Formula
To simplify a fraction: find the GCD (Greatest Common Divisor) of the numerator and denominator, then divide both by the GCD. The result is the fraction in its lowest terms.
- Only the GCD reduces in one step - any common factor shrinks the fraction, but only dividing by the GCD reaches simplest form immediately.
- Simplifying never changes the value - 18/24, 9/12, and 3/4 are the same number (0.75) in different representations.
- GCD = 1 means already simplified - if the numerator and denominator share no common factor, the fraction is in lowest terms and cannot be reduced further.
- Improper fractions simplify the same way - apply the GCD method as normal; the result may still be improper, or it may reduce to a whole number.
- Always verify after simplifying - check that GCD(new numerator, new denominator) = 1; if not, you stopped too early.
How to use our simplified fraction calculator
Enter the numerator and denominator of any fraction. The calculator finds the GCD, divides both parts, and shows the simplified result as a fraction, mixed number (where applicable), decimal, and percentage - along with a three-step GCD breakdown.
Simplified fractions explained to a beginner
Imagine a pizza cut into 24 equal slices and you eat 18 of them. You could say "I ate 18/24 of the pizza" - but "3/4" says the same thing more simply. The GCD (6 in this case) is the largest group size that fits evenly into both numbers: 18 slices become 3 groups of 6, and 24 slices become 4 groups of 6. So 18/24 = 3/4 - same amount of pizza, just described more compactly.
The GCD method gets you there in one step. Dividing by a smaller factor (say, 2) also works - 18/24 becomes 9/12 - but then you have to repeat until GCD = 1. Finding the largest group size first skips all the repetition.
Simplifying fractions formula
A fraction is in its simplest form (lowest terms) when the numerator and denominator share no common factor other than 1. The reduction formula is:
$$\frac{a}{b} = \frac{a \div \gcd(a, b)}{b \div \gcd(a, b)}$$
GCD (Greatest Common Divisor) is also written as GCF (Greatest Common Factor) or HCF (Highest Common Factor) - all refer to the same value: the largest integer that divides both numbers exactly.
How to find the GCD
Three methods exist to find the Greatest Common Divisor, each suited to different situations:
Method 1: Euclidean algorithm (fastest)
Repeatedly divide the larger number by the smaller and take the remainder. Stop when the remainder is 0 - the last non-zero remainder is the GCD:
$$\gcd(48, 36): \quad 48 = 1 \times 36 + 12 \implies \gcd(36, 12) \implies 36 = 3 \times 12 + 0 \implies \gcd = 12$$
This method works efficiently on any pair of integers, regardless of size.
Method 2: Prime factorisation
Write both numbers as products of prime factors. The GCD is the product of all shared prime factors. For 18 and 24: 18 = 2 × 3² and 24 = 2³ × 3. Shared factors: 2¹ × 3¹ = 6. GCD = 6.
Method 3: Listing factors
List all factors of both numbers and find the largest common one. Practical only for small numbers. Factors of 18: 1, 2, 3, 6, 9, 18. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Largest common: 6.
In practice, the Euclidean algorithm is the method worth internalizing. Listing factors works well below 30 but becomes unreliable above 50 in my opinion - it's easy to miss a factor or a prime.
When I do GCDs by hand, I default to the Euclidean algorithm for anything over two digits: it's mechanical, guaranteed to terminate, and doesn't require recognizing all prime factors first.
Worked examples for simplifying fractions
Example 1: straightforward reduction
$$\frac{18}{24}: \quad \gcd(18, 24) = 6 \implies \frac{18 \div 6}{24 \div 6} = \frac{3}{4}$$
GCD(18, 24) = 6. Divide: 18 ÷ 6 = 3, 24 ÷ 6 = 4. Result: 3/4. Check: GCD(3, 4) = 1 ✓
Example 2: different path, same result
$$\frac{36}{48}: \quad \gcd(36, 48) = 12 \implies \frac{36 \div 12}{48 \div 12} = \frac{3}{4}$$
GCD(36, 48) = 12. Both 18/24 and 36/48 simplify to 3/4 - they are equivalent fractions at different scales.
| Fraction | GCD | Simplified | Decimal |
|---|---|---|---|
| 2/4 | 2 | 1/2 | 0.5 |
| 6/9 | 3 | 2/3 | 0.6667 |
| 8/12 | 4 | 2/3 | 0.6667 |
| 15/25 | 5 | 3/5 | 0.6 |
| 14/21 | 7 | 2/3 | 0.6667 |
| 24/36 | 12 | 2/3 | 0.6667 |
| 45/60 | 15 | 3/4 | 0.75 |
| 100/250 | 50 | 2/5 | 0.4 |
How about fractions already in simplest form?
