How to Simplify Fractions
Simplifying a fraction means reducing it to its lowest terms. A fraction is in its simplest form when the numerator and denominator have no common factors other than 1.
The Process
- Find the Greatest Common Divisor (GCD) of the numerator and denominator
- Divide both the numerator and denominator by the GCD
- The result is the fraction in its simplest form
Finding the GCD
The Greatest Common Divisor (GCD) is the largest number that divides both the numerator and denominator without a remainder. You can find it by:
- Prime factorization: Break down both numbers into prime factors and find common factors
- Euclidean algorithm: An efficient method for finding GCD
- Listing factors: List all factors of both numbers and find the largest common one
Examples
Example 1: Simplify 12/18
Step 1: Find GCD of 12 and 18
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
GCD = 6
Step 2: Divide by GCD
12 / 6 = 2, 18 / 6 = 3
Result: 12/18 = 2/3
Example 2: Simplify 45/60
Step 1: Find GCD of 45 and 60
45 = 3^2 × 5
60 = 2^2 × 3 × 5
Common factors: 3 × 5 = 15
GCD = 15
Step 2: Divide by GCD
45 / 15 = 3, 60 / 15 = 4
Result: 45/60 = 3/4
Example 3: Simplify 24/36
Step 1: Find GCD of 24 and 36
Using Euclidean algorithm:
36 = 24 × 1 + 12
24 = 12 × 2 + 0
GCD = 12
Step 2: Divide by GCD
24 / 12 = 2, 36 / 12 = 3
Result: 24/36 = 2/3
Example 4: Simplify 100/250
Step 1: Find GCD of 100 and 250
100 = 2^2 × 5^2
250 = 2 × 5^3
Common factors: 2 × 5^2 = 50
GCD = 50
Step 2: Divide by GCD
100 / 50 = 2, 250 / 50 = 5
Result: 100/250 = 2/5
Example 5: Simplify 7/21
Step 1: Find GCD of 7 and 21
7 is prime
21 = 3 × 7
GCD = 7
Step 2: Divide by GCD
7 / 7 = 1, 21 / 7 = 3
Result: 7/21 = 1/3