Order Of Operations With Fractions

Q: What if there are mixed numbers in the problem?

If your problem includes mixed numbers (like $ 2\frac{1}{4} $), the best first step is to convert them into improper fractions. Once all numbers are in fraction form, you can apply the PEMDAS rules as usual.

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Ever feel stuck when a math problem has fractions, parentheses, and exponents all at once? Don't worry! The rules you know for the order of operations work for fractions, too. This guide will show you how to solve any fractional expression, one step at a time.

What Is the Order of Operations with Fractions?

The order of operations with fractions is the specific sequence of steps, commonly remembered by the acronym PEMDAS, used to solve mathematical expressions that involve fractions and multiple operations like addition, subtraction, multiplication, division, and exponents. These are the very same rules you use for whole numbers, but now we apply them to numerators and denominators. Following this order ensures that everyone who solves the same problem will arrive at the exact same, correct answer. It's the universal language of math that prevents confusion and keeps our calculations consistent.

Think of it like a recipe. If you add ingredients in the wrong order, you might not get the cake you were hoping for. In math, if you perform operations in the wrong order, you will get an incorrect answer. Whether you're working with integers or fractions, the sequence of PEMDAS remains your reliable guide.

Why Does the Order of Operations Matter?

Imagine two students are asked to solve the problem $\frac{1}{2} + \frac{1}{4} \times \frac{2}{3}$ . Without a standard order, the results could be completely different.

Student A adds first:

$(\frac{1}{2} + \frac{1}{4}) \times \frac{2}{3}$
First, find a common denominator for addition: $\frac{1}{2} = \frac{2}{4}$ .
$(\frac{2}{4} + \frac{1}{4}) \times \frac{2}{3} = \frac{3}{4} \times \frac{2}{3}$
Now, multiply: $\frac{3 \times 2}{4 \times 3} = \frac{6}{12}$ , which simplifies to $\frac{1}{2}$ .

Student B multiplies first:

$\frac{1}{2} + (\frac{1}{4} \times \frac{2}{3})$
First, multiply the fractions: $\frac{1 \times 2}{4 \times 3} = \frac{2}{12}$ , which simplifies to $\frac{1}{6}$ .
Now, add: $\frac{1}{2} + \frac{1}{6}$
Find a common denominator: $\frac{1}{2} = \frac{3}{6}$ .
$\frac{3}{6} + \frac{1}{6} = \frac{4}{6}$ , which simplifies to $\frac{2}{3}$ .

We have two different answers: $\frac{1}{2}$ and $\frac{2}{3}$ . Which one is correct? The order of operations tells us that multiplication comes before addition, so Student B is correct. This example shows why having a consistent set of rules is essential for everyone to communicate mathematically and get the right answer every time.

A Quick Review of PEMDAS

PEMDAS is an acronym to help you remember the correct order of operations. Some people use the phrase "Please Excuse My Dear Aunt Sally" to recall the sequence. Another common acronym is BODMAS (Brackets, Orders, Division/Multiplication, Addition/Subtraction), which is used in some countries but represents the exact same mathematical rules.

P - Parentheses
E - Exponents
M - Multiplication
D - Division
A - Addition
S - Subtraction

Here is a more detailed breakdown of what each step means:

Step	Operation	Key Details
P	Parentheses (or any grouping symbols)	Always solve what's inside parentheses, brackets $[]$ , or braces ${}$ first. If there are multiple sets, work from the innermost set outwards.
E	Exponents (or Orders/Roots)	After handling parentheses, calculate any exponents. For a fraction, this means applying the exponent to both the numerator and the denominator.
M/D	Multiplication and Division	This is a crucial step. Multiplication and Division are partners; they have equal priority. You solve them as they appear from left to right in the problem.
A/S	Addition and Subtraction	Like multiplication and division, Addition and Subtraction are partners with equal priority. You solve them as they appear from left to right. Remember to find common denominators before adding or subtracting fractions.

How Do You Apply PEMDAS to Fractions? A Step-by-Step Example

Let's walk through a complete problem to see how PEMDAS guides us when fractions are involved. We will solve each part methodically, following the rules precisely.

