Fractions, Mixed Numbers, and Factors: Foundations for College Algebra

Fractions and Mixed Numbers

Introduction to Fractions

Fractions are a fundamental concept in algebra, representing parts of a whole. Understanding fractions is essential for solving equations, working with ratios, and interpreting mathematical relationships.

• Fraction: A number of the form a/b, where a is the numerator and b is the denominator (with b ≠ 0).

• Numerator: The number of parts being considered.

• Denominator: The total number of equal parts in the whole.

Examples of fractions:

• Proper fraction: $\frac{2}{3}$

• Improper fraction: $\frac{7}{4}$

• Mixed number: $1\ \frac{1}{2}$

Types of Fractions

• Proper Fraction: Numerator is less than the denominator (e.g., $\frac{2}{5}$).

• Improper Fraction: Numerator is greater than or equal to the denominator (e.g., $\frac{7}{4}$).

• Mixed Number: A whole number and a fraction combined (e.g., $2\ \frac{1}{3}$).

Examples and Applications

• In a class of 20 students, if there are 9 boys and 11 girls:

  • Fraction of boys: $\frac{9}{20}$

  • Fraction of girls: $\frac{11}{20}$

• If 3 out of 10 math students are girls: $\frac{3}{10}$

• If 7 out of 10 math students are boys: $\frac{7}{10}$

Writing Improper Fractions as Mixed Numbers or Whole Numbers

To convert an improper fraction to a mixed number:

• Divide the numerator by the denominator.

• The quotient is the whole number part; the remainder over the denominator is the fractional part.

Example:

• $\frac{7}{4} = 1\ \frac{3}{4}$ (since 7 divided by 4 is 1 with a remainder of 3)

Factors and Simplest Form

Prime and Composite Numbers

Understanding factors is crucial for simplifying fractions and solving algebraic equations.

• Prime Number: A natural number greater than 1 that has only two factors: 1 and itself (e.g., 2, 3, 5, 7, 11, 13, ...).

• Composite Number: A natural number greater than 1 that is not prime; it has more than two factors (e.g., 4, 6, 8, 9, 10, ...).

Prime Factorization

• Prime Factorization: Writing a composite number as a product of prime factors.

Example:

• Prime factorization of 12: $2 \times 2 \times 3$ or $2^2 \times 3$

Divisibility Tests

Divisibility tests help determine if one number is divisible by another without performing full division.

• Divisible by 2: The last digit is even (0, 2, 4, 6, 8).

• Divisible by 3: The sum of the digits is divisible by 3.

• Divisible by 5: The last digit is 0 or 5.

• Divisible by 10: The last digit is 0.

Examples

• Is 987 divisible by 3? (9 + 8 + 7 = 24; 24 is divisible by 3, so yes.)

• Is 2,025 divisible by 5? (Last digit is 5, so yes.)