Fractions Overview and Operations

Study Notes

Definition: A fraction represents a part of a whole and is expressed as a/b, where:
- a = numerator (part)
- b = denominator (whole)
Types of Fractions:
- Proper Fractions: Numerator < Denominator (e.g., 3/4)
- Improper Fractions: Numerator ≥ Denominator (e.g., 5/3)
- Mixed Numbers: Combination of a whole number and a proper fraction (e.g., 2 1/2)
Operations:
- Addition:
  - Same Denominator: Add numerators (e.g., 1/4 + 2/4 = 3/4)
  - Different Denominators: Find a common denominator, then add.
- Subtraction:
  - Same Denominator: Subtract numerators (e.g., 3/4 - 1/4 = 2/4 = 1/2)
  - Different Denominators: Find a common denominator, then subtract.
- Multiplication: Multiply numerators and denominators (e.g., 2/3 * 4/5 = 8/15).
- Division: Multiply by the reciprocal (e.g., 2/3 ÷ 4/5 = 2/3 * 5/4 = 10/12 = 5/6).
Simplifying Fractions: Divide numerator and denominator by their greatest common factor (GCF).

Definition: A decimal represents a fraction whose denominator is a power of ten, expressed with a decimal point (e.g., 0.75).
Place Value:
- Tenths (0.1), Hundredths (0.01), Thousandths (0.001), etc.
Operations:
- Addition and Subtraction: Align decimal points before calculating (e.g., 1.5 + 0.75 = 2.25).
- Multiplication: Multiply as whole numbers, then count total decimal places (e.g., 2.5 * 0.4 = 1.00).
- Division: Move decimal points to make the divisor a whole number, then divide (e.g., 0.6 ÷ 0.2 = 3).
Conversion:
- Fractions to Decimals: Divide the numerator by the denominator (e.g., 1/4 = 0.25).
- Decimals to Fractions: Write the decimal over its place value and simplify (e.g., 0.75 = 75/100 = 3/4).

Representation: A fraction denotes a part of a whole, formatted as a/b with 'a' as the numerator and 'b' as the denominator.
Proper Fractions: Occur when the numerator is less than the denominator, such as 3/4.
Improper Fractions: Feature a numerator that is greater than or equal to the denominator, for example, 5/3.
Mixed Numbers: Consist of a whole number combined with a proper fraction, like 2 1/2.
Addition:
- If fractions have the same denominator, simply add the numerators (e.g., 1/4 + 2/4 = 3/4).
- For different denominators, a common denominator must be established before addition.
Subtraction:
- Same denominator allows for straightforward numerator subtraction (e.g., 3/4 - 1/4 = 2/4 = 1/2).
- Different denominators require finding a common denominator first.
Multiplication: Combine by multiplying the numerators together and the denominators together (e.g., 2/3 * 4/5 = 8/15).
Division: Achieved by multiplying by the reciprocal of the second fraction (e.g., 2/3 ÷ 4/5 becomes 2/3 * 5/4 = 10/12 = 5/6).
Simplifying Fractions: Involves dividing both numerator and denominator by their greatest common factor (GCF).

Definition: A decimal signifies a fraction with a denominator that is a power of ten, shown with a decimal point (e.g., 0.75).
Place Value: Includes components such as tenths (0.1), hundredths (0.01), and thousandths (0.001).
Addition and Subtraction: Requires alignment of decimal points for accurate calculations (e.g., 1.5 + 0.75 = 2.25).
Multiplication: Treated as whole numbers initially, then the count of decimal places determines the final placement of the decimal (e.g., 2.5 * 0.4 = 1.00).
Division: Involves adjusting decimal points to convert the divisor into a whole number, after which traditional division occurs (e.g., 0.6 ÷ 0.2 = 3).
Conversion:
- Fractions to Decimals: Achieved by dividing the numerator by the denominator (e.g., 1/4 = 0.25).
- Decimals to Fractions: Written as a fraction over its place value, then simplified if necessary (e.g., 0.75 = 75/100 = 3/4).