Example 1

Solve the expression: $(\frac{4}{5} - \frac{1}{2}) \div \frac{3}{4} + {(\frac{1}{3})}^{2}$

Step 1: P - Parentheses
We start with the operation inside the parentheses: $\frac{4}{5} - \frac{1}{2}$ . To subtract, we need a common denominator. The least common multiple of $5$ and $2$ is $10$ .
$\frac{4}{5} = \frac{4 \times 2}{5 \times 2} = \frac{8}{10}$
$\frac{1}{2} = \frac{1 \times 5}{2 \times 5} = \frac{5}{10}$
Now subtract: $\frac{8}{10} - \frac{5}{10} = \frac{3}{10}$ .
Our expression is now: $\frac{3}{10} \div \frac{3}{4} + {(\frac{1}{3})}^{2}$

Step 2: E - Exponents
Next, we handle the exponent: ${(\frac{1}{3})}^{2}$ . This means we multiply the fraction by itself. We apply the exponent to both the numerator and the denominator.
${(\frac{1}{3})}^{2} = \frac{1^{2}}{3^{2}} = \frac{1 \times 1}{3 \times 3} = \frac{1}{9}$
Our expression becomes: $\frac{3}{10} \div \frac{3}{4} + \frac{1}{9}$

Step 3: M/D - Multiplication and Division (from left to right)
We have one division operation: $\frac{3}{10} \div \frac{3}{4}$ . To divide by a fraction, we multiply by its reciprocal (keep, change, flip).
$\frac{3}{10} \div \frac{3}{4} = \frac{3}{10} \times \frac{4}{3} = \frac{3 \times 4}{10 \times 3} = \frac{12}{30}$
We can simplify this fraction by dividing the numerator and denominator by their greatest common factor, $6$ .
$\frac{12}{30} = \frac{12 \div 6}{30 \div 6} = \frac{2}{5}$
Our expression is now: $\frac{2}{5} + \frac{1}{9}$

Step 4: A/S - Addition and Subtraction (from left to right)
Finally, we perform the addition. We need a common denominator for $5$ and $9$ , which is $45$ .
$\frac{2}{5} = \frac{2 \times 9}{5 \times 9} = \frac{18}{45}$
$\frac{1}{9} = \frac{1 \times 5}{9 \times 5} = \frac{5}{45}$
Now add: $\frac{18}{45} + \frac{5}{45} = \frac{23}{45}$ .
The fraction $\frac{23}{45}$ cannot be simplified further.

Final Answer: $\frac{23}{45}$

How Do You Handle Exponents with Fractions?

Exponents can look intimidating with fractions, but the rule is straightforward. When a fraction inside parentheses is raised to a power, you apply that power to both the numerator and the denominator separately.

{(\frac{a}{b})}^{n} = \frac{a^{n}}{b^{n}}

For example, to calculate ${(\frac{2}{5})}^{3}$ , you would calculate $2^{3}$ and $5^{3}$ .
$2^{3} = 2 \times 2 \times 2 = 8$
$5^{3} = 5 \times 5 \times 5 = 125$
So, ${(\frac{2}{5})}^{3} = \frac{8}{125}$ .

Let's see this in a problem.

Example 2

Solve the expression: $\frac{7}{8} - \frac{1}{4} \times {(\frac{2}{3})}^{2}$

Step 1: P - Parentheses
There are parentheses, but no operations inside them. They are just there to show the exponent applies to the entire fraction. So, we move to the next step.

Step 2: E - Exponents
We calculate ${(\frac{2}{3})}^{2}$ .
${(\frac{2}{3})}^{2} = \frac{2^{2}}{3^{2}} = \frac{4}{9}$
The expression is now: $\frac{7}{8} - \frac{1}{4} \times \frac{4}{9}$

Step 3: M/D - Multiplication and Division
Next, we perform the multiplication: $\frac{1}{4} \times \frac{4}{9}$ .
$\frac{1}{4} \times \frac{4}{9} = \frac{1 \times 4}{4 \times 9} = \frac{4}{36}$
We simplify this fraction. The greatest common factor of $4$ and $36$ is $4$ .
$\frac{4 \div 4}{36 \div 4} = \frac{1}{9}$
The expression becomes: $\frac{7}{8} - \frac{1}{9}$

Step 4: A/S - Addition and Subtraction
Finally, we subtract. The least common denominator for $8$ and $9$ is $72$ .
$\frac{7}{8} = \frac{7 \times 9}{8 \times 9} = \frac{63}{72}$
$\frac{1}{9} = \frac{1 \times 8}{9 \times 8} = \frac{8}{72}$
Now subtract: $\frac{63}{72} - \frac{8}{72} = \frac{55}{72}$ .

Final Answer: $\frac{55}{72}$

The Left-to-Right Rule for Multiplication and Division

One of the most common points of confusion in PEMDAS is the relationship between multiplication and division. Students sometimes think the 'M' in PEMDAS means you must always multiply before you divide. This is incorrect! Multiplication and Division are a team. They have equal priority, so you solve them in the order they appear from left to right.

The same is true for Addition and Subtraction. They are also a team of equal priority, and you solve them from left to right. Let's look at an example where this rule is critical.

