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 that involve multiple operations (addition, subtraction, multiplication, division), it's crucial to follow these rules to get the correct result.
There are two common acronyms used to remember the order of operations:
BODMAS
- B - Brackets
- O - Orders (exponents/indices)
- D - Division
- M - Multiplication
- A - Addition
- S - Subtraction
PEMDAS
- P - Parentheses (brackets)
- E - Exponents (orders)
- MD - Multiplication and Division (left to right)
- AS - Addition and Subtraction (left to right)
For example, in the expression \( 3 + 4 \times 2 \):
Correct: \( 3 + (4 \times 2) = 3 + 8 = 11 \)
Incorrect: \( (3 + 4) \times 2 = 7 \times 2 = 14 \)
Whole Numbers
Key Steps
When simplifying expressions with whole numbers:
1. Brackets First
Always start with operations inside brackets (parentheses).
2. Exponents/Orders
Next, evaluate any exponents or indices.
3. Division & Multiplication
Work from left to right through these operations.
4. Addition & Subtraction
Finally, work from left to right through these operations.
Example 1:
Simplify \( 12 ÷ 3 × 4 + 5 - 2 \)
Solution:
1. Division first: \( 12 ÷ 3 = 4 \)
2. Multiplication: \( 4 × 4 = 16 \)
3. Addition: \( 16 + 5 = 21 \)
4. Subtraction: \( 21 - 2 = 19 \)
Final answer: \( 19 \)
Example 2:
Simplify \( 5 + 2 × (8 - 3)^2 ÷ 5 \)
Solution:
1. Brackets first: \( (8 - 3) = 5 \)
2. Exponents: \( 5^2 = 25 \)
3. Division: \( 25 ÷ 5 = 5 \)
4. Multiplication: \( 2 × 5 = 10 \)
5. Addition: \( 5 + 10 = 15 \)
Final answer: \( 15 \)
Practice Exercise
Question 1
Simplify \( 18 ÷ 3 × 2 + 5 - 1 \)
1. Division first: \( 18 ÷ 3 = 6 \)
2. Multiplication: \( 6 × 2 = 12 \)
3. Addition: \( 12 + 5 = 17 \)
4. Subtraction: \( 17 - 1 = 16 \)
Final answer: \( 16 \)
Question 2
Simplify \( (15 - 3) × 2 + 8 ÷ 4 \)
1. Brackets first: \( (15 - 3) = 12 \)
2. Multiplication: \( 12 × 2 = 24 \)
3. Division: \( 8 ÷ 4 = 2 \)
4. Addition: \( 24 + 2 = 26 \)
Final answer: \( 26 \)
Question 3
Simplify \( 3 + 6 × (5 + 1) ÷ 9 - 2 \)
1. Brackets first: \( (5 + 1) = 6 \)
2. Multiplication: \( 6 × 6 = 36 \)
3. Division: \( 36 ÷ 9 = 4 \)
4. Addition: \( 3 + 4 = 7 \)
5. Subtraction: \( 7 - 2 = 5 \)
Final answer: \( 5 \)
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Full Practice SetFractions
Working with Fractions
The same BODMAS/PEMDAS rules apply when working with fractions, but we need to be careful with fraction operations:
1. Mixed Numbers
Convert mixed numbers to improper fractions first.
2. Division of Fractions
Remember to multiply by the reciprocal when dividing fractions.
3. Common Denominator
For addition/subtraction, find a common denominator.
