We present examples on how to use the rules of fractions
to simplify expressions including fractions. Questions and their solutions with detailed explanations are also included.

Fractions are some of the most important concepts in mathematics and therefore must be very well understood. Students need to have strong skills related to operations on fractions such as adding, dividing, equivalent fractions, ... . The skills discussed here gives you the much needed basics in dealing with fraction in numerical form as well as fractions with variables.

There are rules and definitions, the idea is to go through them in the order that they are presented and understand each one before you move to the next one.

In order to develop strong skills in fractions, there rules and any other rules on fractions, are to be fully understood and used to practice on more examples and questions for a long as you are studying mathematics.

- Numerator and Denominator of a Fraction

A fraction is written in the form \[ {\Large \dfrac{a}{b}} \] where \( a \) and \( b \) are integers. \( a \) is called the numerator and \( b \) the denominator which can never be equal to zero.

- The Denominator of a Fraction is Never Equal to Zero

These fractions are undefined because their denominators are equal to zero and are therefore NOT allowed in mathematics.

\( \quad \ccancel[red]{\dfrac{2}{0}} \) , \( \ccancel[red]{\dfrac{0}{0}} \) , \( \ccancel[red]{\dfrac{1000000}{0}} \)

- The Numerator of a Fraction May be Equal to Zero

Any fraction whose numerator is equal to zero is itself equal to zero as long as its denominator is not equal to zero.

\( \quad \dfrac{0}{3} = 0 \) , \( \dfrac{0}{100000} = 0 \) , \( \dfrac{0}{-8} = 0 \)

The fraction \( \ccancel[red]{\dfrac{0}{0}} \) is undefined because its denominator is equal to zero.

- A Fraction with Denominator Equal to 1 Simplies to an Integer

Any fraction whose denominator is equal to 1 may be written as an integer

\( \quad \dfrac{4}{1} = 4 \) , \( \dfrac{9}{1} = 9 \) , \( \dfrac{2x}{1} = 2x \)

- A Fraction with Denominator Equal its Numerator

Any fraction whose denominator is equal to its denominator simplifies to 1

\( \quad \dfrac{6}{6} = 1 \) , \( \dfrac{3x}{3x} = 1 \) for \( x \ne 0 \)

- Equivalent Fractions

Two fractions \( \dfrac{a}{b} \) and \( \dfrac{c}{d} \) are equivalent and may be written as \( \dfrac{a}{b} = \dfrac{c}{d} \) if and only if \( \quad a \times d = b \times c\)

Example

a) The two fractions \( \dfrac{\color{blue}2}{\color{brown}5} \) and \( \dfrac{\color{red}6}{15} \) are equivalent because \( \quad \color{blue}2 \times 15 = 30\) and \( \color{brown}5 \times \color{red}6 = 30\) and therefore \( \color{blue}2 \times 15 = \color{brown}5 \times \color{red}6 \)

b) The two fractions \( \dfrac{\color{blue}{2x}}{\color{brown}3} \) and \( \dfrac{\color{red}4x}{6} \) are equivalent because \( \quad \color{blue}{2x} \times 6 = 12x\) and \( \color{brown}3 \times \color{red}{4x} = 12 x \) and therefore \( \color{blue}{2x} \times 6 = \color{brown}3 \times \color{red}{4x} \)

- How to Make Equivalent Fractions by Mulitplication

You can make equivalent fractions by multiplying the numerator and denominator of the given fraction by the same number \( k \) , \( k \ne 0 \):

\( \quad \dfrac{a}{b} = \dfrac{a\color{red}{\times k}}{b \color{red} {\times k}} \)

There are many ways to write equivalent fractions by multiplication depending on the values of \( k \).

