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How to Simplify Fractions
To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD). For example, to simplify 18/24, find that the GCD of 18 and 24 is 6, then divide both by 6:
\[\frac{18}{24} = \frac{18 \div 6}{24 \div 6} = \frac{3}{4}\]The fraction 3/4 is in simplest form because 3 and 4 share no common factor other than 1. Every fraction has exactly one simplest form, and the process of reaching it always comes down to finding and dividing by the GCD.
The step-by-step method
For fractions with small numbers, listing factors by hand works well.
Take 12/30. List the factors of each number:
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
The largest factor they share is 6, so the GCD is 6.
\[\frac{12}{30} = \frac{12 \div 6}{30 \div 6} = \frac{2}{5}\]This approach is reliable but becomes tedious with larger numbers. Listing all the factors of 252 and 378 by hand would take a while. For those cases, prime factorization or the Euclidean algorithm is faster.
Prime factorization method
Break both numbers into their prime factors, then multiply the primes they share.
Simplify 84/126:
\[84 = 2 \times 2 \times 3 \times 7\] \[126 = 2 \times 3 \times 3 \times 7\]The shared primes are 2, 3, and 7. Multiply them: 2 x 3 x 7 = 42. That is the GCD.
\[\frac{84}{126} = \frac{84 \div 42}{126 \div 42} = \frac{2}{3}\]This method has the advantage of showing exactly why the GCD is what it is. Each shared prime factor represents a common divisor, and together they form the greatest one. The GCD/LCM calculator finds the greatest common divisor instantly for any pair of numbers.
The Euclidean algorithm
For large numbers, the Euclidean algorithm is the most efficient approach. It works by repeatedly dividing and taking remainders until the remainder is zero. The last nonzero remainder is the GCD.
Simplify 252/378:
| Step | Division | Remainder |
|---|---|---|
| 1 | 378 / 252 = 1 | 126 |
| 2 | 252 / 126 = 2 | 0 |
The last nonzero remainder is 126, so the GCD is 126.
\[\frac{252}{378} = \frac{252 \div 126}{378 \div 126} = \frac{2}{3}\]A longer example: simplify 546/390.
| Step | Division | Remainder |
|---|---|---|
| 1 | 546 / 390 = 1 | 156 |
| 2 | 390 / 156 = 2 | 78 |
| 3 | 156 / 78 = 2 | 0 |
The GCD is 78.
\[\frac{546}{390} = \frac{546 \div 78}{390 \div 78} = \frac{7}{5}\]The result is an improper fraction (numerator larger than denominator), which is perfectly valid as a simplified form. You could also write it as the mixed number 1 2/5, but the fraction itself is fully simplified because 7 and 5 share no common factor.
Worked examples from easy to hard
Example 1: 8/12. The GCD of 8 and 12 is 4. Dividing both by 4 gives 2/3.
Example 2: 15/45. The GCD of 15 and 45 is 15. Dividing both by 15 gives 1/3.
Example 3: 36/48. The GCD of 36 and 48 is 12. Dividing both by 12 gives 3/4.
Example 4: 56/98. The prime factorizations are 56 = 2 x 2 x 2 x 7 and 98 = 2 x 7 x 7. The shared primes are 2 and 7, so the GCD is 14. Dividing both by 14 gives 4/7.
Example 5: 270/450. Using the Euclidean algorithm: 450 / 270 = 1 remainder 180; 270 / 180 = 1 remainder 90; 180 / 90 = 2 remainder 0. The GCD is 90. Dividing both by 90 gives 3/5.
The fraction calculator simplifies fractions and also performs addition, subtraction, multiplication, and division with automatic simplification of the result.
Simplifying in stages
You do not have to find the GCD on the first try. Dividing by any common factor makes progress, and you can repeat until nothing works.
Take 72/108. You might notice both are even, so divide by 2: 36/54. Both are still even, so divide by 2 again: 18/27. Now both are divisible by 9: 2/3. You arrived at the same answer you would have reached by dividing by 36 (the GCD) in one step. The path is longer, but the destination is the same.
This approach is often the fastest in practice, especially during exams or mental math. Spotting small common factors like 2, 3, or 5 is quick, and a few rounds of division reach the simplest form without ever computing the GCD explicitly.
When a fraction is already simplified
A fraction is in simplest form when the numerator and denominator share no common factor other than 1. This means they are coprime. For example, 7/12 is already simplified because 7 is prime and does not divide 12. Similarly, 11/30 is in simplest form because 11 is prime and 30 is not a multiple of 11.
A quick check: if the numerator is prime and does not divide the denominator evenly, the fraction is already as simple as it gets. If the numerator is not prime, you need to verify that no prime factor of the numerator also divides the denominator.
Common mistakes
The most frequent error is dividing only the numerator or only the denominator. Simplifying means dividing both by the same number. Dividing just the numerator changes the value of the fraction entirely.
Another mistake is stopping too early. Dividing 24/36 by 2 gives 12/18, which is simpler but not simplest. Both 12 and 18 are still divisible by 6. The fully simplified result is 2/3. Always check whether the result can be reduced further.
A third common error involves negative fractions. If the numerator or denominator is negative, simplify the absolute values first, then attach the negative sign to the numerator by convention. The fraction -18/24 simplifies to -3/4, not 3/-4 (though both are mathematically equivalent, the standard form places the sign in front).