Example 1:
Simplify \( \frac{1}{2} + \frac{3}{4} × \frac{2}{3} \)
Solution:
1. Multiplication first: \( \frac{3}{4} × \frac{2}{3} = \frac{6}{12} = \frac{1}{2} \)
2. Addition: \( \frac{1}{2} + \frac{1}{2} = 1 \)
Final answer: \( 1 \)
Example 2:
Simplify \( \frac{5}{6} ÷ \left(\frac{1}{2} + \frac{1}{3}\right) \)
Solution:
1. Brackets first (find common denominator 6):
\( \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \)
2. Division (multiply by reciprocal):
\( \frac{5}{6} ÷ \frac{5}{6} = \frac{5}{6} × \frac{6}{5} = 1 \)
Final answer: \( 1 \)
Practice Exercise
Question 1
Simplify \( \frac{2}{3} × \frac{3}{4} + \frac{1}{2} \)
1. Multiplication first: \( \frac{2}{3} × \frac{3}{4} = \frac{6}{12} = \frac{1}{2} \)
2. Addition: \( \frac{1}{2} + \frac{1}{2} = 1 \)
Final answer: \( 1 \)
Question 2
Simplify \( \frac{3}{5} - \frac{1}{4} × \frac{4}{5} \)
1. Multiplication first: \( \frac{1}{4} × \frac{4}{5} = \frac{4}{20} = \frac{1}{5} \)
2. Subtraction (common denominator 5): \( \frac{3}{5} - \frac{1}{5} = \frac{2}{5} \)
Final answer: \( \frac{2}{5} \)
Question 3
Simplify \( \left(\frac{1}{2} + \frac{1}{3}\right) ÷ \frac{5}{6} \)
1. Brackets first (common denominator 6): \( \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \)
2. Division (multiply by reciprocal): \( \frac{5}{6} ÷ \frac{5}{6} = \frac{5}{6} × \frac{6}{5} = 1 \)
Final answer: \( 1 \)
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Full Practice SetMixed Operations
Combining Whole Numbers and Fractions
When expressions contain both whole numbers and fractions, we follow the same BODMAS/PEMDAS rules but need to carefully convert between forms when necessary.
Example 1:
Simplify \( 2 + \frac{1}{2} × \frac{4}{5} \)
Solution:
1. Multiplication first: \( \frac{1}{2} × \frac{4}{5} = \frac{4}{10} = \frac{2}{5} \)
2. Convert whole number to fraction: \( 2 = \frac{10}{5} \)
3. Addition: \( \frac{10}{5} + \frac{2}{5} = \frac{12}{5} \)
Final answer: \( \frac{12}{5} \) or \( 2\frac{2}{5} \)
Example 2:
Simplify \( \frac{3}{4} × (2 - \frac{1}{2}) + \frac{1}{8} \)
Solution:
1. Brackets first: \( 2 - \frac{1}{2} = \frac{4}{2} - \frac{1}{2} = \frac{3}{2} \)
2. Multiplication: \( \frac{3}{4} × \frac{3}{2} = \frac{9}{8} \)
3. Addition: \( \frac{9}{8} + \frac{1}{8} = \frac{10}{8} = \frac{5}{4} \)
Final answer: \( \frac{5}{4} \) or \( 1\frac{1}{4} \)
Practice Exercise
Question 1
Simplify \( 3 × \frac{1}{2} + \frac{3}{4} ÷ \frac{1}{2} \)
1. Multiplication first: \( 3 × \frac{1}{2} = \frac{3}{2} \)
2. Division: \( \frac{3}{4} ÷ \frac{1}{2} = \frac{3}{4} × \frac{2}{1} = \frac{6}{4} = \frac{3}{2} \)
3. Addition: \( \frac{3}{2} + \frac{3}{2} = \frac{6}{2} = 3 \)
Final answer: \( 3 \)
Question 2
Simplify \( \frac{1}{2} + \left(1\frac{1}{2} × \frac{2}{3}\right) - \frac{1}{4} \)
1. Convert mixed number: \( 1\frac{1}{2} = \frac{3}{2} \)
2. Brackets first (multiplication): \( \frac{3}{2} × \frac{2}{3} = \frac{6}{6} = 1 \)
3. Addition: \( \frac{1}{2} + 1 = \frac{3}{2} \)
4. Subtraction: \( \frac{3}{2} - \frac{1}{4} = \frac{6}{4} - \frac{1}{4} = \frac{5}{4} \)
Final answer: \( \frac{5}{4} \) or \( 1\frac{1}{4} \)
Question 3
Simplify \( \left(\frac{2}{3} + \frac{1}{6}\right) × (2 - \frac{1}{2}) ÷ \frac{1}{4} \)
1. First brackets (common denominator 6): \( \frac{4}{6} + \frac{1}{6} = \frac{5}{6} \)
2. Second brackets: \( 2 - \frac{1}{2} = \frac{4}{2} - \frac{1}{2} = \frac{3}{2} \)
3. Multiplication: \( \frac{5}{6} × \frac{3}{2} = \frac{15}{12} = \frac{5}{4} \)
4. Division: \( \frac{5}{4} ÷ \frac{1}{4} = \frac{5}{4} × \frac{4}{1} = \frac{20}{4} = 5 \)
Final answer: \( 5 \)
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Timed Test in 7 Levels
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