Lowest terms and the GCF
A fraction like 12/18 is a rational number — a ratio of integers. To write it in lowest terms we divide top and bottom by their greatest common factor (GCF), the largest number dividing both. The cleanest way to find the GCF is prime factorization: 12 = 2·2·3 and 18 = 2·3·3 share one 2 and one 3, so the GCF is 6. Dividing gives 12/18 = 2/3.
Reduce 12/18 using the GCF 12 = 2 * 2 * 3 18 = 2 * 3 * 3 common factors: 2 and 3 -> GCF = 2 * 3 = 6 12 / 18 = (12 / 6) / (18 / 6) = 2 / 3
Adding fractions and the LCM
You can only add fractions once they share a common denominator. The best one is the least common multiple (LCM) of the denominators — the smallest number that is a multiple of both. From the prime factorizations, take the highest power of each prime that appears. For 4 = 2·2 and 6 = 2·3, the LCM is 2·2·3 = 12.
Add 1/4 + 1/6 4 = 2 * 2 6 = 2 * 3 LCM = 2 * 2 * 3 = 12 (common denominator) 1/4 = 3/12 (multiply top & bottom by 3) 1/6 = 2/12 (multiply top & bottom by 2) 3/12 + 2/12 = 5/12 (already lowest terms)