Example

a) \( \quad \dfrac{3}{7} = \dfrac{3 \color{red}{\times 5}}{7 \color{red}{\times 5} } = \dfrac{15}{35} \)

b) \( \quad \dfrac{2}{3} = \dfrac{2 \color{red}{\times 2x}}{3 \color{red}{\times 2x} } = \dfrac{4x}{6x} \) for \( x \ne 0\)

- How to Make Equivalent Fractions by Division

You can make equivalent fractions by dividing the numerator and denominator of the given fraction by the same number \( k \) , \( k \ne 0 \):

\( \quad \dfrac{a}{b} = \dfrac{a \color{red}{\div k}}{b \color{red}{\div k}} \)

Example

a) \( \quad \dfrac{8}{12} = \dfrac{8 \color{red}{\div 4}}{12 \color{red}{\div 4} } = \dfrac{2}{3} \)

b) \( \quad \dfrac{x}{14x} = \dfrac{x \color{red}{\div x}}{14x \color{red}{\div x} } = \dfrac{1}{14} \) for \( x \ne 0\)

- Reciprocal of a Fraction

The reciprocal of a fraction \( \dfrac{a}{b} \) is equal to \( \quad \color{red}{ \dfrac{b}{a} } \) for \( a \ne 0 \).

Note: The product of the fraction \( \dfrac{a}{b} \) and its reciprocal \( \color{red}{ \dfrac{b}{a} } \) is equal to 1.

\[ \dfrac{a}{b} \times \color{red}{ \dfrac{b}{a} } = 1 \]

Note: The reciprocal of a fraction whose denominator is equal to zero is undefined.

Example

a) The reciprocal of \( \dfrac{7}{9} \) is \( \dfrac{9}{7} \)

and \( \dfrac{7}{9} \times \dfrac{9}{7} = \dfrac{7 \times 9}{9 \times 7} = \dfrac{63}{63} = 1\)

b) The reciprocal of \( \dfrac{0}{7} \) is undefined because the numeartor of the given fraction is equal to zero.

c) The reciprocal of \( \dfrac{x}{2} \) is \( \dfrac{2}{x} \) , for \( x \ne 0 \)

and \( \dfrac{x}{2} \times \dfrac{2}{x} = \dfrac{x \times 2}{2 \times x} = \dfrac{2x}{2x} = 1\)

- Write an Integer as a Fraction

Any integer \( a \) may be written as a fraction as follows:

\( \quad a = \dfrac{a \color{red}{\times k}}{ \color{red}k} \) for any integer \( k \ne 0\)

There are many ways to write an integer as a fraction

Example

a) \( \quad 3 = \dfrac{3 \times 1}{1} = \dfrac{3}{1} \)

b) \( \quad 3 = \dfrac{3 \times 4}{4} = \dfrac{12}{4} \)

c) \( \quad 5 = \dfrac{5 \times x}{x} = \dfrac{5x}{x} \) for \( x \ne 0 \)

- Write a Decimal Number as a Fraction

A decimal number \( a \) may be written as a fraction by first writing it as a division by 1 and then mutliply the top and bottom of the division by a multiple of 10 such that the decimal number becomes an integer.

Example

a) \( \quad 0.1 = \dfrac{0.1 }{1} = \dfrac{0.1 \times 10}{1 \times 10 } = \dfrac{1}{10} \)

b) \( \quad 2.09 = \dfrac{2.09 }{1} = \dfrac{2.09 \times 100 }{1 \times 100} = \dfrac{209}{100}\)

c) \( \quad 0.0137 = \dfrac{0.0137}{1} = \dfrac{0.0137 \times 10000}{1 \times 10000} = \dfrac{137}{10000} \)

- Add Fractions

a)__Add Fractions with Common Denominators__

We add fractions with common denominators by keeping the common denominator and adding the numerators as follows:

\( \quad \dfrac{a}{c} + \dfrac{b}{c} = \dfrac{ \color{red}{a+b}}{c}\)

Example

\( \quad \dfrac{2}{8} + \dfrac{1}{8} = \dfrac{2+1}{8} = \dfrac{3}{8} \)

b)__Add Fractions with Different Denominators__

We add fractions with different denominators by first writing the two fractions to add with a common denominator then add them.

