Topics
2 Fractions
topic badge
AustraliaVIC
Level 7

2.07 Order of operations with fractions

Lesson
Worksheet
Practice
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

VCMNA244

Multiply and divide fractions and decimals using efficient written strategies and digital technologies.

What is Mathspace

About Mathspace