# Order of Operations with Fractions

Just like solving equations with whole numbers, solving equations with fractions follows the same PEMDAS rule.

Let us recap the PEMDAS rule before we solve some expressions involving fractions.

• Step 1: P for Parentheses
• Step 2: E for Exponent
• Step 3: MD for Multiplication and Division
• Step 4: AS for Addition and Subtraction

We can simplify expressions having both like and unlike fractions involving PEMDAS.

### With Like Fractions

Let us consider a simple example with like fractions (having the same denominators)

Simplify: {\begin{aligned}\left( \dfrac{2}{3}\times \dfrac{4}{3}\right) +\dfrac{6}{3}\\ .\end{aligned}}

Solving this expression using PEMDAS rule, we will first perform the operation within the parenthesis involving multiplication and then we will add the result with 6/3

= ${\dfrac{8}{3}+\dfrac{6}{3}}$ (∵ Multiplication of {\begin{aligned}\dfrac{2}{3}\times \dfrac{4}{3}=\dfrac{8}{3}\\ .\end{aligned}})

= ${\dfrac{14}{3}}$ (∵ Addition of ${\dfrac{8}{3}+\dfrac{6}{3}=\dfrac{14}{3}}$)

Let us solve some more expressions using PEMDAS having like fraction

Simplify: ${\dfrac{1}{3}\times \left( \dfrac{2}{3}\right) ^{2}-\dfrac{1}{3}}$

Solution:

Using PEMDAS,
= ${\dfrac{1}{3}\times \dfrac{4}{9}-\dfrac{1}{3}}$ (Evaluating the exponents)
= ${\dfrac{4}{27}-\dfrac{1}{3}}$ (Multiplying)
= ${\dfrac{4}{27}-\dfrac{9}{27}}$ (Making the fractions equivalent)
= ${-\dfrac{5}{27}}$ (Subtracting)
Answer: ${-\dfrac{5}{27}}$

Solve: ${\dfrac{1}{14}\div \dfrac{2}{14}-\dfrac{1}{2}}$

Solution:

Using PEMDAS,
= ${\dfrac{1}{14}\times \dfrac{14}{2}-\dfrac{1}{2}}$ (Dividing: Inverting the 2nd fraction)
= ${\dfrac{1}{2}-\dfrac{1}{2}}$ (Multiplying)
= 0 (Subtracting)

### With Unlike Fractions

When, we consider an expression involving improper fractions, we follow the same rule but an extra step is involved, making the fractions equivalent

Simplify: ${-\dfrac{1}{3}+\dfrac{1}{3}\left( -\dfrac{1}{5}\right)}$

Here, we will use the rule of PEMDAS

${-\dfrac{1}{3}-\dfrac{1}{15}}$ (∵ Multiplication of ${\dfrac{1}{3}\left( -\dfrac{1}{5}\right) =-\dfrac{1}{15}}$)

= ${-1\times \dfrac{5}{3}\times 5-\dfrac{1}{15}}$ (Making the fractions equivalent, Lowest Common Denominator (LCD) = 15)

= ${-\dfrac{5}{15}-\dfrac{1}{15}}$ (Subtracting the like fractions)

= ${-\dfrac{6}{15}}$

Let us solve another expression using PEMDAS having unlike fraction

Simplify: ${\dfrac{1}{3}\div \dfrac{4}{5}-\dfrac{1}{7}}$

Solution:

Using PEMDAS,
= ${\dfrac{1}{3}\times \dfrac{5}{4}-\dfrac{1}{6}}$ (Dividing: Inverting the 2nd fraction)
= ${\dfrac{5}{12}-\dfrac{1}{6}}$ (Multiplying)
= ${\dfrac{5}{12}-1\times \dfrac{2}{6}\times 2}$ (Making the fractions equivalent)
= ${\dfrac{5}{12}-\dfrac{2}{12}}$ (Subtracting
= ${\dfrac{3}{12}}$