2.07 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\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}$56−1110×29.
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}$56−2290. 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}$7590−2290. To evaluate the subtraction we subtract the numerators over the common denominator. This gives us $\frac{75-22}{90}=\frac{53}{90}$75−2290=5390. So $\frac{5}{6}-\frac{11}{10}\times\frac{2}{9}=\frac{53}{90}$56−1110×29=5390.
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−(67−45).
Question 3
Evaluate and simplify $\frac{2}{3}\div\frac{3}{4}+\frac{7}{9}$23÷34+79.