Fraction Rules, Examples, and Detailed Solutions

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, and more. The skills discussed here give you the much-needed basics in dealing with fractions 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. In order to develop strong skills in fractions, these rules and any other rules on fractions must be fully understood and used to practice on more examples and questions for as long as you are studying mathematics.

Fraction Rules and Properties

1. 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.

2. 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.

\[ \dfrac{2}{0} \quad (\text{undefined}), \quad \dfrac{0}{0} \quad (\text{undefined}), \quad \dfrac{1000000}{0} \quad (\text{undefined}) \]

3. 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.

\[ \dfrac{0}{3} = 0, \quad \dfrac{0}{100000} = 0, \quad \dfrac{0}{-8} = 0 \]

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

4. A Fraction with Denominator Equal to 1 Simplifies to an Integer

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

\[ \dfrac{4}{1} = 4, \quad \dfrac{9}{1} = 9, \quad \dfrac{2x}{1} = 2x \]

5. A Fraction with Denominator Equal to its Numerator

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

\[ \dfrac{6}{6} = 1, \quad \dfrac{3x}{3x} = 1 \quad \text{for } x \ne 0 \]

6. 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 \( a \times d = b \times c \).

Example:
a) The two fractions \( \dfrac{2}{5} \) and \( \dfrac{6}{15} \) are equivalent because \( 2 \times 15 = 30 \) and \( 5 \times 6 = 30 \), therefore \( 2 \times 15 = 5 \times 6 \).
b) The two fractions \( \dfrac{2x}{3} \) and \( \dfrac{4x}{6} \) are equivalent because \( 2x \times 6 = 12x \) and \( 3 \times 4x = 12x \), therefore \( 2x \times 6 = 3 \times 4x \).

7. How to Make Equivalent Fractions by Multiplication

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

\[ \dfrac{a}{b} = \dfrac{a \times k}{b \times k} \]

Example:
a) \( \dfrac{3}{7} = \dfrac{3 \times 5}{7 \times 5} = \dfrac{15}{35} \)
b) \( \dfrac{2}{3} = \dfrac{2 \times 2x}{3 \times 2x} = \dfrac{4x}{6x} \) for \( x \ne 0 \)

8. 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 \( k \), where \( k \ne 0 \):

\[ \dfrac{a}{b} = \dfrac{a \div k}{b \div k} \]

Example:
a) \( \dfrac{8}{12} = \dfrac{8 \div 4}{12 \div 4} = \dfrac{2}{3} \)
b) \( \dfrac{x}{14x} = \dfrac{x \div x}{14x \div x} = \dfrac{1}{14} \) for \( x \ne 0 \)

9. Reciprocal of a Fraction

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

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

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

Example:
a) The reciprocal of \( \dfrac{7}{9} \) is \( \dfrac{9}{7} \), and \( \dfrac{7}{9} \times \dfrac{9}{7} = \dfrac{63}{63} = 1 \).
b) The reciprocal of \( \dfrac{0}{7} \) is undefined because the numerator of the given fraction is equal to zero.
c) The reciprocal of \( \dfrac{x}{2} \) is \( \dfrac{2}{x} \) for \( x \ne 0 \), and \( \dfrac{x}{2} \times \dfrac{2}{x} = \dfrac{2x}{2x} = 1 \).

10. Write an Integer as a Fraction

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

\[ a = \dfrac{a \times k}{k} \quad \text{for any integer } k \ne 0 \]

Example:
a) \( 3 = \dfrac{3 \times 1}{1} = \dfrac{3}{1} \)
b) \( 3 = \dfrac{3 \times 4}{4} = \dfrac{12}{4} \)
c) \( 5 = \dfrac{5 \times x}{x} = \dfrac{5x}{x} \) for \( x \ne 0 \)

11. Write a Decimal Number as a Fraction

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

Example:
a) \( 0.1 = \dfrac{0.1}{1} = \dfrac{0.1 \times 10}{1 \times 10} = \dfrac{1}{10} \)
b) \( 2.09 = \dfrac{2.09}{1} = \dfrac{2.09 \times 100}{1 \times 100} = \dfrac{209}{100} \)
c) \( 0.0137 = \dfrac{0.0137}{1} = \dfrac{0.0137 \times 10000}{1 \times 10000} = \dfrac{137}{10000} \)

