}$
- $4\frac{1}{2}$
- $\frac{5}{7}4$
- $8\frac{19}{4}$

- Write down the like fractions from the following. Also, write down pairs of unlike fractions
- $\frac{2}{5}$, $\frac{1}{2}$, $\frac{3}{5}$, $\frac{2}{3}$, $\frac{1}{7}$,$\frac{4}{5}$, $\frac{5}{7}$, $\frac{1}{3}$

Good