Fractions: Definitions, Types, and Examples

Fraction ​

we shall learn about the types of fractions, ordering of fractions and the basic arithmetic operations in fractions, etc.

• Imagine that you have invited five friends to your birthday party. You have cut the cake into six equal pieces. Each person gets one piece out of a total of six pieces. In other words, each person gets one-sixth of the cake. We know that ${\displaystyle \frac{1}{6}}$ is a fraction. Here, cake is considered as whole and a piece is a part. ${\displaystyle \frac{3}{4}}$, ${\displaystyle \frac{4}{5}}$, ${\displaystyle \frac{4}{9}}$ and ${\displaystyle \frac{21}{2}}$ are some examples of fractions.
• A mark of 9 out of 14 in an examination may be written as ${\displaystyle \frac{9}{14}}$ or 9/14.
• Suppose you cute a whole pizza into 6 pieces and eat 5 of the pieces.

${\displaystyle \frac{9}{14}}$ is an example of a fraction. The number above the line, i.e. 9, is called the numerator. The number below the line, i.e. 14 is called the denominator.

A number of the form ${\displaystyle \frac{a}{b}}$, where a and b are whole numbers and b ≠ 0 is a fraction. The number a is the numerator and b is the denominator

A Fraction is a number representing a part of a whole. This whole may be a single object or a group of objects

Types of Fraction ​

1. Proper Fraction ​

A fraction that represents a part of whole is a proper fraction.

A fraction whose numerator is less than its denominator is called a proper fraction.

2. Improper Fraction ​

A fraction that represents a combination of a whole and a part of the whole is an improper fraction.

A fraction whose numerator is greater than or equal to its denominator is called an improper fraction.

A fraction whose numerator is less than the denominator is called a proper fraction, otherwise it is called an improper fraction.

3. Mixed Fraction ​

When an improper fraction is written as an integer followed by a proper fraction, it is a mixed fraction.

A combination of a whole number and a proper fraction is called a mixed fraction.

Converting Mixed Fractions and Improper Fractions ​

To convert a mixed fraction into an improper fraction ​

Multiply the whole number with the denominator of the fraction and to this product add the numerator of the fraction. This gives the numerator of the required improper fraction. Its denominator is the same as the denominator of the fractional part.

Equivalent Fractions ​

Two or more fractions representing the same part of a whole are called equivalent fractions.

RULE To get a fraction equivalent to a given fraction, we multiply or divide the numerator and the denominator of the given fraction by the same non-zero number.

Comparing Fractions ${\displaystyle \frac{a}{b}}$ and ${\displaystyle \frac{c}{d}}$ ​

1. If ad > bc, then ${\displaystyle \frac{a}{b}}$ > ${\displaystyle \frac{c}{d}}$
2. If ad = bc, then ${\displaystyle \frac{a}{b}}$ = ${\displaystyle \frac{c}{d}}$
3. If ad < bc, then ${\displaystyle \frac{a}{b}}$ < ${\displaystyle \frac{c}{d}}$

Like and Unlike Fractions ​

Like Fractions ​

Fractions with the same denominators are called like fractions.

Thus, ${\displaystyle \frac{2}{9}}$, ${\displaystyle \frac{4}{9}}$, ${\displaystyle \frac{5}{9}}$, ${\displaystyle \frac{7}{9}}$ are all like fractions.

Unlike Fractions ​

Fractions with different denominators are called unlike fractions.

Thus, ${\displaystyle \frac{1}{2}}$, ${\displaystyle \frac{3}{4}}$, ${\displaystyle \frac{7}{8}}$, ${\displaystyle \frac{2}{3}}$ are all unlike fractions.

Operation of Fractions ​

1. Addition of Fractions ​

Addition of Like Fractions ​

When the denominators of two (or more) fractions to be added are the same, the fractions can be added ‘on sight’.

Example :

• ${\displaystyle \frac{3}{7}}$ + ${\displaystyle \frac{5}{7}}$ = ${\displaystyle \frac{3+5}{7}}$ = ${\displaystyle \frac{8}{7}}$
• ${\displaystyle \frac{3}{7}}$ + 2 = ${\displaystyle \frac{3+2\times 7}{7}}$ = ${\displaystyle \frac{3+14}{7}}$ = ${\displaystyle \frac{17}{7}}$
• 2 + ${\displaystyle \frac{3}{7}}$ = ${\displaystyle \frac{2\times 7+3}{7}}$ = ${\displaystyle \frac{14+3}{7}}$ = ${\displaystyle \frac{17}{7}}$

2. Subtraction of Fraction ​

• ${\displaystyle \frac{5}{7}}$ − ${\displaystyle \frac{3}{7}}$ = ${\displaystyle \frac{5-3}{7}}$ = ${\displaystyle \frac{2}{7}}$
• ${\displaystyle \frac{17}{7}}$ − 2 = ${\displaystyle \frac{17-2\times 7}{7}}$ = ${\displaystyle \frac{17-14}{7}}$ = ${\displaystyle \frac{3}{7}}$
• 2 − ${\displaystyle \frac{3}{7}}$ = ${\displaystyle \frac{2\times 7-3}{7}}$ = ${\displaystyle \frac{14-3}{7}}$ = ${\displaystyle \frac{11}{7}}$

3. Multiplication of Fraction ​

• ${\displaystyle \frac{5}{7}}$ × ${\displaystyle \frac{3}{7}}$ = ${\displaystyle \frac{5\times 3}{7\times 7}}$ = ${\displaystyle \frac{15}{49}}$
• ${\displaystyle \frac{17}{7}}$ × 2 = ${\displaystyle \frac{17\times 7}{7}}$ = ${\displaystyle \frac{119}{7}}$
• 2 × ${\displaystyle \frac{3}{7}}$ = ${\displaystyle \frac{2\times 3}{7}}$ = ${\displaystyle \frac{6}{7}}$

4. Division of Fraction ​

• ${\displaystyle \frac{5}{7}}$ ÷ ${\displaystyle \frac{3}{7}}$ = ${\displaystyle \frac{5}{7}}$ × ${\displaystyle \frac{7}{3}}$ = ${\displaystyle \frac{5\times \cancel{7}}{\cancel{7}\times 3}}$ = ${\displaystyle \frac{5}{3}}$
• ${\displaystyle \frac{17}{7}}$ ÷ 2 = ${\displaystyle \frac{17}{7}}$ × ${\displaystyle \frac{1}{2}}$ = ${\displaystyle \frac{17}{7\times 2}}$ = ${\displaystyle \frac{17}{14}}$
• 2 ÷ ${\displaystyle \frac{3}{7}}$ = 2 × ${\displaystyle \frac{7}{3}}$ = ${\displaystyle \frac{2\times 7}{3}}$ = ${\displaystyle \frac{14}{3}}$

Conclusion ​

Here is a summary of the fractions:

• Number of the form ${\displaystyle \frac{a}{b}}$ where b ≠ 0 are called fractions.
• Types of fractions:
  1. Proper fraction : A fraction $\frac{a}{b}$ in which b ≠ 0 and a < b
  2. Improper fraction : A fraction $\frac{a}{b}$ in which b ≠ 0 and a > b
  3. Mixed fraction : A fraction which can be expressed as the sum of a natural number and a fraction.
• Operation of fraction: Addition, Subtraction, Product, Division