Arithmetic operations
60 flashcards covering Arithmetic operations for the GMAT Quantitative section.
Arithmetic operations are the basic building blocks of math, involving simple actions like adding, subtracting, multiplying, and dividing numbers. These operations form the foundation for handling everyday calculations, such as balancing a budget or measuring ingredients, and they extend to more complex problems in algebra and statistics. If you're new to this, think of them as the tools you use to manipulate numbers accurately and efficiently.
On the GMAT Quantitative section, arithmetic operations frequently appear in problem-solving and data-sufficiency questions, often combined with word problems or real-world scenarios. Common traps include errors in the order of operations, mishandling fractions or decimals, and overlooking negative numbers, which can lead to incorrect answers under time pressure. Focus on practicing speed and accuracy, as these skills help you identify patterns and avoid careless mistakes. For a quick tip: Memorize key shortcuts, like multiplying by 10, to save valuable time.
Terms (60)
- 01
Addition
Addition is the arithmetic operation of combining two or more numbers to get a sum, such as 2 + 3 = 5.
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Subtraction
Subtraction is the arithmetic operation of taking one number away from another to find the difference, such as 5 - 2 = 3.
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Multiplication
Multiplication is the arithmetic operation of repeated addition, represented by the product of two numbers, such as 4 × 3 = 12.
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Division
Division is the arithmetic operation of splitting a number into equal parts, resulting in a quotient, such as 12 ÷ 3 = 4.
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Commutative Property of Addition
The commutative property of addition states that the order of addends does not affect the sum, so a + b equals b + a.
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Commutative Property of Multiplication
The commutative property of multiplication states that the order of factors does not affect the product, so a × b equals b × a.
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Associative Property of Addition
The associative property of addition states that the grouping of addends does not affect the sum, so (a + b) + c equals a + (b + c).
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Associative Property of Multiplication
The associative property of multiplication states that the grouping of factors does not affect the product, so (a × b) × c equals a × (b × c).
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Distributive Property
The distributive property states that multiplying a number by a sum is the same as multiplying the number by each addend and then adding the products, such as a × (b + c) = (a × b) + (a × c).
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Order of Operations
The order of operations is the rule that dictates the sequence for performing arithmetic calculations, starting with parentheses, then exponents, multiplication and division from left to right, and finally addition and subtraction from left to right.
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Parentheses in Order of Operations
In the order of operations, parentheses indicate that expressions inside them must be calculated first, such as in (2 + 3) × 4, where 2 + 3 is done before multiplying.
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Exponents in Order of Operations
Exponents are calculated after parentheses but before multiplication or division, as in 2^3 × 4, where 2^3 equals 8 before multiplying by 4.
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Multiplication and Division in Order of Operations
Multiplication and division are performed from left to right after parentheses and exponents, as in 10 ÷ 2 × 3, which is calculated as (10 ÷ 2) × 3 = 5 × 3 = 15.
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Addition and Subtraction in Order of Operations
Addition and subtraction are performed from left to right after all other operations, as in 10 + 5 - 2, which equals 15 - 2 = 13.
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Fractions
Fractions represent parts of a whole, consisting of a numerator over a denominator, such as 1/2 meaning one part out of two equal parts.
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Proper Fractions
Proper fractions are fractions where the numerator is less than the denominator, such as 3/4, which is less than 1.
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Improper Fractions
Improper fractions are fractions where the numerator is greater than or equal to the denominator, such as 5/3, which is greater than 1.
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Mixed Numbers
Mixed numbers combine a whole number and a proper fraction, such as 2 1/2, which equals 5/2.
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Simplifying Fractions
Simplifying fractions involves dividing both the numerator and denominator by their greatest common divisor to get the lowest terms, such as reducing 4/8 to 1/2.
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Adding Fractions with Like Denominators
Adding fractions with the same denominator means adding the numerators and keeping the denominator, such as 1/4 + 2/4 = 3/4.
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Adding Fractions with Unlike Denominators
Adding fractions with different denominators requires finding a common denominator first, then adding the numerators, such as 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
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Subtracting Fractions
Subtracting fractions involves ensuring they have a common denominator, then subtracting the numerators, such as 5/6 - 1/6 = 4/6, which simplifies to 2/3.
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Multiplying Fractions
Multiplying fractions means multiplying the numerators together and the denominators together, such as (2/3) × (3/4) = 6/12, which simplifies to 1/2.
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Dividing Fractions
Dividing fractions involves multiplying the first fraction by the reciprocal of the second, such as (2/3) ÷ (3/4) = (2/3) × (4/3) = 8/9.
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Reciprocals
Reciprocals are pairs of numbers that, when multiplied, equal 1, such as the reciprocal of 2/3 being 3/2.