12. Add Fractions

a) Add Fractions with Common Denominators: Keep the common denominator and add the numerators:

\[ \dfrac{a}{c} + \dfrac{b}{c} = \dfrac{a + b}{c} \]

b) Add Fractions with Different Denominators: Rewrite the fractions with a common denominator before adding:

\[ \dfrac{a}{c} + \dfrac{b}{d} = \dfrac{a \times d}{c \times d} + \dfrac{b \times c}{d \times c} = \dfrac{a \times d + b \times c}{c \times d} \]

13. Subtract Fractions

a) Subtract Fractions with Common Denominators: Keep the common denominator and subtract the numerators:

\[ \dfrac{a}{c} - \dfrac{b}{c} = \dfrac{a - b}{c} \]

b) Subtract Fractions with Different Denominators: Rewrite the fractions with a common denominator before subtracting:

\[ \dfrac{a}{c} - \dfrac{b}{d} = \dfrac{a \times d}{c \times d} - \dfrac{b \times c}{d \times c} = \dfrac{a \times d - b \times c}{c \times d} \]

14. Multiply Fractions

Multiply numerator by numerator and denominator by denominator:

\[ \dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{a \times c}{b \times d} \]

15. Divide Fractions

Multiply the first fraction by the reciprocal of the second fraction:

\[ \dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c} = \dfrac{a \times d}{b \times c} \]

16. Add a Number and a Fraction

Rewrite the number as a fraction with the same denominator as the fraction:

\[ a + \dfrac{b}{c} = a \times \dfrac{c}{c} + \dfrac{b}{c} = \dfrac{ac + b}{c} \]

17. Multiply a Number by a Fraction

Rewrite the number with a denominator of 1:

\[ a \times \dfrac{b}{c} = \dfrac{a}{1} \times \dfrac{b}{c} = \dfrac{ab}{c} \]

18. Divide a Number by a Fraction

Rewrite the number as a fraction and multiply by the reciprocal:

\[ a \div \dfrac{b}{c} = \dfrac{a}{1} \div \dfrac{b}{c} = \dfrac{a}{1} \times \dfrac{c}{b} = \dfrac{ac}{b} \]

19. Divide a Fraction by a Number

Rewrite the number with a denominator of 1 and multiply by its reciprocal:

\[ \dfrac{a}{b} \div c = \dfrac{a}{b} \div \dfrac{c}{1} = \dfrac{a}{b} \times \dfrac{1}{c} = \dfrac{a}{bc} \]

20. Signed Fractions

a) Placement of negative signs:

\[ - \dfrac{a}{b} = \dfrac{-a}{b} = \dfrac{a}{-b} \]

b) Division of negative signs:

\[ \dfrac{-a}{-b} = \dfrac{a}{b} \]

Questions

Write the following as a single fraction and reduce it to lowest terms if possible.

  1. \( \dfrac{0}{3} + \dfrac{1}{3} \)
  2. \( \dfrac{2}{0} + 5 \)
  3. \( \dfrac{3}{5} + 2 \)
  4. \( \dfrac{3}{2} + 2.1 \)
  5. \( 0.1 x + \dfrac{2x}{3} \)
  6. \( 3 x + \dfrac{x}{4} \)
  7. \( 3x - \dfrac{5 x}{4} \)
  8. \( \dfrac{3}{5} \times \dfrac{4}{9} \)
  9. \( 6 \times \dfrac{3}{7} \)
  10. \( \dfrac{2x}{5} \times \dfrac{1}{2} \)
  11. \( \dfrac{6}{7} \div 3 \)
  12. \( x \div \dfrac{1}{9} \)
  13. \( \dfrac{2x}{5} \div \dfrac{1}{9} \)
  14. \( - \dfrac{-3}{5} + \dfrac{-3}{5} \)
  15. \( \dfrac{2}{-9} + \dfrac{7}{9} \)
  16. \( \dfrac{-5}{-2} - \dfrac{7}{2} \)
  17. \( \dfrac{-2x}{3} - \dfrac{ - 5x}{- 3} \)
  18. \( 2 - \dfrac{ 4 + \dfrac{1}{3}}{1+\dfrac{1}{2}} \)
  19. \( x - \dfrac{ 2x + \dfrac{x}{2}}{x - \dfrac{2x}{3}} \)

Step-by-Step Solutions to the Questions

  1. \( \dfrac{0}{3} + \dfrac{1}{3} = \dfrac{0+1}{3} = \dfrac{1}{3} \)

  2. \( \dfrac{2}{0} + 5 \) is undefined because the fraction \( \dfrac{2}{0} \) has a denominator equal to zero and is therefore undefined.

