Order of Operations with Fractions

The order of operations (PEMDAS) works exactly the same way with fractions as it does with whole numbers. The operations do not change — you still handle Parentheses first, then Exponents, then Multiplication/Division (left to right), then Addition/Subtraction (left to right). The difference is that each step involves fraction arithmetic instead of whole-number arithmetic.

Quick PEMDAS Review

Example 1: Basic Two-Step

Evaluate: $\frac{1}{2} + \frac{3}{4} \times \frac{2}{3}$

Step 1: Multiplication before addition.

$\frac{3}{4} \times \frac{2}{3} = \frac{6}{12} = \frac{1}{2}$

Step 2: Add:

$\frac{1}{2} + \frac{1}{2} = 1$

Answer: $1$

Example 2: Parentheses Change the Order

Evaluate: $\left(\frac{1}{2} + \frac{3}{4}\right) \times \frac{2}{3}$

Step 1: Parentheses first. LCD of 2 and 4 is 4:

$\frac{1}{2} + \frac{3}{4} = \frac{2}{4} + \frac{3}{4} = \frac{5}{4}$

Step 2: Multiply:

$\frac{5}{4} \times \frac{2}{3} = \frac{10}{12} = \frac{5}{6}$

Answer: $\frac{5}{6}$

Notice how parentheses changed the answer from $1$ to $\frac{5}{6}$.

Example 3: Exponents with Fractions

Evaluate: $\left(\frac{2}{3}\right)^2 + \frac{1}{9}$

Step 1: Exponent first. Squaring a fraction means squaring both the numerator and denominator:

$\left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2} = \frac{4}{9}$

Step 2: Add (denominators already match):

$\frac{4}{9} + \frac{1}{9} = \frac{5}{9}$

Answer: $\frac{5}{9}$

Example 4: Multiple Operations

Evaluate: $\frac{3}{4} - \frac{1}{2} \times \frac{1}{3} + \frac{1}{6}$

Step 1: Multiplication first:

$\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$

Step 2: Now evaluate left to right — the expression is $\frac{3}{4} - \frac{1}{6} + \frac{1}{6}$

LCD of 4 and 6 is 12:

$\frac{3}{4} - \frac{1}{6} + \frac{1}{6} = \frac{9}{12} - \frac{2}{12} + \frac{2}{12} = \frac{9}{12} = \frac{3}{4}$

Answer: $\frac{3}{4}$

Example 5: Division and Subtraction

Evaluate: $\frac{5}{6} \div \frac{5}{3} - \frac{1}{4}$

Step 1: Division first (Keep-Change-Flip):

$\frac{5}{6} \div \frac{5}{3} = \frac{5}{6} \times \frac{3}{5} = \frac{15}{30} = \frac{1}{2}$

Step 2: Subtract. LCD of 2 and 4 is 4:

$\frac{1}{2} - \frac{1}{4} = \frac{2}{4} - \frac{1}{4} = \frac{1}{4}$

Answer: $\frac{1}{4}$

Example 6: Nested Parentheses

Evaluate: $\frac{1}{2} \times \left(\frac{3}{4} - \left(\frac{1}{4} + \frac{1}{8}\right)\right)$

Step 1: Innermost parentheses first. LCD of 4 and 8 is 8:

$\frac{1}{4} + \frac{1}{8} = \frac{2}{8} + \frac{1}{8} = \frac{3}{8}$

Step 2: Outer parentheses. LCD of 4 and 8 is 8:

$\frac{3}{4} - \frac{3}{8} = \frac{6}{8} - \frac{3}{8} = \frac{3}{8}$

Step 3: Multiply:

$\frac{1}{2} \times \frac{3}{8} = \frac{3}{16}$

Answer: $\frac{3}{16}$

Common Mistakes

Mistake 1: Ignoring left-to-right rule. Multiplication and division have equal priority — evaluate whichever comes first from left to right. The same applies to addition and subtraction.

Mistake 2: Distributing incorrectly. When you see $\frac{1}{2} \times \left(\frac{3}{4} + \frac{1}{4}\right)$, you can either add inside the parentheses first or distribute. Both should give the same answer. If your answers differ, check your fraction arithmetic.

Mistake 3: Forgetting to simplify. After every step, simplify fractions before moving to the next operation — it keeps numbers small and reduces errors.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• PEMDAS applies to fractions exactly as it does to whole numbers
• Parentheses first, then exponents, then multiplication/division (left to right), then addition/subtraction (left to right)
• To square a fraction: square both the numerator and denominator
• Simplify after each step to keep numbers manageable
• When in doubt about order, add parentheses to make the grouping explicit

Return to Arithmetic for more foundational math topics.