Order of Operations with Fractions and Decimals

Grade: 6-7 | Topic: Arithmetic

What You Will Learn

After this lesson you will be able to apply the order of operations to expressions that contain fractions and decimals. You will understand that a fraction bar acts as a grouping symbol, know how to evaluate complex numerators and denominators separately, and handle mixed expressions that combine fractions, decimals, and whole numbers.

Theory

PEMDAS Still Applies — No Exceptions

When an expression includes fractions or decimals, the same four priority levels apply:

1. Parentheses (and grouping symbols, including fraction bars)
2. Exponents
3. Multiplication and Division (left to right)
4. Addition and Subtraction (left to right)

The only new idea is that fractions introduce an additional grouping symbol.

The Fraction Bar as a Grouping Symbol

A fraction bar is not just a division sign — it also acts like parentheses. The expression:

$\frac{8 + 4}{3}$

means the same as:

$(8 + 4) \div 3 = 12 \div 3 = 4$

You must simplify the entire numerator and the entire denominator before dividing:

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

Rule: Treat the numerator as one group and the denominator as another. Apply PEMDAS inside each group, then divide.

Working with Mixed Numbers

When an expression includes mixed numbers (like $2\frac{1}{3}$), convert them to improper fractions first:

$2\frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{7}{3}$

Then proceed with normal fraction arithmetic.

Decimals in Order of Operations

Decimals follow the exact same PEMDAS rules. Two helpful strategies:

1. Work with decimals directly — line up decimal points for addition/subtraction and count decimal places for multiplication.
2. Convert to fractions — $0.5 = \frac{1}{2}$, $0.25 = \frac{1}{4}$, $0.75 = \frac{3}{4}$.

Choose whichever method makes the problem easier.

Worked Examples

Example 1: Fraction Bar as Grouping (Easy)

Problem: Evaluate $\frac{6 + 10}{4}$

Step 1: Simplify the numerator (it is a group):

$6 + 10 = 16$

Step 2: Divide by the denominator:

$\frac{16}{4} = 4$

Answer: $\boxed{4}$

Example 2: Operations in Both Numerator and Denominator (Medium)

Problem: Evaluate $\frac{3 \times 4 + 8}{2^2 + 1}$

Step 1 — Numerator (apply PEMDAS: multiply first, then add):

$3 \times 4 = 12$

$12 + 8 = 20$

Step 2 — Denominator (exponent first, then add):

$2^2 = 4$

$4 + 1 = 5$

Step 3 — Divide:

$\frac{20}{5} = 4$

Answer: $\boxed{4}$

Example 3: Fraction Arithmetic with PEMDAS (Medium)

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

Step 1 — Multiplication before addition (PEMDAS Level 3 before Level 4):

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

Step 2 — Addition (find a common denominator):

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

Answer: $\boxed{2}$

Example 4: Decimals with PEMDAS (Medium)

Problem: Evaluate $0.5 + 0.3 \times 4 - 0.2^2$

Step 1 — Exponent:

$0.2^2 = 0.04$

Step 2 — Multiplication:

$0.3 \times 4 = 1.2$

Step 3 — Addition and subtraction left to right:

$0.5 + 1.2 = 1.7$

$1.7 - 0.04 = 1.66$

Answer: $\boxed{1.66}$

Example 5: Complex Expression with Fractions and Parentheses (Challenging)

Problem: Evaluate $\frac{(2 + 3)^2 - 5}{\frac{1}{2} + \frac{3}{2}} \times \frac{1}{4}$

Step 1 — Numerator of the big fraction:

Parentheses: $2 + 3 = 5$

Exponent: $5^2 = 25$

Subtraction: $25 - 5 = 20$

Step 2 — Denominator of the big fraction:

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

Step 3 — Divide numerator by denominator:

$\frac{20}{2} = 10$

Step 4 — Multiply by $\frac{1}{4}$:

$10 \times \frac{1}{4} = \frac{10}{4} = \frac{5}{2} = 2.5$

Answer: $\boxed{\frac{5}{2}}$ or $2.5$

Common Mistakes

Mistake 1: Not treating the fraction bar as a grouping symbol

Expression: $\frac{6 + 2}{4}$

❌ Dividing first: $6 + \frac{2}{4} = 6 + 0.5 = 6.5$

✅ Treating the bar as grouping: $\frac{6 + 2}{4} = \frac{8}{4} = 2$

Why this matters: The fraction bar groups the entire numerator and the entire denominator. You must simplify each part before dividing.

Mistake 2: Adding fractions before multiplying

Expression: $\frac{1}{3} + \frac{1}{2} \times 6$

❌ Adding first: $\frac{1}{3} + \frac{1}{2} = \frac{5}{6}$, then $\frac{5}{6} \times 6 = 5$.

✅ Multiplying first (PEMDAS): $\frac{1}{2} \times 6 = 3$, then $\frac{1}{3} + 3 = \frac{1}{3} + \frac{9}{3} = \frac{10}{3}$.

Why this matters: Multiplication has higher priority than addition, even when fractions are involved. Do not let fractions trick you into abandoning PEMDAS.

Mistake 3: Forgetting to convert mixed numbers before calculating

Expression: $1\frac{1}{2} \times 4$

❌ Multiplying the whole number and fraction separately: $1 \times 4 = 4$ and $\frac{1}{2} \times 4 = 2$, then $4 + 2 = 6$.

✅ Convert first: $1\frac{1}{2} = \frac{3}{2}$, then $\frac{3}{2} \times 4 = \frac{12}{2} = 6$.

In this case both approaches give 6, but the "split" method fails for more complex expressions (especially with subtraction or exponents). Always convert to an improper fraction first.

Practice Problems

Try these on your own before checking the answers:

1. $\frac{15 - 3}{4}$
2. $\frac{2 + 4 \times 3}{7}$
3. $\frac{1}{4} + \frac{1}{2} \times 3$
4. $0.6 + 0.2 \times 5 - 0.1^2$
5. $\frac{(1 + 2)^3}{9} + \frac{1}{3} \times 6$

Click to see answers

1. $\frac{15 - 3}{4} = \frac{12}{4} = \mathbf{3}$ — simplify numerator first.
2. Numerator: $2 + 12 = 14$. Then $\frac{14}{7} = \mathbf{2}$.
3. Multiply first: $\frac{1}{2} \times 3 = \frac{3}{2}$. Then $\frac{1}{4} + \frac{3}{2} = \frac{1}{4} + \frac{6}{4} = \frac{7}{4} = \mathbf{1\frac{3}{4}}$.
4. Exponent: $0.01$. Multiply: $1.0$. Then $0.6 + 1.0 - 0.01 = \mathbf{1.59}$.
5. Parentheses: $3^3 = 27$. Then $\frac{27}{9} = 3$. Then $\frac{1}{3} \times 6 = 2$. Finally $3 + 2 = \mathbf{5}$.

Summary

• PEMDAS/BODMAS rules apply the same way whether the expression has whole numbers, fractions, or decimals.
• A fraction bar is a grouping symbol — simplify the numerator and denominator separately before dividing.
• When expressions include mixed numbers, convert to improper fractions before applying operations.
• For decimals, either work with them directly or convert to fractions — choose whichever makes the calculation easier.
• The most common error is ignoring the grouping effect of the fraction bar or abandoning PEMDAS when fractions appear.

Related Topics

• Order of Operations (PEMDAS/BODMAS) — Rules and Examples
• PEMDAS vs BODMAS — What's the Difference and How to Use Them
• Order of Operations Practice Problems with Answers
• Fractions — Complete Guide

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