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Decimals
Decimals are numbers expressed in base-10 with a decimal point, such as 0.5 representing half.
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Converting Decimals to Fractions
Converting decimals to fractions involves expressing the decimal as a fraction, such as 0.75 becoming 75/100, which simplifies to 3/4.
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Percentages
Percentages express a number as a part per hundred, such as 50% meaning 50 per 100, or 0.5.
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Converting Percentages to Decimals
Converting percentages to decimals involves dividing by 100, such as 25% becoming 0.25.
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Percentage Change
Percentage change calculates the increase or decrease relative to the original amount, such as (new value - original value) / original value × 100%.
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Ratios
Ratios compare two quantities by division, often written as a:b or a/b, such as 2:3 meaning two parts to three parts.
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Solving Proportions
Solving proportions involves finding the missing value in two equal ratios, such as in 2/3 = x/6, where x equals 4.
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Exponents
Exponents indicate how many times a base is multiplied by itself, such as 2^3 meaning 2 × 2 × 2 = 8.
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Product Rule for Exponents
The product rule for exponents states that when multiplying powers with the same base, add the exponents, such as a^m × a^n = a^(m+n).
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Quotient Rule for Exponents
The quotient rule for exponents states that when dividing powers with the same base, subtract the exponents, such as a^m / a^n = a^(m-n).
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Power Rule for Exponents
The power rule for exponents states that when raising a power to another power, multiply the exponents, such as (a^m)^n = a^(m×n).
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Negative Exponents
Negative exponents indicate the reciprocal of the base raised to the positive exponent, such as 2^-3 = 1 / 2^3 = 1/8.
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Square Roots
Square roots are the values that, when multiplied by themselves, give the original number, such as the square root of 9 being 3.
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Cube Roots
Cube roots are the values that, when multiplied by themselves three times, give the original number, such as the cube root of 8 being 2.
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Absolute Value
Absolute value is the distance of a number from zero on the number line, always positive, such as | -5 | = 5.
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Even Numbers
Even numbers are integers divisible by 2 with no remainder, such as 2, 4, and 6.
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Odd Numbers
Odd numbers are integers not divisible by 2, leaving a remainder of 1 when divided by 2, such as 1, 3, and 5.
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Positive Numbers
Positive numbers are greater than zero on the number line, such as 1, 2, or 3.
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Negative Numbers
Negative numbers are less than zero on the number line, such as -1, -2, or -3.
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Zero in Arithmetic
Zero is the additive identity, meaning adding zero to any number leaves it unchanged, and it is neither positive nor negative.
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Greatest Common Factor
The greatest common factor is the largest number that divides evenly into two or more numbers, such as 6 for 12 and 18.
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Least Common Multiple
The least common multiple is the smallest number that is a multiple of two or more numbers, such as 12 for 4 and 6.
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Prime Factorization
Prime factorization breaks down a number into its prime number components, such as 12 = 2 × 2 × 3.
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Divisibility by 2
A number is divisible by 2 if it is even, meaning its last digit is 0, 2, 4, 6, or 8.
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Divisibility by 3
A number is divisible by 3 if the sum of its digits is divisible by 3, such as 12 because 1 + 2 = 3.
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Remainders
Remainders are the amounts left over after division, such as 5 when 17 is divided by 6, since 6 × 2 = 12 and 17 - 12 = 5.
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Modular Arithmetic Example
Modular arithmetic deals with remainders, such as 17 mod 6 equals 5, which is the remainder when 17 is divided by 6.
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Sign Errors in Arithmetic
Sign errors occur when incorrectly handling positive and negative numbers, such as mistakenly adding signs instead of subtracting them in subtraction.
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Fractional Exponents
Fractional exponents represent roots and powers, such as 4^(1/2) meaning the square root of 4, which is 2.
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Percentage of a Number
Finding a percentage of a number involves multiplying the number by the percentage in decimal form, such as 20% of 50 is 0.20 × 50 = 10.
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Ratio Simplification
Ratio simplification reduces a ratio to its lowest terms by dividing both parts by their greatest common factor, such as simplifying 4:6 to 2:3.
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Exponent Zero Rule
Any non-zero number raised to the power of zero equals 1, such as 5^0 = 1.
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Decimal Rounding
Decimal rounding adjusts a number to a specified place value, such as rounding 3.1416 to two decimal places gives 3.14.
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Arithmetic Sequence
An arithmetic sequence is a series of numbers where the difference between consecutive terms is constant, such as 2, 4, 6, 8.
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Weighted Average
A weighted average is the mean of values adjusted by their weights, such as (3 × 2 + 4 × 3) / (2 + 3) = (6 + 12) / 5 = 3.6.