Introduction
Order of operations is a fundamental concept in mathematics that determines the sequence in which operations should be performed in an expression. When working with fractions, it's especially important to follow these rules to get correct results.
In this lesson, we'll learn how to apply the order of operations (BODMAS/PEMDAS) to simplify expressions containing fractions with multiple operations. We'll start with simple expressions and progress to more complex ones involving addition, subtraction, multiplication, and division of fractions.
Example expression with fractions:
\[ \left(\frac{1}{2} + \frac{3}{4}\right) \times \frac{2}{5} - \frac{1}{10} \]BODMAS/PEMDAS Rules
Understanding the Order
BODMAS and PEMDAS are acronyms that help remember the order of operations:
- BODMAS: Brackets, Orders (exponents), Division/Multiplication (left to right), Addition/Subtraction (left to right)
- PEMDAS: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right)
Both acronyms lead to the same order of operations.
Step-by-Step Application
When simplifying expressions with fractions, follow these steps:
1. Brackets/Parentheses
First, perform operations inside brackets or parentheses.
\[ \left(\frac{1}{2} + \frac{1}{4}\right) \times 2 = \frac{3}{4} \times 2 \]2. Exponents/Orders
Next, evaluate any exponents or powers.
\[ \left(\frac{2}{3}\right)^2 = \frac{4}{9} \]3. Division & Multiplication
Then perform division and multiplication from left to right.
\[ \frac{1}{2} \times \frac{3}{4} \div \frac{1}{2} = \frac{3}{8} \div \frac{1}{2} = \frac{3}{4} \]4. Addition & Subtraction
Finally, perform addition and subtraction from left to right.
\[ \frac{1}{2} + \frac{1}{3} - \frac{1}{6} = \frac{5}{6} - \frac{1}{6} = \frac{4}{6} = \frac{2}{3} \]Practice Exercise
Question 1
Simplify: \(\frac{1}{2} + \frac{3}{4} \times \frac{2}{3}\)
Solution:
First multiply, then add:
\[ \frac{3}{4} \times \frac{2}{3} = \frac{6}{12} = \frac{1}{2} \] \[ \frac{1}{2} + \frac{1}{2} = 1 \]Question 2
Simplify: \(\left(\frac{1}{2} + \frac{1}{3}\right) \div \frac{5}{6}\)
Solution:
First brackets, then division:
\[ \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \] \[ \frac{5}{6} \div \frac{5}{6} = 1 \]Question 3
Simplify: \(\frac{2}{3} \times \left(\frac{1}{4} + \frac{3}{8}\right) - \frac{1}{6}\)
Solution:
First brackets, then multiplication, finally subtraction:
\[ \frac{1}{4} + \frac{3}{8} = \frac{2}{8} + \frac{3}{8} = \frac{5}{8} \] \[ \frac{2}{3} \times \frac{5}{8} = \frac{10}{24} = \frac{5}{12} \] \[ \frac{5}{12} - \frac{1}{6} = \frac{5}{12} - \frac{2}{12} = \frac{3}{12} = \frac{1}{4} \]Ready for more challenges?
Full Practice SetFraction Operations
Addition & Subtraction
To add or subtract fractions, they must have the same denominator (common denominator). If they don't, find equivalent fractions with a common denominator first.
Example:
\[ \frac{1}{2} + \frac{1}{3} - \frac{1}{6} \]Find common denominator (6):
\[ \frac{3}{6} + \frac{2}{6} - \frac{1}{6} = \frac{4}{6} = \frac{2}{3} \]Multiplication & Division
For multiplication, multiply numerators together and denominators together. For division, multiply by the reciprocal of the divisor.
Multiplication example:
\[ \frac{2}{3} \times \frac{5}{7} = \frac{10}{21} \]Division example:
\[ \frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8} \]Mixed Operations
Combining Operations
When expressions contain multiple operations, carefully apply BODMAS/PEMDAS rules while performing each fraction operation correctly.
