Lowest Term Calculator
Solve lowest term problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
Simplified = (a / GCD(a,b)) / (b / GCD(a,b))
Divide both numerator (a) and denominator (b) by their Greatest Common Divisor (GCD). The GCD is found using the Euclidean algorithm or prime factorization. The result is the unique fraction in lowest terms equal to a/b.
Worked Examples
Example 1: Simplifying 48/64
Problem:Reduce the fraction 48/64 to its lowest terms.
Solution:Euclidean Algorithm:\n64 = 1 * 48 + 16\n48 = 3 * 16 + 0\nGCD = 16\n\n48 / 16 = 3\n64 / 16 = 4\n\nPrime factorization check:\n48 = 2^4 * 3\n64 = 2^6\nCommon: 2^4 = 16\nResult: 3/4\n\nVerification: 3/4 = 0.75 and 48/64 = 0.75
Result:48/64 = 3/4 (GCD = 16)
Example 2: Simplifying 105/135
Problem:Reduce 105/135 to lowest terms and express as a decimal.
Solution:Euclidean Algorithm:\n135 = 1 * 105 + 30\n105 = 3 * 30 + 15\n30 = 2 * 15 + 0\nGCD = 15\n\n105 / 15 = 7\n135 / 15 = 9\n\nPrime factorization:\n105 = 3 * 5 * 7\n135 = 3^3 * 5\nCommon: 3 * 5 = 15\nResult: 7/9\nDecimal: 0.777... (repeating)
Result:105/135 = 7/9 (GCD = 15) = 0.7777...
Frequently Asked Questions
What does it mean to reduce a fraction to its lowest terms?
Reducing a fraction to lowest terms (also called simplifying) means finding the equivalent fraction where the numerator and denominator are as small as possible and share no common factors other than 1. This is done by dividing both the numerator and denominator by their Greatest Common Divisor (GCD). For example, 48/64 reduced to lowest terms is 3/4, because GCD(48,64) = 16, and 48/16 = 3, 64/16 = 4. The reduced fraction is mathematically equal to the original but uses smaller numbers. A fraction is in lowest terms when the GCD of its numerator and denominator is 1. Every fraction has exactly one representation in lowest terms, making it the canonical form for that rational number.
When is a fraction already in lowest terms?
A fraction is already in lowest terms when its numerator and denominator share no common factors other than 1, meaning their GCD is 1. Such pairs of numbers are called coprime or relatively prime. For example, 7/12 is in lowest terms because 7 is prime and does not divide 12. Quick checks include: if the numerator is prime and does not divide the denominator, the fraction is in lowest terms. If both numbers are odd, they share no factor of 2. If both are not divisible by 3, they share no factor of 3. However, these checks are not sufficient alone; you need to verify that no prime divides both. Lowest Term Calculator automatically checks and reports whether the input fraction is already simplified.
What is the relationship between lowest terms and prime factorization?
The Fundamental Theorem of Arithmetic states that every integer has a unique prime factorization. When simplifying a fraction, we identify prime factors common to both numerator and denominator and cancel them. For 48/64: 48 = 2^4 * 3 and 64 = 2^6. Common factor is 2^4 = 16. After cancellation: (2^4 * 3)/(2^6) = 3/2^2 = 3/4. A fraction is in lowest terms when the prime factorizations of numerator and denominator share no common primes. This connection between fractions and prime factorization is deeply important in number theory and forms the basis for understanding rational numbers, Diophantine equations, and the structure of the integers.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy