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How to Perform Fraction Operations
Fractions represent parts of a whole. A fraction has two parts: the numerator (top number) and the denominator (bottom number). The denominator tells you how many equal parts the whole is divided into; the numerator tells you how many of those parts you have. Understanding fraction arithmetic is essential for algebra, cooking measurements, engineering ratios, and many everyday calculations.
Adding and Subtracting Fractions
Fractions can only be added or subtracted when they share the same denominator (a "common denominator"). The process: (1) Find the Least Common Denominator (LCD) — the smallest number divisible by both denominators. (2) Rewrite each fraction with the LCD as the denominator by multiplying numerator and denominator by the same factor. (3) Add or subtract the numerators, keeping the denominator unchanged. (4) Simplify the result by dividing numerator and denominator by their Greatest Common Factor (GCF).
Example: 1/2 + 1/3. LCD = 6. Rewrite: 3/6 + 2/6 = 5/6. The GCF of 5 and 6 is 1, so the fraction is already simplified.
Multiplying Fractions
Multiplication is the simplest operation: multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator. Formula: (a/b) × (c/d) = (a×c)/(b×d). Then simplify. Example: 2/3 × 3/4 = 6/12 = 1/2 (simplified by dividing by GCF of 6).
Dividing Fractions
To divide fractions, multiply by the reciprocal of the second fraction. The reciprocal simply swaps numerator and denominator. Formula: (a/b) ÷ (c/d) = (a/b) × (d/c) = (a×d)/(b×c). Example: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6.
Simplifying Fractions
A fraction is in its simplest form (lowest terms) when the GCF of the numerator and denominator is 1. To simplify, find the GCF and divide both numbers by it. Example: 12/18 — GCF is 6 — simplified to 2/3.
Mixed Numbers
A mixed number like 2½ combines a whole number and a fraction. To convert to an improper fraction: (whole × denominator + numerator) / denominator. So 2½ = (2×2+1)/2 = 5/2. To convert an improper fraction back: divide numerator by denominator to get the whole number, with the remainder as the new numerator.
Frequently Asked Questions
Find the least common denominator (LCD), convert both fractions to use it, then add the numerators. Example: 1/4 + 1/6 — LCD is 12 — becomes 3/12 + 2/12 = 5/12.
Multiply the numerators together and the denominators together: (a/b) × (c/d) = (a×c)/(b×d). Then simplify by dividing by the GCF. Example: 2/3 × 3/4 = 6/12 = 1/2.
Multiply by the reciprocal of the second fraction (flip it): (a/b) ÷ (c/d) = (a/b) × (d/c). Example: 3/4 ÷ 3/8 = 3/4 × 8/3 = 24/12 = 2.
A proper fraction has a numerator smaller than the denominator (e.g., 3/4). An improper fraction has a numerator equal to or greater than the denominator (e.g., 5/4 or 4/4). Improper fractions are often converted to mixed numbers.
Find the Greatest Common Factor (GCF) of the numerator and denominator, then divide both by it. Example: 18/24 — GCF is 6 — simplifies to 3/4.