\( \quad \dfrac{a}{\color{red} c} + \dfrac{b}{\color{blue} d} = \dfrac{a}{c} \color{blue}{ \times \dfrac{d}{d}} + \dfrac{b}{d} \color{red}{ \times \dfrac{c}{c}} \)

\( = \dfrac{a \times d}{c \times d} + \dfrac{b \times c}{d \times c} = \dfrac{a \times d + b \times c}{c\times d} \)

Example

\( \quad \dfrac{2}{3} + \dfrac{1}{5} \)

\( = \dfrac{2}{3} \color{red}{ \times \dfrac{5}{5}} + \dfrac{1}{5} \color{red}{ \times \dfrac{3}{3}} \)

\( = \quad \dfrac{2 \times 5 + 1 \times 3 }{3 \times 5} = \dfrac{13}{15}\)

- Subtract Fractions

a)__Subtract Fractions with Common Denominators__

We subtract fractions with common denominators by keeping the common denominator and subtracting the numerator as follows:

\( \quad \dfrac{a}{c} - \dfrac{b}{c} = \dfrac{ \color{red}{ a - b}}{c}\)

Example

\( \quad \dfrac{11}{7} - \dfrac{6}{7} = \dfrac{11-6}{7} = \dfrac{5}{7} \)

b)__Subtract Fractions with Different Denominator__

We subtract fractions with different denominators by first writing the two fractions to subtract with a common denominator and then subtract them.

\( \quad \dfrac{a}{\color{red}c} - \dfrac{b}{ \color{blue} d} = \dfrac{a}{c} \color{blue}{ \times \dfrac{d}{d}} - \dfrac{b}{d} \color{red}{ \times \dfrac{c}{c} } \)

\( = \dfrac{a \times d}{c \times d} - \dfrac{b \times c}{d \times c} = \dfrac{a \times d - b \times c}{c\times d} \)

Example

\( \quad \dfrac{4}{3} - \dfrac{2}{5} \)

\( = \dfrac{4}{3} \color{red}{ \times \dfrac{5}{5}} - \dfrac{2}{5} \color{red}{ \times \dfrac{3}{3} } \)

\( \quad =\dfrac{20}{15} - \dfrac{6}{15} = \dfrac{14}{15} \)

- Multiply Fractions

We multiply fractions by multiplying numerator by numerator and denominator by denominator as follows:

\( \quad \dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{\color{red}{a \times c}}{\color{red}{b \times d}}\)

Example

\( \quad \dfrac{2}{5} \times \dfrac{4}{7} = \dfrac{2 \times 4}{5 \times 7} = \dfrac{8}{35} \)

- Divide Fractions

We divide fractions by mutliplying the first fraction by the reciprocal of the second fraction as follows:

\( \quad \dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \color{red}{ \times \dfrac{d}{c}} \)

Example

\( \quad \dfrac{7}{2} \div \dfrac{2}{5} = \dfrac{7}{2} \color{red}{\times \dfrac{5}{2}} = \dfrac{7 \times 5}{2 \times 2} = \dfrac{35}{4} \)

- Add a Number and a Fraction

We add a number and a fraction by rewriting the number as a fraction with common denominator then add as follows.

\( \quad a + \dfrac{b}{c} = \color{red} {a \times \dfrac{c}{c}} + \dfrac{b}{c} = \dfrac{a c }{c} + \dfrac{b}{c} = \dfrac{a c + b}{c} \)

Example

\( \quad 2 + \dfrac{3}{4} = \color{red} {2 \times \dfrac{4}{4}} + \dfrac{3}{4} \)

\( \quad \quad = \dfrac{2 \times 4 }{4} + \dfrac{3}{4} = \dfrac{8}{4} + \dfrac{3}{4} \)

\( \quad \quad \quad = \dfrac{8 + 3}{4} = \dfrac{11}{4} \)

- Multiply a Number by a Fraction

We multiply a number by a fraction by rewriting the number as a fraction then multiply as follows .