If the GCD of numerator and denominator equals 1, the fraction cannot be reduced further - it is already in its simplest form:
$$\frac{7}{13}: \quad \gcd(7, 13) = 1 \implies \text{already in simplest form}$$
7 and 13 are both prime and share no common factor. GCD = 1, so 7/13 is already simplified. Any fraction where both numerator and denominator are prime numbers (and unequal) is automatically in lowest terms.
Simplifying improper fractions and mixed numbers
An improper fraction (numerator ≥ denominator) is simplified exactly like a proper fraction. The simplified result may still be improper:
$$\frac{22}{8}: \quad \gcd(22, 8) = 2 \implies \frac{22 \div 2}{8 \div 2} = \frac{11}{4} = 2\frac{3}{4}$$
22/8 → GCD(22, 8) = 2 → 11/4. As a mixed number: 11 ÷ 4 = 2 remainder 3, so 2 3/4.
Note: simplifying and converting to a mixed number are two separate operations. Simplifying reduces the fraction; converting to a mixed number changes its representation. This calculator performs both.
Simplify negative fractions
For negative fractions, the GCD is applied to the absolute values. The sign is preserved in the numerator:
$$\frac{-12}{16}: \quad \gcd(12, 16) = 4 \implies \frac{-12 \div 4}{16 \div 4} = \frac{-3}{4}$$
Convention: the negative sign is placed on the numerator, not the denominator. −3/4 is standard; 3/−4 is avoided in final answers.
Common mistakes for fractions simplification
Dividing by a common factor that is not the GCD
Any common factor reduces the fraction, but only dividing by the GCD reduces it in one step to its simplest form. Dividing 18/24 by 2 gives 9/12 - correct but not fully simplified. Dividing by GCD(18, 24) = 6 reaches 3/4 directly.
Thinking that simplification changes the value
Simplifying a fraction does not change its value - it changes its representation. 18/24, 9/12, 6/8, 3/4 are all the same number (0.75). Simplest form is the most compact representation.
Stopping too early
A common error is to divide by a factor that is not the GCD, resulting in a fraction that can still be reduced. Always verify the result: if GCD(new numerator, new denominator) = 1, the simplification is complete.
This is the error I see most often. People divide by 2, see a smaller fraction, and consider the job done. The reliable fix is a two-second verification: after simplifying, check that GCD(new numerator, new denominator) = 1. If it isn't, there's still a common factor to remove. Building this habit catches every incomplete simplification.
Simplifying across addition or subtraction
You cannot "cancel" numerators and denominators across a + or − sign. For example, (2 + 3)/(2 + 5) ≠ (1 + 3)/(1 + 5). Simplification applies only to multiplication and division of fractions, not to their components separately.
FAQs about simplifying fractions
How do you simplify a fraction?
Find the GCD of the numerator and denominator, then divide both by it. The result is the fraction in its lowest terms.
What does simplest form mean?
A fraction is in simplest form when its numerator and denominator share no common factor other than 1 - their GCD is 1.
What is the GCD?
The Greatest Common Divisor is the largest integer that divides both numbers exactly. Dividing both parts of a fraction by their GCD reduces it to simplest form in one step.
How do you find the GCD?
The Euclidean algorithm is the fastest method: divide the larger by the smaller, take the remainder, and repeat until the remainder is 0. The last non-zero remainder is the GCD.
Can an improper fraction be simplified?
Yes - apply the same GCD method. The simplified result may remain improper, or it may convert to a whole number or mixed number.
What if the GCD is 1?
The fraction is already in simplest form and cannot be reduced further. This applies to all fractions where numerator and denominator are coprime.
Quiz: how well do you know simplifying fractions?
1. What does it mean for a fraction to be in its simplest form (lowest terms)?
2. Using the Euclidean algorithm shown on the page: GCD(48, 36). Step 1: 48 = 1×36 + 12. Step 2: 36 = 3×12 + 0. What is the GCD?
3. The page shows that both 18/24 and 36/48 simplify to 3/4. What does this tell us?
4. From the reference table on the page, what does 100/250 simplify to?
5. You simplify 36/48 by dividing both parts by 4, getting 9/12. What is the problem?