Fractions Overview and Operations

Study Notes

Fractions

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).

Decimals

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).

Fractions

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).

Decimals

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).

Description

This quiz covers the fundamental concepts of fractions, including their definition, types, and operations such as addition, subtraction, multiplication, and division. Learn how to simplify fractions and understand the difference between proper, improper fractions, and mixed numbers. Test your knowledge on how to handle fractions in various mathematical operations.