Order of operations
54 flashcards covering Order of operations for the GMAT Quantitative section.
Order of operations is a fundamental math rule that dictates the sequence for performing calculations in an expression, ensuring everyone arrives at the same result. For instance, in a problem like 5 + 6 × 2, you don't add first; instead, you multiply 6 and 2 before adding 5, following the standard acronym PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right). This concept is essential because it prevents errors in more complex equations and forms the backbone of algebraic problem-solving.
On the GMAT Quantitative section, order of operations appears in problem-solving and data sufficiency questions, often within algebra or arithmetic problems where you must simplify expressions quickly and accurately. Common traps include overlooking parentheses or performing operations out of sequence, which can lead to wrong answers in multi-step calculations. Focus on mastering PEMDAS and practicing with varied expressions to spot potential pitfalls. Always remember to follow PEMDAS step by step for reliable results.
Terms (54)
- 01
Order of Operations
The standard sequence for evaluating mathematical expressions, ensuring that operations are performed in a specific order to avoid ambiguity.
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PEMDAS
An acronym representing the order of operations: Parentheses first, then Exponents, followed by Multiplication and Division from left to right, and finally Addition and Subtraction from left to right.
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Parentheses in Expressions
Grouping symbols that indicate operations inside them must be performed before operations outside, allowing for custom order in complex expressions.
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Nested Parentheses
Parentheses within parentheses, where the innermost set is evaluated first, then outward, to handle layered operations in an expression.
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Exponents in Order of Operations
Exponents are calculated after parentheses but before multiplication, division, addition, or subtraction in an expression.
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Square Roots as Exponents
Square roots are treated as exponents of one-half and must be evaluated after parentheses but before multiplication or division.
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Multiplication in Order of Operations
Multiplication is performed after parentheses and exponents, and it holds equal precedence with division, so they are done from left to right.
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Division in Order of Operations
Division is performed after parentheses and exponents, sharing equal precedence with multiplication, evaluated from left to right.
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Left-to-Right Rule for Multiplication and Division
When multiplication and division appear in the same expression, they are carried out in the order they appear from left to right, regardless of which comes first.
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Addition in Order of Operations
Addition is performed after parentheses, exponents, multiplication, and division, and it shares equal precedence with subtraction.
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Subtraction in Order of Operations
Subtraction is performed after parentheses, exponents, multiplication, and division, evaluated from left to right alongside addition.
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Left-to-Right Rule for Addition and Subtraction
When addition and subtraction are in the same expression, they are executed in the order they appear from left to right.
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Implied Multiplication
A form of multiplication without an explicit symbol, such as a number next to a parenthesis, which is treated the same as explicit multiplication in the order of operations.
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Fractions and Order of Operations
In fractions, the numerator and denominator are treated as grouped expressions, so operations within them are resolved before the fraction is simplified.
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Order with Absolute Values
Absolute values are treated like parentheses, meaning the expression inside must be evaluated first before applying the absolute value.
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Exponents with Negative Bases
When raising a negative number to a power, the base is negative, and the exponent is applied as per order, but care must be taken with even and odd exponents.
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Order in Algebraic Expressions
In expressions with variables, operations follow the same order, so exponents on variables are calculated before multiplication by coefficients.
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Common Mistake: Addition Before Multiplication
A frequent error where addition is performed before multiplication, which violates the order and leads to incorrect results.
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Common Mistake: Exponents After Multiplication
An error where multiplication is done before exponents, reversing the correct order and altering the expression's value.
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Using Parentheses to Override Order
Parentheses can be added to an expression to change the default order, forcing certain operations to occur earlier.
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Evaluating Expressions with Decimals
Decimals in expressions are handled by following the standard order, with no special rules, ensuring accuracy in calculations.
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Order with Percentages
Percentages are converted to decimals or fractions and then evaluated according to the order of operations in the expression.
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Scientific Notation and Order
In scientific notation, exponents are evaluated first, followed by multiplication, as part of the overall order of operations.
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Order in Equations vs. Expressions
In equations, order of operations is used to simplify expressions on both sides before solving, but it does not change the equality.
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Simplifying Complex Expressions
A process that applies order of operations step by step to reduce an expression to its simplest form, often involving multiple levels.
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Example: Simple Expression
For an expression like 2 + 3 × 4, multiplication is done first, so it equals 2 + 12 = 14.
Evaluate 2 + 3 × 4: First, 3 × 4 = 12, then 2 + 12 = 14.
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Example: With Exponents
For 2 × 3^2 + 1, exponents are calculated first, then multiplication, and finally addition.
