Chapter 1: Variables, Real Numbers, and Mathematical Models

Fractions in Algebra

This section introduces the foundational concepts of fractions as they are used in algebra, including definitions, operations, and applications. Mastery of these concepts is essential for success in college algebra and subsequent mathematical studies.

Definitions and Basic Concepts

• Numerator: The number above the fraction bar, representing how many parts are considered.

• Denominator: The number below the fraction bar, indicating the total number of equal parts in the whole.

• Fraction: Expresses a part of a whole, such as "five sevenths" (), meaning 5 out of 7 equal parts.

• Natural Numbers: The counting numbers: 1, 2, 3, 4, 5, ...

Mixed Numbers and Improper Fractions

• Mixed Number: The sum of a natural number and a fraction, written without an addition sign (e.g., ).

• Improper Fraction: A fraction where the numerator is greater than or equal to the denominator (e.g., ).

Converting a Mixed Number to an Improper Fraction:

1. Multiply the denominator by the whole number part and add the numerator.

2. Place the result over the original denominator.

Formula:

Converting an Improper Fraction to a Mixed Number:

1. Divide the denominator into the numerator to get the quotient and remainder.

2. Write as: quotient

Prime and Composite Numbers

• Prime Number: A natural number greater than 1 with only two factors: 1 and itself. The first ten primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

• Composite Number: A natural number greater than 1 that is not prime; it can be written as a product of prime numbers.

Example: The prime factorization of 36 is or .

Reducing Fractions (Simplifying)

• Fundamental Principle of Fractions: The value of a fraction does not change if both numerator and denominator are multiplied or divided by the same nonzero number.

• Steps to Reduce:

  1. Write the prime factorization of numerator and denominator.

  2. Divide both by their greatest common factor (GCF).

Example: after dividing by 6 (the GCF).

Multiplying Fractions

• Multiply the numerators together and the denominators together.

Formula:

Example:

Dividing Fractions

• Multiply the first fraction by the reciprocal of the second.

Formula:

Example:

Adding and Subtracting Fractions with Identical Denominators

• Add or subtract the numerators and keep the common denominator.

Formula:

Example:

Adding and Subtracting Fractions with Unlike Denominators

1. Find the least common denominator (LCD).

2. Rewrite each fraction as an equivalent fraction with the LCD.

3. Add or subtract the numerators over the common denominator.

Example:

Fractions in Algebraic Equations and Applications

• Fractions can be used to represent solutions to equations and to translate verbal statements into algebraic expressions.

• To determine if a fraction is a solution, substitute it into the equation and check if the statement is true.

Example: Is a solution to ? Substitute: (True).

Evaluating Formulas Containing Fractions

• Many real-world formulas involve fractions. Substitute the given values and simplify.

Example: To convert 77°F to Celsius using :