  3. Given: \( \dfrac{3}{5} + 2 \)
    Rewrite the integer \( 2 \) as a fraction with denominator \( 5 \):
    \( \quad \dfrac{3}{5} + 2 = \dfrac{3}{5} + 2 \times \dfrac{5}{5} \)
    Add the fractions and simplify:
    \( \quad = \dfrac{3 + 2 \times 5}{5} = \dfrac{13}{5} \)

  4. Given: \( \dfrac{3}{2} + 2.1 \)
    Rewrite the decimal number \( 2.1 \) as a fraction:
    \( \quad \dfrac{3}{2} + 2.1 = \dfrac{3}{2} + \dfrac{2.1}{1} \)
    \( \quad = \dfrac{3}{2} + \dfrac{2.1 \times 10}{1 \times 10} \)
    \( \quad = \dfrac{3}{2} + \dfrac{21}{10} \)
    Rewrite the two fractions with a common denominator:
    \( \quad = \dfrac{3}{2} \times \dfrac{10}{10} + \dfrac{21}{10} \times \dfrac{2}{2} \)
    Multiply fractions and simplify:
    \( \quad = \dfrac{30}{20} + \dfrac{42}{20} \)
    Add the fractions:
    \( \quad = \dfrac{30 + 42}{20} = \dfrac{72}{20} \)
    Which may be reduced to:
    \( \quad = \dfrac{18}{5} \)

  5. Given: \( 0.1 x + \dfrac{2x}{3} \)
    Rewrite the term \( 0.1 x \) as a fraction:
    \( \quad 0.1 x + \dfrac{2x}{3} = \dfrac{1}{10} x + \dfrac{2x}{3} = \dfrac{x}{10} + \dfrac{2x}{3} \)
    Rewrite the fractions with a common denominator:
    \( \quad = \dfrac{x}{10} \times \dfrac{3}{3} + \dfrac{2x}{3} \times \dfrac{10}{10} \)
    Simplify:
    \( \quad = \dfrac{3 x}{30} + \dfrac{20 x}{30} \)
    Add the above fractions:
    \( \quad = \dfrac{3 x + 20 x}{30} = \dfrac{23 x}{30} \)

  6. Given: \( 3 x + \dfrac{x}{4} \)
    Rewrite the terms with a common denominator:
    \( \quad 3 x + \dfrac{x}{4} = 3 x \times \dfrac{4}{4} + \dfrac{x}{4} \)
    Simplify:
    \( \quad = \dfrac{12 x}{4} + \dfrac{x}{4} \)
    Add the fractions:
    \( \quad = \dfrac{12x + x}{4} = \dfrac{13 x }{4} \)

  7. Given: \( 3x - \dfrac{5 x}{4} \)
    Rewrite with a common denominator:
    \( \quad 3x - \dfrac{5 x}{4} = 3 x \times \dfrac{4}{4} - \dfrac{5 x}{4} \)
    Simplify:
    \( \quad = \dfrac{12 x}{4} - \dfrac{5 x}{4} \)
    Subtract the fractions:
    \( \quad = \dfrac{12x - 5 x}{4} = \dfrac{7 x }{4} \)

  8. Given: \( \dfrac{3}{5} \times \dfrac{4}{9} \)
    Apply multiplication rule of fractions:
    \( \quad \dfrac{3}{5} \times \dfrac{4}{9} = \dfrac{3 \times 4}{5 \times 9} = \dfrac{3 \times 4}{5 \times 3 \times 3} \)
    The numerator and denominator have a common factor \( 3 \) and therefore the fraction may be reduced by dividing numerator and denominator by \( 3 \):
    \( \quad = \dfrac{ (3 \times 4) \div 3 }{ (5 \times 9) \div 3} \)
    Simplify:
    \( \quad = \dfrac{4}{15} \)

  9. Given: \( 6 \times \dfrac{3}{7} \)
    Rewrite \( 6 \) as a fraction:
    \( \quad 6 \times \dfrac{3}{7} = \dfrac{6}{1} \times \dfrac{3}{7} \)
    Apply multiplication rule of fractions:
    \( \quad = \dfrac{6 \times 3}{ 1 \times 7 } \)
    Simplify:
    \( \quad = \dfrac{18}{7} \)

