Simplify Fractions Calculator
Reduce any fraction to its simplest form (lowest terms) using the GCD. Shows the GCD used and the decimal equivalent.
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How to use this calculator
Dividing numerator and denominator by their Greatest Common Divisor produces the equivalent fraction in lowest terms.
- 1
Enter the numerator and denominator of the fraction you want to simplify.
- 2
The GCD of the two numbers is found and both are divided by it.
- 3
The simplified fraction and decimal equivalent are displayed.
Frequently asked questions
Can I enter decimals in the numerator or denominator?
No. Only whole numbers are accepted, and decimals return an error rather than being silently rounded. To reduce a fraction like 12.5/16, multiply top and bottom by the same power of ten, or by 2 here, to get 25/32, then enter that. The calculator confirms that 25/32 is already in lowest terms, with a GCD of 1.
What happens with negative numbers?
The sign is moved to the numerator. 49/−14 simplifies to −7/2, and −18/−45 simplifies to 2/5 because two negatives cancel. The GCD shown is always positive.
What does it show for a numerator of 0?
Any non-zero denominator reduces 0 to 0/1. For 0/9 the GCD displayed is 9, so the status line says the fraction was divided by 9 rather than “already simplified”, but the final answer is 0/1 with a decimal of 0.
When is a mixed number shown?
When the simplified fraction is improper, meaning the numerator’s size exceeds the denominator. −7/2 is shown with the mixed form −3 1/2. When the fraction simplifies to a whole number, such as 144/12 becoming 12/1, the mixed-number line shows just 12. Proper fractions like 3/4 have no mixed-number line.
What does a GCD of 1 mean?
The numerator and denominator share no common factor greater than 1, so the fraction is already in lowest terms. 17/23 is an example: the calculator returns it unchanged with “Already simplified? Yes”. The numbers do not both need to be prime for this to hold.
Why is the decimal sometimes rounded?
The decimal equivalent is shown to eight decimal places with trailing zeros removed. 84/126 reduces to 2/3 and displays 0.66666667. The simplified fraction itself is exact; only the decimal is rounded.
Is there a limit on the size of the numbers?
Very large integers beyond 9,007,199,254,740,991 cannot be represented exactly by the browser, so GCD results above that size may be wrong. For ordinary school and kitchen fractions this never comes up.
Simplify fractions calculator — reduce to lowest terms with the GCD
Dividing by the greatest common divisor
A fraction is in lowest terms when its numerator and denominator have no common factor except 1. To reach that form, divide both by their greatest common divisor, found here with the Euclidean algorithm. The calculator shows the reduced fraction, the GCD it used, the original, the decimal equivalent and a mixed number when the value exceeds one.
| Input | GCD | Lowest terms | Extra |
|---|---|---|---|
| 12/16 | 4 | 3/4 | Decimal 0.75 |
| 84/126 | 42 | 2/3 | Decimal 0.66666667 |
| −18/−45 | 9 | 2/5 | Signs cancel |
| 49/−14 | 7 | −7/2 | Mixed −3 1/2 |
| 17/23 | 1 | 17/23 | Already simplified |
| 144/12 | 12 | 12/1 | Mixed 12 |
Worked example: a large pair
For 84/126, it is not obvious by eye how far to reduce. The calculator finds the GCD of 42 and returns 2/3 in a single step. Dividing by smaller common factors one at a time, such as 2 then 3 then 7, would reach the same answer, but using the GCD avoids missing a factor.
Worked example: signs and improper fractions
The input 49/−14 has a negative denominator. After dividing by 7 the result is −7/2, with the sign on the numerator, which is the standard way to write it. Since 7 exceeds 2, the mixed-number form −3 1/2 is also shown. A fraction that reduces to a whole number, like 144/12, appears as 12/1 and the mixed line simply reads 12.
Checking by hand
You can verify any result by multiplying the simplified numerator and denominator back by the GCD; you should recover the original pair. For example, 3/4 times 4 over 4 gives 12/16. If two numbers are consecutive integers, their GCD is always 1, so a fraction like 17/18 cannot be reduced. The calculator’s GCD line is a quick way to confirm this. Likewise, if the denominator is a prime number that does not divide the numerator, the fraction is already in lowest terms, which is why 17/23 comes back unchanged.
Input requirements
Enter whole numbers in both boxes and keep the denominator non-zero. A zero denominator and any decimal input both return an error message rather than a guess. Convert decimals first by scaling both parts: 12.5/16 becomes 25/32, which the calculator reports as already in lowest terms.
Learn more from an authoritative source:
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