Introduction to Basic Arithmetic Operations

• Arithmetic is the branch of mathematics dealing with numerical calculations
• Arithmetic includes the study of numbers, especially the properties of basic operations

Basic Operations

• The basic arithmetic operations are addition, subtraction, multiplication, and division

Addition

• Addition is the process of combining two or more numbers
• The result of addition is called the sum or total
• Represented by the symbol "+"
• Example: 5 + 3 = 8

Subtraction

• Subtraction is the process of finding the difference between two numbers
• The result of subtraction is called the difference
• Represented by the symbol "−"
• Example: 7 − 2 = 5

Multiplication

• Multiplication is the process of repeated addition of the same number
• The result of multiplication is called the product
• Represented by the symbol "×" or "*"
• Example: 4 × 6 = 24

Division

• Division is the process of splitting a number into equal parts
• The result of division is called the quotient
• Represented by the symbol "÷" or "/"
• Example: 10 ÷ 2 = 5

Order of Operations

• The order of operations is a convention used to standardize how expressions are evaluated

• Often remembered by the acronym PEMDAS

• PEMDAS stands for Parentheses, Exponents, Multiplication and Division, Addition and Subtraction

  • Parentheses: Expressions inside parentheses are evaluated first
  • Exponents: Exponents are evaluated next
  • Multiplication and Division: These operations are performed from left to right
  • Addition and Subtraction: These operations are performed from left to right

• Example: Evaluate 9 + (5 × 2) − 12 ÷ 3

  • First, evaluate the expression inside the parentheses: 5 × 2 = 10
  • Next, perform the division: 12 ÷ 3 = 4
  • Now, the expression is: 9 + 10 − 4
  • Perform addition and subtraction from left to right: 9 + 10 = 19, then 19 − 4 = 15
  • So, 9 + (5 × 2) − 12 ÷ 3 = 15

Percentages

• A percentage is a way of expressing a number as a fraction of 100
• Represented by the symbol "%"
• Percentages are used to express how large one quantity is relative to another quantity
• Used to express changes in a value

Converting Percentages to Decimals

• To convert a percentage to a decimal, divide the percentage by 100
• Example: 75% = 75/100 = 0.75

Converting Decimals to Percentages

• To convert a decimal to a percentage, multiply the decimal by 100
• Example: 0.25 = 0.25 × 100 = 25%

Converting Percentages to Fractions

• To convert a percentage to a fraction, divide the percentage by 100 and simplify
• Example: 60% = 60/100 = 3/5

Converting Fractions to Percentages

• To convert a fraction to a percentage, first convert the fraction to a decimal by dividing the numerator by the denominator, then multiply by 100
• Example: 1/4 = 0.25, then 0.25 × 100 = 25%

Calculating Percentage of a Number

• To find a percentage of a number, convert the percentage to a decimal and multiply by the number
• Example: Find 20% of 80
  • 20% = 0.20
  • 0.20 × 80 = 16
  • 20% of 80 is 16

Calculating Percentage Increase

• To calculate the percentage increase:
  • Find the difference between the new value and the original value
  • Divide the difference by the original value
  • Multiply the result by 100
• Formula: Percentage Increase = [(New Value − Original Value) / Original Value] × 100
• Example: If a price increases from $50 to $60
  • Difference = $60 − $50 = $10
  • Percentage Increase = ($10 / $50) × 100 = 0.2 × 100 = 20%
  • The price increased by 20%

Calculating Percentage Decrease

• To calculate the percentage decrease:
  • Find the difference between the original value and the new value
  • Divide the difference by the original value
  • Multiply the result by 100
• Formula: Percentage Decrease = [(Original Value − New Value) / Original Value] × 100
• Example: If a price decreases from $50 to $40
  • Difference = $50 − $40 = $10
  • Percentage Decrease = ($10 / $50) × 100 = 0.2 × 100 = 20%
  • The price decreased by 20%

Finding the Original Value

• If you know a percentage of a value and the result, you can find the original value
• Divide the result by the percentage (in decimal form)
• Example: 30% of a number is 60, find the original number
  • Original Number = 60 / 0.30 = 200
  • The original number is 200