  10. Given: \( \dfrac{2x}{5} \times \dfrac{1}{2} \)
    Apply rule of multiplication of fractions:
    \( \quad \dfrac{2x}{5} \times \dfrac{1}{2} = \dfrac{2x \times 1}{5 \times 2} \)
    The numerator and denominator have a common factor \( 2 \) and the fraction may then be reduced by dividing numerator and denominator by \( 2 \):
    \( \quad = \dfrac{(2x \times 1) \div 2}{(5 \times 2)\div 2} \)
    Simplify:
    \( \quad = \dfrac{x}{5} \)

  11. Given: \( \dfrac{6}{7} \div 3 \)
    Rewrite \( 3 \) as a fraction:
    \( \quad \dfrac{6}{7} \div 3 = \dfrac{6}{7} \div \dfrac{3}{1} \)
    Apply division rule of fraction (change into a multiplication of the first fraction by the reciprocal of the second fraction):
    \( \quad = \dfrac{6}{7} \times \dfrac{1}{3} \)
    Apply the multiplication rule of fractions:
    \( \quad = \dfrac{6 \times 1}{7 \times 3} \)
    Factor the numerator:
    \( \quad = \dfrac{3 \times 2 \times 1}{7 \times 3} \)
    Divide numerator and denominator by the common factor \( 3 \) to reduce the fraction:
    \( \quad = \dfrac{ (3 \times 2 \times 1) \div 3}{ (7 \times 3) \div 3} \)
    Simplify:
    \( \quad = \dfrac{2}{7} \)

  12. Given: \( x \div \dfrac{1}{9} \)
    Use division rule of fractions:
    \( \quad x \div \dfrac{1}{9} = x \times \dfrac{9}{1} \)
    Rewrite \( x \) as a fraction:
    \( \quad = \dfrac{x}{1} \times \dfrac{9}{1} \)
    Apply the multiplication rule of fractions:
    \( \quad = \dfrac{x \times 9}{1 \times 1} \)
    Simplify:
    \( \quad = \dfrac{9 x}{1} \)
    Simplify fraction with denominator equal to 1:
    \( \quad = 9x \)

  13. Given: \( \dfrac{2x}{5} \div \dfrac{1}{9} \)
    Use division rule of fractions:
    \( \quad \dfrac{2x}{5} \div \dfrac{1}{9} = \dfrac{2x}{5} \times \dfrac{9}{1} \)
    Apply the multiplication rule of fractions:
    \( \quad = \dfrac{2x \times 9}{5 \times 1} \)
    Simplify:
    \( \quad = \dfrac{18 x}{5} \)

  14. Given: \( - \dfrac{-3}{5} + \dfrac{-3}{5} \)
    Apply rule of signs to rewrite the fraction \( \dfrac{-3}{5} \) included in the given expression as \( - \dfrac{3}{5} \):
    \( \quad - \dfrac{-3}{5} + \dfrac{-3}{5} = - ( - \dfrac{3}{5}) + \dfrac{-3}{5} \)
    Simplify:
    \( \quad = \dfrac{3}{5} + \dfrac{-3}{5} \)
    Add the fractions and simplify:
    \( \quad = \dfrac{3 +(- 3)}{5} = \dfrac{0}{5} = 0 \)

  15. Given: \( \dfrac{2}{-9} + \dfrac{7}{9} \)
    Apply rule of signs to rewrite the fraction \( \dfrac{2}{-9} \) included in the given expression as \( \dfrac{-2}{9} \):
    \( \quad \dfrac{2}{-9} + \dfrac{7}{9} = \dfrac{- 2}{9} + \dfrac{7}{9} \)
    Add the fractions and simplify:
    \( \quad = \dfrac{-2 + 7}{9} = \dfrac{5}{9} \)

  16. Given: \( \dfrac{-5}{-2} - \dfrac{7}{2} \)
    Apply rule of signs to rewrite the fraction \( \dfrac{-5}{-2} \) included in the given expression as \( \dfrac{5}{2} \):
    \( \quad \dfrac{-5}{-2} - \dfrac{7}{2} = \dfrac{5}{2} - \dfrac{7}{2} \)
    Subtract the fractions:
    \( \quad = \dfrac{5 - 7}{2} \)
    Apply rule of signs to simplify:
    \( \quad = \dfrac{-2}{2} = - \dfrac{2}{2} = -1 \)

