Fraction Simplification

Number Theory · Axiom Academy

REAL WORLD Fraction Simplification How the GCD makes fractions work in everyday life You type 48 ÷ 72 into your calculator. Instead of showing "0.666666...", it instantly displays 2/3 . How does it know? The answer lies in a powerful mathematical tool: the Greatest Common Divisor (GCD) . Let's discover how this ancient algorithm powers modern technology! To simplify any fraction , we use this powerful principle: Divide both numerator and denominator by their GCD Key insight: A fraction is in lowest terms when . This means the numerator and denominator share no common factors other than 1. Let's simplify 48/72. What's gcd(48, 72)? Interactive Fraction Simplifier Try it yourself! Enter any fraction and watch the GCD work its magic. You're cooking for 6 people, but your recipe serves 4. You need to multiply all ingredients by 6/4 . Working with 3/2 is much easier than 6/4! ➕ Adding Fractions: The LCM Connection To add fractions with different denominators, we need a common denominator . The best choice is the Least Common Denominator (LCD) , which equals the LCM of the denominators . Connection: gcd(a, b) × lcm(a, b) = a × b Application: Music Time Signatures Musicians use fractions constantly! Time signatures tell us how many beats per measure. Common Time Signatures (all simplified fractions): Why simplified? 4/4 means "4 quarter notes per measure." If we used 8/8 (unsimplified), musicians would have to think in eighth notes instead—much harder to read!

This is the written version of the interactive lesson above. See the full Number Theory course.