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How Fraction Arithmetic Works

A fraction represents a part of a whole. The top number is the numerator (how many parts you have) and the bottom is the denominator (how many equal parts the whole is divided into). To add or subtract fractions they must share a common denominator — the lowest common denominator (LCD) is the smallest number both denominators divide into evenly. For example, to compute 1/3 + 1/4, the LCD is 12, giving 4/12 + 3/12 = 7/12. Multiplication is simpler: multiply numerators together, then denominators together. Division flips the second fraction to its reciprocal, then multiplies.

Simplification and Lowest Terms

A fraction is fully simplified (in lowest terms) when its numerator and denominator share no common factor other than 1. To simplify, find the greatest common divisor (GCD) of both numbers and divide each by it. For example, 12/18 simplifies to 2/3 because the GCD of 12 and 18 is 6. The Euclidean algorithm is a fast way to find the GCD by hand: divide the larger number by the smaller, then repeat with the remainder until the remainder is zero. The last non-zero remainder is the GCD.

Mixed Numbers vs Improper Fractions

An improper fraction has a numerator larger than or equal to its denominator (e.g., 7/4). A mixed number combines a whole number and a proper fraction (e.g., 1¾). To convert an improper fraction to a mixed number, divide the numerator by the denominator: the quotient is the whole number and the remainder becomes the new numerator. So 7 ÷ 4 = 1 remainder 3, giving 1¾. To go the other way, multiply the whole number by the denominator and add the numerator: 1 × 4 + 3 = 7, yielding 7/4. Mixed numbers are easier to read at a glance; improper fractions are more convenient for arithmetic.