Example 1
Simplify: \(\frac{1}{2} + \frac{3}{4} \times \frac{2}{3} - \frac{1}{6}\)
Step 1: Multiplication first
\[ \frac{3}{4} \times \frac{2}{3} = \frac{6}{12} = \frac{1}{2} \]Step 2: Now addition and subtraction (left to right)
\[ \frac{1}{2} + \frac{1}{2} = 1 \] \[ 1 - \frac{1}{6} = \frac{5}{6} \]Example 2
Simplify: \(\left(\frac{2}{3} - \frac{1}{4}\right) \div \frac{5}{6} + \frac{1}{2}\)
Step 1: Brackets first
\[ \frac{2}{3} - \frac{1}{4} = \frac{8}{12} - \frac{3}{12} = \frac{5}{12} \]Step 2: Division next
\[ \frac{5}{12} \div \frac{5}{6} = \frac{5}{12} \times \frac{6}{5} = \frac{30}{60} = \frac{1}{2} \]Step 3: Addition last
\[ \frac{1}{2} + \frac{1}{2} = 1 \]Practice Exercise
Question 1
Simplify: \(\frac{3}{4} \times \frac{2}{3} + \frac{1}{2} \div \frac{1}{4}\)
Solution:
Multiplication and division first (left to right):
\[ \frac{3}{4} \times \frac{2}{3} = \frac{6}{12} = \frac{1}{2} \] \[ \frac{1}{2} \div \frac{1}{4} = \frac{1}{2} \times \frac{4}{1} = 2 \]Then addition:
\[ \frac{1}{2} + 2 = \frac{5}{2} \]Question 2
Simplify: \(\left(\frac{2}{5} + \frac{1}{3}\right) \times \frac{15}{11} - \frac{1}{2}\)
Solution:
Brackets first:
\[ \frac{2}{5} + \frac{1}{3} = \frac{6}{15} + \frac{5}{15} = \frac{11}{15} \]Then multiplication:
\[ \frac{11}{15} \times \frac{15}{11} = 1 \]Finally subtraction:
\[ 1 - \frac{1}{2} = \frac{1}{2} \]Question 3
Simplify: \(\frac{1}{2} + \frac{3}{4} \times \frac{8}{9} - \frac{1}{3} \div \frac{2}{5}\)
Solution:
Multiplication and division first (left to right):
\[ \frac{3}{4} \times \frac{8}{9} = \frac{24}{36} = \frac{2}{3} \] \[ \frac{1}{3} \div \frac{2}{5} = \frac{1}{3} \times \frac{5}{2} = \frac{5}{6} \]Then addition and subtraction (left to right):
\[ \frac{1}{2} + \frac{2}{3} = \frac{3}{6} + \frac{4}{6} = \frac{7}{6} \] \[ \frac{7}{6} - \frac{5}{6} = \frac{2}{6} = \frac{1}{3} \]Ready for more challenges?
Full Practice SetWord Problems
Problem-Solving Approach
When solving word problems involving fractions and multiple operations:
- Read the problem carefully and identify what is being asked
- Determine which operations are needed
- Write the mathematical expression
- Apply BODMAS/PEMDAS rules to solve
- Check if the answer makes sense in the context of the problem
Example 1
A recipe requires \(\frac{1}{2}\) cup of sugar for the cake and \(\frac{1}{3}\) cup for the frosting. If you want to make \(\frac{3}{4}\) of the recipe, how much sugar will you need in total?
Solution:
First find total sugar for full recipe:
\[ \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \text{ cup} \]Then calculate \(\frac{3}{4}\) of this amount:
\[ \frac{5}{6} \times \frac{3}{4} = \frac{15}{24} = \frac{5}{8} \text{ cup} \]Example 2
A tank is \(\frac{2}{5}\) full. After adding 30 liters, it becomes \(\frac{3}{4}\) full. What is the total capacity of the tank?
Solution:
The difference between \(\frac{3}{4}\) and \(\frac{2}{5}\) represents 30 liters:
\[ \frac{3}{4} - \frac{2}{5} = \frac{15}{20} - \frac{8}{20} = \frac{7}{20} \]So \(\frac{7}{20}\) of the tank = 30 liters
Total capacity = \(30 \div \frac{7}{20} = 30 \times \frac{20}{7} = \frac{600}{7} \approx 85.71\) liters
Practice Exercise
Question 1
John walks \(\frac{3}{4}\) km to school and \(\frac{2}{5}\) km to the library. What is the total distance he walks if he goes to school and then to the library?
Solution:
Simply add the two distances:
\[ \frac{3}{4} + \frac{2}{5} = \frac{15}{20} + \frac{8}{20} = \frac{23}{20} = 1\frac{3}{20} \text{ km} \]Question 2
A piece of cloth is \(4\frac{1}{2}\) meters long. If \(\frac{2}{3}\) of it is cut off, how much cloth remains?
Solution:
First convert mixed number to improper fraction:
\[ 4\frac{1}{2} = \frac{9}{2} \]Calculate \(\frac{2}{3}\) of the cloth:
\[ \frac{9}{2} \times \frac{2}{3} = 3 \text{ meters (cut off)} \]Remaining cloth:
\[ \frac{9}{2} - 3 = \frac{9}{2} - \frac{6}{2} = \frac{3}{2} = 1\frac{1}{2} \text{ meters} \]Question 3
A recipe calls for \(\frac{3}{4}\) cup of flour. If you want to make \(\frac{1}{2}\) of the recipe, plus an additional \(\frac{1}{8}\) cup, how much flour will you need?
Solution:
Calculate \(\frac{1}{2}\) of the recipe:
\[ \frac{3}{4} \times \frac{1}{2} = \frac{3}{8} \text{ cup} \]Add the additional \(\frac{1}{8}\) cup:
\[ \frac{3}{8} + \frac{1}{8} = \frac{4}{8} = \frac{1}{2} \text{ cup} \]Ready for more challenges?
Full Practice SetMath Challenge
Timed Test in 7 Levels
Test your fraction operation skills through progressively challenging levels