Order of operations with fractions I
In the chapter Maths in Order, we saw that some operations need to be completed before others.
Order of operations is the same for all topics within mathematics,
- operations inside grouping symbols such as parentheses (...), brackets [...] and braces {...}
- multiplication (including powers) and division (including roots) going from left to right
- addition and subtraction going from left to right
I'm not about to tell you that fractions are special and behave differently. But you do need to know that sometimes fractions are a bit sneaky. There are brackets in fractions that you can't see. Invisible brackets around everything in the numerator and similarly around the denominator.
So, $\frac{2+3}{7}$2+37 is really $\frac{\left(2+3\right)}{7}$(2+3)7
Under the order of operations we would work on the numerator addition first, in brackets.
and $\frac{20-4}{3+5}$20−43+5 is really $\frac{\left(20-4\right)}{\left(3+5\right)}$(20−4)(3+5) and under the order of operations we would work on the numerator and denominator addition/subtraction, before doing the fraction (which is like a division)
$\frac{\left(20-4\right)}{\left(3+5\right)}$(20−4)(3+5)$=$=$\frac{16}{8}=2$168=2
The order goes:
Step 1: Do operations inside grouping symbols such as parentheses (...), brackets [...] and braces {...}
With fractions, the numerator and denominator are in unwritten brackets, so we work with these first.
Step 2: Do multiplication (including powers) and division (including roots) going from left to right.
Step 3: Do addition and subtraction going from left to right.
These rules apply to all of mathematics, yes - even fractions! So order of operations with fractions is just following the order of operations, with fractions in the picture.
Worked Examples
QUESTION 1
Evaluate $\frac{2}{3}+\frac{3}{4}\times\frac{5}{6}$23+34×56 and simplify, writing your answer as an improper or proper fraction.
QUESTION 2
Evaluate $\frac{3}{5}\times\left(\frac{3}{4}+\frac{6}{5}\right)$35×(34+65), writing your answer as an improper fraction if necessary.
QUESTION 3
Evaluate $\frac{7}{3}-\frac{2}{5}\div\frac{2}{7}$73−25÷27
Write your answer as a fraction in simplest form possible.