\( \quad a \times \dfrac{b}{c} = \color{red}{ \dfrac{a}{1}} \times \dfrac{b}{c} = \dfrac{a b}{c} \)

Example

\( \quad 7 \times \dfrac{2}{5} = \color{red}{ \dfrac{7}{1} } \times \dfrac{2}{5} = \dfrac{7 \times 2}{1 \times 5} = \dfrac{14}{5} \)

- Divide a Number by a Fraction

We divide a number by a fraction by first rewriting the number as a fraction then divide as follows .

\( \quad a \div \dfrac{b}{c} = \color{red}{ \dfrac{a}{1} } \div \dfrac{b}{c} = \dfrac{a}{1} \times \dfrac{c}{b} = \dfrac{a c}{b} \)

Example

\( \quad 2 \div \dfrac{3}{11} = \color{red}{ \dfrac{2}{1}} \times \dfrac{11}{3} = \dfrac{2 \times 11}{1 \times 3} = \dfrac{22}{3} \)

- Divide a Fraction by a Number

We divide a fraction by a number by first rewriting the number as a fraction then divide as follows .

\( \quad \dfrac{a}{b} \div c = \dfrac{a}{b} \div \color{red}{ \dfrac{c}{1}} = \dfrac{a}{b} \times \dfrac{1}{c} = \dfrac{a}{b c} \)

Example

\( \quad \dfrac{2}{7} \div 3 = \dfrac{2}{7} \div \color{red}{ \dfrac{3}{1} } = \dfrac{2}{7} \times \dfrac{1}{3} = \dfrac{2}{21} \)

- Signed Fractions

Signed fractions may be written in any of the following forms:

a)

\( \quad - \dfrac{a}{b} = \dfrac{- a}{b} = \dfrac{a}{-b} \)

Example

\( \quad - \dfrac{5}{12} = \dfrac{-5}{12} = \dfrac{5}{-12} \)

b)

\( \quad \dfrac{ - a}{ - b} = \dfrac{a}{b} \)

Example

\( \quad \dfrac{-2}{-7} = \dfrac{2}{7} \)

- Write the follwoing as a single fraction and reduce it to lowest terms if possible. Solutions are included.
- ) \( \dfrac{0}{3} + \dfrac{1}{3} \)

- ) \( \dfrac{2}{0} + 5 \)

- ) \( \dfrac{3}{5} + 2 \)

- ) \( \dfrac{3}{2} + 2.1 \)

- ) \( 0.1 x + \dfrac{2x}{3}\)

- ) \( 3 x + \dfrac{x}{4} \)

- ) \( 3x - \dfrac{5 x}{4} \)

- ) \( \dfrac{3}{5} \times \dfrac{4}{9} \)

- ) \( 6 \times \dfrac{3}{7} \)

- ) \( \dfrac{2x}{5} \times \dfrac{1}{2} \)

- ) \( \dfrac{6}{7} \div 3 \)

- ) \( x \div \dfrac{1}{9} \)

- ) \( \dfrac{2x}{5} \div \dfrac{1}{9} \)

- ) \( - \dfrac{-3}{5} + \dfrac{-3}{5} \)

- ) \( \dfrac{2}{-9} + \dfrac{7}{9} \)

- ) \( \dfrac{-5}{-2} - \dfrac{7}{2} \)

- ) \( \dfrac{-2x}{3} - \dfrac{ - 5x}{- 3} \)

- ) \( 2 - \dfrac{ 4 + \dfrac{1}{3}}{1+\dfrac{1}{2}} \)

- ) \( x - \dfrac{ 2x + \dfrac{x}{2}}{x - \dfrac{2x}{3}} \)

- ) \( \dfrac{0}{3} + \dfrac{1}{3} \)

Equivalent Fractions Examples and Questions

Reduce Fractions

Fractions with Variables

Adding Fractions

Divide Fractions

Multiply Fractions