Simplify Fraction Calculator
Reduce a fraction to lowest terms by dividing by the greatest common factor, with Euclid’s steps, the mixed number and the decimal.
Result
Simplest form: 2/3
Exact whole-number arithmetic; only the decimal is rounded.
- Greatest common factor
- 42
- As a decimal, to ten significant figures
- 0.6666666667
- Numerator in prime factors
- 2^2 × 3 × 7
- Denominator in prime factors
- 2 × 3^2 × 7
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Results follow the mathematical operation and values shown. Check that you selected the intended operation, convention and dataset type before relying on the result. Read the full disclaimer
How to use the Simplify Fraction Calculator
- Type the numerator and the denominator as whole numbers. A minus sign on either is fine.
- Read the fraction in simplest form, and the greatest common factor both terms were divided by.
- Use the mixed number or the decimal if that is the form you need.
The rule
A fraction keeps its value when both terms are divided by the same non-zero number. Dividing by their greatest common factor, g, is the one division that leaves no common factor behind:
g = gcd(numerator, denominator)simplest form = (numerator ÷ g) / (denominator ÷ g)Euclid’s algorithm
The page finds g without listing factors. Each step divides and keeps the remainder, and the pair shrinks quickly: for 1071 and 462, 1071 = 462 × 2 + 147, then 462 = 147 × 3 + 21, then 147 = 21 × 7 + 0, so g = 21 and 1071/462 = 51/22. The working under the result shows these steps for your own numbers.
The prime factorisations are shown too, while both terms are at most a trillion, because cancelling shared primes is how most people are taught to see the answer. For longer numbers the page still simplifies exactly; it only stops factorising.
Exact, not rounded
Both terms may be up to 30 digits long and are handled as exact whole numbers, so a fraction such as 123456789012345678901234567890/987654321098765432109876543210 reduces to 13717421/109739369 with nothing lost. Only the decimal is rounded, to ten significant figures. To add, subtract, multiply or divide two fractions, use the fraction calculator, which simplifies its answer the same way.
With your numbers
- 84 = 126 × 0 + 84
- 126 = 84 × 1 + 42
- 84 = 42 × 2 + 0
- The last non-zero remainder is the greatest common factor: 42
- 84 ÷ 42 = 2, and 126 ÷ 42 = 3
Quick reference
Exact whole-number arithmetic; only the decimal is rounded.
| Example | Simplest form |
|---|---|
| 84/126 | 2/3 |
| 6/8 | 3/4 |
| 12/16 | 3/4 |
| 45/60 | 3/4 |
| −6/8 | −3/4 |
| 100/7 | 100/7 |
| 1071/462 | 51/22 |
| 17/17 | 1 |
Common questions about the Simplify Fraction Calculator
- How do I simplify a fraction by hand?
- Find the largest number that divides both the numerator and the denominator, their greatest common factor, and divide both by it. For 84/126 that number is 42, so 84 ÷ 42 = 2 and 126 ÷ 42 = 3, and the fraction is 2/3. Dividing by smaller common factors one at a time (2, then 3, then 7) gets to the same place in more steps.
- How do I find the greatest common factor quickly?
- Divide the larger number by the smaller and keep the remainder, then divide the smaller number by that remainder, and repeat until the remainder is zero. The last divisor is the greatest common factor. This is Euclid’s algorithm, and it needs no factor lists: 126 = 84 × 1 + 42, then 84 = 42 × 2 + 0, so the answer is 42.
- How do I know a fraction is already in simplest form?
- When the numerator and denominator have no common factor except 1. That does not require either number to be prime: 8/9 is in simplest form because 8 = 2³ and 9 = 3² share no prime. The page says so when the greatest common factor is 1.
- Where does the minus sign go?
- On the numerator. −6/8, 6/−8 and −(6/8) are the same number, and all three simplify to −3/4. A negative over a negative is positive, so −6/−8 is 3/4.
- Why can the denominator not be zero?
- Because a fraction is a division, and division by zero has no answer: no number times 0 gives 5. The page says the value is not defined rather than printing anything. A zero numerator is fine; 0/5 is simply 0.