Australian Curriculum Year 7 - 2020 Edition 2.07 Order of operations with fractions
Lesson

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\frac{3}{4}$5÷​34 is the same as $5\div\left(3\div4\right)$5÷​(3÷​4) and not $5\div3\div4$5÷​3÷​4.

Second, there is effectively a pair of brackets around both the numerator and the denominator of a fraction. So $\frac{2+7}{9+6}$2+79+6 is the same as $\left(2+7\right)\div\left(9+6\right)$(2+7)÷​(9+6) and not $2+7\div9+6$2+7÷​9+6.

#### Worked Example

Evaluate $\frac{5}{6}-\frac{11}{10}\times\frac{2}{9}$561110×29.

Think: Following the order of operations, we evaluate the multiplication first, followed by the subtraction.

Do: First we find $\frac{11}{10}\times\frac{2}{9}$1110×29. We can evaluate this by multiplying the numerators and the denominators separately. This gives us $\frac{11\times2}{10\times9}=\frac{22}{90}$11×210×9=2290. We could simplify this fraction now, but it will be easier to evaluate the addition first.

Now we have $\frac{5}{6}-\frac{22}{90}$562290. To evaluate the subtraction we rewrite the fractions with the same denominator. Since $90=6\times15$90=6×15, we multiply both the numerator and denominator by $15$15 which gives $\frac{5\times15}{6\times15}=\frac{75}{90}$5×156×15=7590.

Now we have $\frac{75}{90}-\frac{22}{90}$75902290. To evaluate the subtraction we subtract the numerators over the common denominator. This gives us $\frac{75-22}{90}=\frac{53}{90}$752290=5390. So $\frac{5}{6}-\frac{11}{10}\times\frac{2}{9}=\frac{53}{90}$561110×29=5390.

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.

#### Practice questions

##### Question 1

Evaluate and simplify $\frac{3}{40}+\frac{4}{5}\times\frac{7}{8}$340+45×78.

##### Question 2

Evaluate and simplify $\frac{4}{35}-\left(\frac{6}{7}-\frac{4}{5}\right)$435(6745).

##### Question 3

Evaluate and simplify $\frac{2}{3}\div\frac{3}{4}+\frac{7}{9}$23÷​34+79.

### Outcomes

#### ACMNA154

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