  17. Given: \( \dfrac{-2x}{3} - \dfrac{ - 5x}{- 3} \)
    Apply rule of signs to rewrite the fraction \( \dfrac{ - 5x}{- 3} \) included in the given expression as \( \dfrac{ 5x}{3} \):
    \( \quad \dfrac{-2x}{3} - \dfrac{ - 5x}{- 3} = \dfrac{-2x}{3} - \dfrac{ 5x}{3} \)
    Subtract:
    \( \quad = \dfrac{-2x - 5x}{3} \)
    Simplify:
    \( \quad = \dfrac{-7x}{3} \)
    Which may also be written as (using rule of signs):
    \( \quad = - \dfrac{7x}{3} \)

  18. Given: \( 2 - \dfrac{ 4 + \dfrac{1}{3}}{1+\dfrac{1}{2}} \)

    Rewrite \( 4 \) in the numerator \( 4 + \dfrac{1}{3} \) as a fraction with denominator equal to \( 3 \) and \( 1 \) in the denominator \( 1+\dfrac{1}{2} \) as a fraction with denominator equal to \( 2 \):

    \( \quad 2 - \dfrac{ 4 + \dfrac{1}{3}}{1+\dfrac{1}{2}} = 2 - \dfrac{ 4 \times \dfrac{3}{3} + \dfrac{1}{3}}{1 \times \dfrac{2}{2} +\dfrac{1}{2}} \)
    Simplify:
    \( \quad = 2 - \dfrac{ \dfrac{12}{3} + \dfrac{1}{3}} {\dfrac{2}{2} +\dfrac{1}{2}} \)
    Add fractions with common denominator:
    \( \quad = 2 - \dfrac{ \dfrac{13}{3}} {\dfrac{3}{2}} \)
    Use rule of division of fractions:
    \( \quad = 2 - \dfrac{13}{3} \times {\dfrac{2}{3}} \)
    Multiply fractions:
    \( \quad = 2 - \dfrac{26}{9} \)
    Rewrite \( 2 \) as a fraction with denominator equal to \( 9 \):
    \( \quad = 2 \times \dfrac{9}{9} - \dfrac{26}{9} \)
    Simplify:
    \( \quad = \dfrac{18}{9} - \dfrac{26}{9} \)
    Subtract and simplify:
    \( \quad = \dfrac{18-26}{9} = \dfrac{-8}{9} = - \dfrac{8}{9} \)

  19. Given: \( x - \dfrac{ 2x + \dfrac{x}{2}}{x - \dfrac{2x}{3}} \)

    Rewrite \( 2x \) in \( 2x + \dfrac{x}{2} \) as a fraction with denominator equal to \( 2 \) and \( x \) in \( x - \dfrac{2x}{3} \) as a fraction with denominator equal to \( 3 \):

    \( \quad x - \dfrac{ 2x + \dfrac{x}{2}}{x - \dfrac{2x}{3}} = x - \dfrac{ 2x \dfrac {2}{2} + \dfrac{x}{2}}{x \dfrac{3}{3} - \dfrac{2x}{3}} \)
    Simplify:
    \( \quad = x - \dfrac{ \dfrac {4x}{2} + \dfrac{x}{2}}{\dfrac{3 x}{3} - \dfrac{2x}{3}} \)
    Add and subtract fractions with common denominator:

    \( \quad = x - \dfrac{ \dfrac {4x + x}{2} }{\dfrac{3 x - 2x}{3} } \)
    Simplify:
    \( \quad = x - \dfrac{ \dfrac {5x}{2} }{\dfrac{x}{3} } \)
    Use rule of division of fractions:
    \( \quad = x - \dfrac {5x}{2} \times \dfrac{3}{x} \)
    Simplify:
    \( \quad = x - \dfrac {15 x}{2 x} \)
    Reduce the fraction \( \dfrac {15 x}{2 x} \) dividing its numerator and denominator by \( x \):
    \( \quad = x - \dfrac {15 x \div x}{2 x \div x} \)
    Simplify:
    \( \quad = x - \dfrac {15}{2} \)
    Rewrite \( x \) as a fraction with denominator equal to \( 2 \):
    \( \quad = x \times \dfrac{2}{2} - \dfrac {15}{2} \)
    Simplify:
    \( \quad = \dfrac{2x }{2} - \dfrac {15}{2} \)
    Subtract fractions:
    \( \quad = \dfrac{2x - 15}{2} \)

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