FRACTIONS

Fractions
Definition

A fraction represents a part of a whole or, more generally, any number of equal parts. It is written as \frac{a}{b}, where:

  • a is the numerator (the top number), representing the number of parts taken.

  • b is the denominator (the bottom number), representing the total number of parts.

Types of Fractions

  1. Proper Fraction: The numerator is less than the denominator. For example, \frac{3}{4}.

  2. Improper Fraction: The numerator is greater than or equal to the denominator. For example, \frac{5}{3}.

  3. Mixed Number: A whole number and a proper fraction combined. For example, 2\frac{1}{2}.

Equivalent Fractions

Fractions that represent the same value, even though they have different numerators and denominators. For example, \frac{1}{2} = \frac{2}{4} = \frac{4}{8}.

Simplifying Fractions

Reducing a fraction to its simplest form by dividing both the numerator and the denominator by their greatest common divisor (GCD).

  • Example: Simplify \frac{12}{18}.

    • The GCD of 12 and 18 is 6.

    • \frac{12 \div 6}{18 \div 6} = \frac{2}{3}

Operations with Fractions

  1. Addition and Subtraction: Fractions must have a common denominator before they can be added or subtracted.

    • \frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}

    • \frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}

    • If the denominators are different, find the least common multiple (LCM) and convert the fractions to equivalent fractions with the LCM as the common denominator.

      • Example: \frac{1}{4} + \frac{2}{3}

      • LCM of 4 and 3 is 12.

      • \frac{1}{4} = \frac{3}{12}, \frac{2}{3} = \frac{8}{12}

      • \frac{3}{12} + \frac{8}{12} = \frac{11}{12}

  2. Multiplication: Multiply the numerators and multiply the denominators.

    • \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}

    • Example: \frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15}

  3. Division: Multiply by the reciprocal of the divisor.

    • \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c}

    • Example: \frac{1}{2} \div \frac{3}{4} = \frac{1}{2} \times \frac{4}{3} = \frac{1 \times 4}{2 \times 3} = \frac{4}{6} = \frac{2}{3}

Converting Improper Fractions to Mixed Numbers

Divide the numerator by the denominator. The quotient is the whole number, the remainder is the new numerator, and the denominator stays the same.

  • Example: Convert \frac{17}{5} to a mixed number.

    • 17 \div 5 = 3 with a remainder of 2.

    • So, \frac{17}{5} = 3\frac{2}{5}

Converting Mixed Numbers to Improper