Example 3

Solve the expression: $\frac{5}{6} \div \frac{2}{3} \times \frac{1}{4}$

In this problem, we only have division and multiplication. We must work from left to right.

Step 1: Perform the leftmost operation (Division)
The first operation is $\frac{5}{6} \div \frac{2}{3}$ . We use the 'keep, change, flip' method.
$\frac{5}{6} \div \frac{2}{3} = \frac{5}{6} \times \frac{3}{2} = \frac{5 \times 3}{6 \times 2} = \frac{15}{12}$
Let's simplify this improper fraction. The greatest common factor is $3$ .
$\frac{15 \div 3}{12 \div 3} = \frac{5}{4}$
Our expression is now: $\frac{5}{4} \times \frac{1}{4}$

Step 2: Perform the next operation (Multiplication)
Now we multiply the result from Step 1 by the remaining fraction.
$\frac{5}{4} \times \frac{1}{4} = \frac{5 \times 1}{4 \times 4} = \frac{5}{16}$

Final Answer: $\frac{5}{16}$

If we had incorrectly multiplied first ( $\frac{2}{3} \times \frac{1}{4} = \frac{2}{12} = \frac{1}{6}$ ) and then divided ( $\frac{5}{6} \div \frac{1}{6} = \frac{5}{6} \times \frac{6}{1} = 5$ ), we would get a completely different and wrong answer. Always remember: left to right for M/D and A/S!

Common Mistakes to Avoid

When working with fractions and the order of operations, a few common errors can trip you up. Being aware of them is the best way to avoid making them!

Forgetting Common Denominators: You can only add or subtract fractions after you have found a common denominator. It's a required step that doesn't apply to multiplication or division.
Ignoring the Left-to-Right Rule: Remember that Multiplication/Division and Addition/Subtraction are pairs. You don't always do multiplication before division; you do whichever comes first when reading the problem from left to right.
Incorrectly Applying Exponents: An exponent outside parentheses applies to the entire fraction. A common mistake is to apply it only to the numerator. Remember to raise both the numerator and the denominator to the power.
Errors in Division: When dividing fractions, students sometimes flip the wrong fraction or forget to change the operation from division to multiplication. Remember: Keep the first fraction, Change the sign to multiply, and Flip the second fraction.
Not Simplifying: Your final answer should always be in its simplest form. This includes converting improper fractions to mixed numbers if requested by your teacher, and reducing fractions by dividing the numerator and denominator by their greatest common factor.

Quick Summary: Your PEMDAS Checklist for Fractions

Feeling confident? Here is a quick checklist to use as a reference when you're solving problems involving the order of operations with fractions. Follow these steps every time for the correct answer.

Parentheses: Start by solving all operations inside any grouping symbols, from the innermost set outwards.
Exponents: Next, calculate any exponents. Remember to apply the exponent to both the numerator and the denominator.
Multiplication and Division: Work through the problem from left to right, performing any multiplication or division as you encounter it. Don't prioritize one over the other.
Addition and Subtraction: Finally, work through the problem again from left to right, performing any addition or subtraction. Remember to find common denominators first!
Simplify: Check your final answer to see if it can be simplified. Reduce the fraction to its lowest terms.

Frequently Asked Questions

What does PEMDAS stand for?

PEMDAS is an acronym that stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. It's a memory tool to help you remember the correct order for solving math problems with multiple operations.

Is BODMAS the same as PEMDAS?

Yes, they represent the same mathematical rules. BODMAS stands for Brackets, Orders, Division, Multiplication, Addition, and Subtraction. 'Brackets' are the same as 'Parentheses,' and 'Orders' are the same as 'Exponents.' Both systems work the same way.

Do I always do multiplication before division?

No, this is a common misconception. Multiplication and division have equal priority. You should solve them in the order they appear as you read the problem from left to right.

What's the first step when adding or subtracting fractions?

Before you can add or subtract fractions, you must find a common denominator. This means converting each fraction into an equivalent fraction so that they share the same bottom number (denominator).

How do you handle a fraction with an exponent?

When a fraction in parentheses is raised to a power, you apply that exponent to both the numerator (top number) and the denominator (bottom number). For example, $(\frac{2}{3})^{2}$ becomes $\frac{2^{2}}{3^{2}}$ , which is $\frac{4}{9}$ .

What if there are mixed numbers in the problem?

If your problem includes mixed numbers (like $2 \frac{1}{4}$ ), the best first step is to convert them into improper fractions. Once all numbers are in fraction form, you can apply the PEMDAS rules as usual.

Why is learning order of operations with fractions important?

This skill is a fundamental building block for more advanced math, like algebra. It helps you accurately solve complex problems and ensures that you can communicate your mathematical reasoning clearly and correctly with others.