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2 Fractions
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Stage 4

2.07 Order of operations with fractions

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Lesson

Order of operations with fractions

The order of operations with fractions is the same as the order of operations with whole numbers:

  • Any operations inside brackets are evaluated first

  • Then multiplications and divisions are evaluated from left to right

  • Then additions and subtractions are evaluated from left to right

There are two things to keep in mind with fractions.

First, while a fraction is a way of writing a division, the fraction takes precedence over other divisions. For example, 5\div\dfrac{3}{4} is the same as 5\div(3\div4) and not 5\div3\div4.

Second, there is effectively a pair of brackets around both the numerator and the denominator of a fraction. So \dfrac{2+7}{9+6} is the same as (2+7)\div(9+6) and not 2+7\div9+6.

Examples

Example 1

Evaluate and simplify \dfrac{3}{40}+\dfrac{4}{5}\times\dfrac{7}{8}.

Worked Solution
Create a strategy

Evaluate the multiplication, then evaluate the addition.

Apply the idea
\displaystyle \dfrac{3}{40}+\dfrac{4}{5}\times\dfrac{7}{8}\displaystyle =\displaystyle \dfrac{3}{40}+\dfrac{4\times7}{5\times8}Evaluate the multiplication
\displaystyle =\displaystyle \dfrac{3}{40}+\dfrac{28}{40}Multiply the numerators and the denominators
\displaystyle =\displaystyle \dfrac{31}{40}Evaluate the addition

Example 2

Evaluate and simplify \dfrac{4}{35}-\left(\dfrac{6}{7}-\dfrac{4}{5}\right).

Worked Solution
Create a strategy

Follow the order of operations. Evaluate the operation in the brackets, then evaluate the subtraction (going from left to right).

Apply the idea
\displaystyle \dfrac{4}{35}-\left(\dfrac{6}{7}-\dfrac{4}{5}\right)\displaystyle =\displaystyle \dfrac{4}{35}-\left(\dfrac{6\times5}{7\times5}-\dfrac{4\times7}{5\times7}\right)Rewrite both fractions with same denominator
\displaystyle =\displaystyle \dfrac{4}{35}-\left(\dfrac{30}{35}-\dfrac{28}{35}\right)Evaluate all multiplications
\displaystyle =\displaystyle \dfrac{4}{35}-\dfrac{2}{35}Evaluate the subtraction inside bracket
\displaystyle =\displaystyle \dfrac{2}{35}Evaluate the subtraction

Example 3

Evaluate and simplify \dfrac{2}{3}\div\dfrac{3}{4}+\dfrac{7}{9}.

Worked Solution
Create a strategy

Evaluate the division, then evaluate the addition.

Apply the idea
\displaystyle \dfrac{2}{3}\div\dfrac{3}{4}+\dfrac{7}{9}\displaystyle =\displaystyle \dfrac{2}{3}\times\dfrac{4}{3}+\dfrac{7}{9}Multiply by the reciprocal
\displaystyle =\displaystyle \dfrac{2\times4}{3\times3}+\dfrac{7}{9}Multiply numerators and denominators
\displaystyle =\displaystyle \dfrac{8}{9}+\dfrac{7}{9}Evaluate the multiplication
\displaystyle =\displaystyle \dfrac{15}{9}Evaluate the addition
\displaystyle =\displaystyle \dfrac{15\div 3}{9\div 3}Divide the numerator and denominator by 3
\displaystyle =\displaystyle \dfrac{5}{3}Simplify
Idea summary

The order of operations with fractions is the same as the order of operations with whole numbers.

Operations inside fractions take precedence over other operations.

Outcomes

MA4-5NA

operates with fractions, decimals and percentages

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