Evaluate 2 × 3^2 + 1: First, 3^2 = 9, then 2 × 9 = 18, then 18 + 1 = 19.
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Example: With Parentheses
For (2 + 3) × 4, the parentheses require addition first, then multiplication.
Evaluate (2 + 3) × 4: First, 2 + 3 = 5, then 5 × 4 = 20.
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Example: Fractions in Expressions
For 1 + 1/2 × 3, the fraction is part of multiplication, so division and multiplication precede addition.
Evaluate 1 + 1/2 × 3: First, 1/2 = 0.5, then 0.5 × 3 = 1.5, then 1 + 1.5 = 2.5.
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Example: Multiple Operations
For 10 - 2 × 3 + 4 ÷ 2, operations are performed in order: multiplication and division first, from left to right, then subtraction and addition.
Evaluate 10 - 2 × 3 + 4 ÷ 2: First, 2 × 3 = 6 and 4 ÷ 2 = 2, so 10 - 6 + 2 = 6.
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Strategy: Break Down Expressions
To avoid errors, break an expression into steps by identifying and evaluating the highest precedence operations first.
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Order with Roots and Exponents
Roots, like square roots, are exponents and must be evaluated after parentheses but before multiplication.
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Handling Negative Exponents
Negative exponents indicate reciprocals and are evaluated after parentheses but before multiplication.
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Order in Inequalities
The same order applies to inequalities as expressions, ensuring correct simplification before solving.
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Common Trap: Division and Multiplication Order
Students often mistakenly prioritize division over multiplication if it appears later, but both are done left to right.
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Example: With Absolute Values
For |3 - 5| × 2, the absolute value is evaluated first, then multiplication.
Evaluate |3 - 5| × 2: First, 3 - 5 = -2, then |-2| = 2, then 2 × 2 = 4.
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Example: Algebraic Expression
For 2x + 3 × 4, multiplication is done before addition, so it's 2x + 12.
If x = 1, evaluate 2(1) + 3 × 4: First, 3 × 4 = 12, then 2(1) + 12 = 2 + 12 = 14.
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Order with Variables and Constants
In expressions like 2 + x^2, exponents on variables are calculated before addition.
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Common Mistake: Parentheses Omission
Forgetting that implied grouping, like in fractions, requires evaluating the numerator and denominator first.
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Strategy: Parenthesize for Clarity
Rewriting expressions with extra parentheses can help verify the correct order without changing the value.
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Example: Nested Operations
For 2 + (3 × 4)^2, parentheses and exponents are handled in sequence.
Evaluate 2 + (3 × 4)^2: First, 3 × 4 = 12, then 12^2 = 144, then 2 + 144 = 146.
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Order in Decimal Expressions
Decimals do not alter the order; for example, 1.5 × 2 + 3 is multiplication before addition.
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Handling Zero in Exponents
Any non-zero number to the power of zero is 1, and this is evaluated after parentheses.
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Example: With Division and Multiplication
For 10 ÷ 2 × 3, division and multiplication are done left to right.
Evaluate 10 ÷ 2 × 3: First, 10 ÷ 2 = 5, then 5 × 3 = 15.
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Common Trap: Subtraction as Negative
Subtraction is not the same as a negative sign; it follows addition in order.
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Strategy: Use a Calculator Mindfully
On the GMAT, remember that calculators follow order of operations, so input expressions correctly.
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Example: Percentages in Expressions
For 100 + 10% of 50, percentages are calculated as multiplication by 0.10 first.
Evaluate 100 + 10% of 50: First, 10% of 50 = 0.10 × 50 = 5, then 100 + 5 = 105.
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Order with Multiple Exponents
Each exponent is evaluated independently after parentheses.
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Common Mistake: Left-to-Right Ignorance
Failing to apply left-to-right for equal precedence operations can lead to errors in complex expressions.
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Example: Fractions with Operations
For 1 / (2 + 3) × 4, parentheses are evaluated first, then division and multiplication.
Evaluate 1 / (2 + 3) × 4: First, 2 + 3 = 5, then 1 / 5 = 0.2, then 0.2 × 4 = 0.8.
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Strategy: Check for Ambiguity
If an expression seems unclear, add parentheses based on intended order to ensure accurate evaluation.
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Order in Word Problems
Translating word problems into expressions requires applying order of operations to the resulting math.
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Example: With Negative Numbers
For -2^2 + 3, the exponent applies to 2 first, then the negative sign is handled.
Evaluate -2^2 + 3: First, 2^2 = 4, then -4 + 3 = -1.
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Handling Implied Exponents
In some contexts, like roots, implied exponents must be recognized and evaluated correctly.