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Patterns and Algebra - Order of Operations (PEMDAS/BODMAS)

Grade 6IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The Order of Operations is a fundamental set of rules used to solve mathematical expressions with multiple operations, ensuring everyone arrives at the same answer. Imagine a priority pyramid where the most powerful operations sit at the top and are performed first.

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Brackets or Parentheses ()() are the highest priority. If you see nested brackets like [()][( )], you must work from the innermost set to the outermost set, effectively 'peeling' the expression like an onion.

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Exponents (also called Orders or Indices) involve numbers raised to a power, such as 525^{2}, or square roots like 16\sqrt{16}. These are calculated immediately after resolving the contents of all brackets.

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Multiplication and Division are equal in rank. When both appear in an expression, you must solve them by moving from left to right, similar to how you read a sentence. Do not assume multiplication always comes before division.

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Addition and Subtraction are the final operations to be performed and are also equal in rank. Like multiplication and division, they are solved using the left-to-right rule. Visually, these are the 'base' of our operation pyramid.

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The PEMDAS/BODMAS acronyms serve as a mental checklist: Parentheses/Brackets, Exponents/Orders, Multiplication/Division (left to right), and Addition/Subtraction (left to right).

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Fraction bars act as a grouping symbol. When you see a large horizontal line a+bc\frac{a + b}{c}, you must treat the numerator and the denominator as if they are inside invisible brackets, solving the top and bottom completely before dividing.

📐Formulae

PEMDAS = Parentheses, Exponents, Multiplication, Division, Addition, Subtraction

BODMAS = Brackets, Orders, Division, Multiplication, Addition, Subtraction

xn (Exponents are solved after Brackets)x^{n} \text{ (Exponents are solved after Brackets)}

x (Roots are treated as Orders/Exponents)\sqrt{x} \text{ (Roots are treated as Orders/Exponents)}

NumeratorDenominator (The division bar groups terms)\frac{\text{Numerator}}{\text{Denominator}} \text{ (The division bar groups terms)}

💡Examples

Problem 1:

Evaluate the expression: 15+(12÷22)×3−515 + (12 \div 2^{2}) \times 3 - 5

Solution:

Step 1: Solve inside the parentheses, starting with the exponent: 22=42^{2} = 4. The expression becomes 15+(12÷4)×3−515 + (12 \div 4) \times 3 - 5. Step 2: Complete the parentheses: 12÷4=312 \div 4 = 3. The expression becomes 15+3×3−515 + 3 \times 3 - 5. Step 3: Perform multiplication: 3×3=93 \times 3 = 9. The expression becomes 15+9−515 + 9 - 5. Step 4: Perform addition and subtraction from left to right: 15+9=2415 + 9 = 24, then 24−5=1924 - 5 = 19.

Explanation:

We follow PEMDAS by prioritizing the exponent inside the bracket first, then clearing the bracket, then multiplying, and finally adding/subtracting from left to right.

Problem 2:

Calculate: 10×[20÷(2+3)]−410 \times [20 \div (2 + 3)] - 4

Solution:

Step 1: Solve the innermost parentheses: (2+3)=5(2 + 3) = 5. The expression becomes 10×[20÷5]−410 \times [20 \div 5] - 4. Step 2: Solve the outer square brackets: [20÷5]=4[20 \div 5] = 4. The expression becomes 10×4−410 \times 4 - 4. Step 3: Perform multiplication: 10×4=4010 \times 4 = 40. The expression becomes 40−440 - 4. Step 4: Perform subtraction: 40−4=3640 - 4 = 36.

Explanation:

This problem demonstrates nested grouping symbols. We must work from the 'inside out' (round brackets then square brackets) before applying the remaining operations.