Questions and Answers
What is the set of numbers that includes whole numbers and their negatives?
Which property of operations states that a + (b + c) = (a + b) + c?
What is the result of multiplying 2/3 by 3/4?
What is the first step in the Order of Operations (PEMDAS/BODMAS)?
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What is the value of 0.75 as a percentage?
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What is the result of multiplying two fractions: 1/2 × 1/3?
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What is the ratio of 4 to 6?
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What is the simplified form of the fraction 6/8?
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What is the definition of an acute angle?
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What is the result of dividing 2/3 by 3/4?
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Study Notes
Basic Arithmetic
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Numbers and Types of Numbers:
- Natural Numbers: positive integers (1, 2, 3, …)
- Whole Numbers: natural numbers including zero (0, 1, 2, 3, …)
- Integers: whole numbers and their negatives (…, -3, -2, -1, 0, 1, 2, 3, …)
- Rational Numbers: numbers can be expressed as a fraction of two integers (e.g., 1/2, 3/4)
- Irrational Numbers: numbers cannot be expressed as a fraction of two integers (e.g., √2, π)
- Real Numbers: all rational and irrational numbers
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Basic Operations:
- Addition (+): combining two numbers to get a sum
- Subtraction (−): finding the difference between two numbers
- Multiplication (×): repeated addition of a number
- Division (÷): splitting a number into equal parts
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Properties of Operations:
- Commutative Property:
- Addition: a + b = b + a
- Multiplication: a × b = b × a
- Associative Property:
- Addition: (a + b) + c = a + (b + c)
- Multiplication: (a × b) × c = a × (b × c)
- Distributive Property: a × (b + c) = (a × b) + (a × c)
- Identity Property:
- Addition: a + 0 = a
- Multiplication: a × 1 = a
- Inverse Property:
- Addition: a + (−a) = 0
- Multiplication: a × (1/a) = 1 (a ≠ 0)
- Commutative Property:
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Order of Operations (PEMDAS/BODMAS):
- Parentheses/Brackets: solve expressions inside parentheses first
- Exponents/Orders: calculate powers and roots
- Multiplication and Division: from left to right
- Addition and Subtraction: from left to right
Fractions
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Fractions:
- Numerator: the top part of a fraction
- Denominator: the bottom part of a fraction
- Equivalent Fractions: different fractions that represent the same value
- Simplifying Fractions: reducing fractions to their simplest form
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Operations with Fractions:
- Addition/Subtraction: same denominator, add/subtract numerators
- Addition/Subtraction: different denominators, find a common denominator
- Multiplication: multiply numerators and denominators
- Division: multiply by the reciprocal
Decimals
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Decimals:
- Place Value: tenths, hundredths, thousandths, etc.
- Addition/Subtraction: align decimal points
- Multiplication: multiply as whole numbers, then place the decimal
- Division: move the decimal point to make the divisor a whole number
Percentages
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Percentages:
- Definition: a fraction of 100
- Conversion:
- Fraction to percentage: multiply by 100
- Decimal to percentage: multiply by 100
- Finding Percentages: e.g., 20% of 50 = 0.20 × 50 = 10
Ratios and Proportions
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Ratios and Proportions:
- Ratio: a comparison of two quantities
- Proportion: an equation stating that two ratios are equal
Word Problems
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Word Problems:
- Identify Keywords: sum, difference, product, quotient, etc.
- Set Up Equation: translate the problem into an equation
- Solve: use appropriate arithmetic operations to find the solution
- Check: verify the solution by substituting back into the original problem
Area, Perimeter, and Volume of Basic Shapes
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Notes:
- π (Pi): approximately 3.14159
- Units: always ensure units are consistent
- Dimensions:
- Area: measured in square units (e.g., cm², m²)
- Volume: measured in cubic units (e.g., cm³, m³)
Trigonometry and Angles
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Angles:
- Definition: an angle is formed by two rays with a common endpoint called the vertex
- Types of Angles:
- Acute Angle: less than 90°
- Right Angle: exactly 90°
- Obtuse Angle: more than 90° but less than 180°
- Straight Angle: exactly 180°
- Reflex Angle: more than 180°
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Description
Learn the basics of arithmetic, including types of numbers and basic operations. Covers natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.