Fractions

Fractions

A common fraction a/b (where b ≠ 0) represents parts of a whole. Equivalent fractions represent the same value: a/b = (a·k)/(b·k) for k ≠ 0. Reducing a fraction means dividing numerator and denominator by their greatest common divisor (GCD). For example, 18/24 = (18÷6)/(24÷6) = 3/4.

Decimal Fractions

Decimal fractions use place values: 0.75 = 75/100 = 3/4. To compare decimal fractions, align decimal points and compare digit by digit. A terminating decimal is a fraction with denominator a power of 10; a repeating decimal like 0.3̄ = 3/9 = 1/3. The general rule: 0.ᾱ = a/9, 0.āb̄ = (10a + b)/99.

Addition and Subtraction of Fractions

To add or subtract fractions, find the least common denominator (LCD) — the LCM of the denominators. For example: 1/6 + 3/8: LCM(6,8) = 24, so 1/6 + 3/8 = 4/24 + 9/24 = 13/24.

Multiplication and Division

To multiply: (a/b)·(c/d) = (ac)/(bd). To divide: (a/b) ÷ (c/d) = (ad)/(bc), where c ≠ 0.

Converting Fractions to Common Denominators

Converting fractions to a common denominator allows us to compare and add fractions with different denominators. For example: 9/12 ? 15/20. Reduce first: 9/12 = 3/4 and 15/20 = 3/4, so 9/12 = 15/20.