Fraction Calculator
Add, subtract, multiply, and divide fractions with simplified answers, decimal results, formulas, examples, and step-by-step calculations.
Calculate with fractions
Enter two fractions, choose an operation, and get the exact simplified result. The calculator also shows the decimal value and explains how the answer was calculated.
Your result
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| First fraction | — |
| Operation | — |
| Second fraction | — |
| Decimal result | — |
| Simplified result | — |
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Step-by-step calculation
What is a fraction?
A fraction represents a number as a relationship between a numerator and a denominator. The numerator is the number above the fraction bar and tells you how many parts are being considered. The denominator is the number below the bar and describes how many equal parts make up the whole.
For example, 3/4 represents three of four equal parts. Fractions are used to describe quantities, measurements, ratios, probabilities, proportions, and mathematical values that do not need to be written as whole numbers.
A fraction is in simplest form when its numerator and denominator have no common factor greater than 1. Reducing a fraction changes its form but not its value.
How to use this fraction calculator
This calculator is designed for four common fraction operations: addition, subtraction, multiplication, and division. Enter the two fractions as whole-number numerators and denominators, select the required operation, and calculate the result.
- Enter the numerator and denominator of the first fraction.
- Enter the numerator and denominator of the second fraction.
- Select addition, subtraction, multiplication, or division.
- Choose Calculate Fraction.
- Review the exact simplified fraction, decimal result, formula, and calculation steps.
The quick-example buttons can be used to test common fraction operations without entering the values manually.
Fraction formulas
Let the two fractions be a/b and c/d, where the denominators are not zero. Each operation follows a different rule.
a/b + c/d = (a × d + c × b) / (b × d)
Subtraction
a/b − c/d = (a × d − c × b) / (b × d)
Multiplication
a/b × c/d = (a × c) / (b × d)
Division
a/b ÷ c/d = (a × d) / (b × c), where c ≠ 0
After the operation is performed, the calculator finds the greatest common divisor of the numerator and denominator and divides both by that value. This produces the simplified form of the answer.
How to add and subtract fractions
Fractions with the same denominator can be added or subtracted by combining their numerators while keeping the denominator unchanged. When the denominators are different, the fractions are first converted into equivalent fractions with a common denominator.
A common denominator is 6.
1/2 = 3/6
1/3 = 2/6
3/6 + 2/6 = 5/6
Subtraction follows the same principle. Find a common denominator, rewrite both fractions, subtract the numerators, and simplify the final result when possible.
How to multiply fractions
Multiplying fractions does not require a common denominator. Multiply the two numerators together and the two denominators together. The resulting fraction can then be reduced to simplest form.
Numerator: 2 × 3 = 6
Denominator: 3 × 5 = 15
Result: 6/15 = 2/5
In some calculations, common factors can be cancelled before multiplication. This is an optional simplification technique that can make the intermediate arithmetic easier.
How to divide fractions
To divide by a fraction, multiply the first fraction by the reciprocal of the second fraction. The reciprocal is obtained by switching the numerator and denominator.
Reciprocal of 2/3 = 3/2
5/6 × 3/2 = 15/12
15/12 = 5/4
The divisor cannot be zero. In fraction form, this means the second fraction cannot have a zero numerator when it is used as the divisor.
What are proper, improper, and mixed fractions?
Proper fractions
A proper fraction has a numerator smaller in absolute value than its denominator, such as 3/5. For positive values, a proper fraction is less than 1.
Improper fractions
An improper fraction has a numerator whose absolute value is greater than or equal to the denominator, such as 7/4. Its value can be greater than or equal to 1.
Mixed numbers
A mixed number combines a whole number with a proper fraction, such as 1 3/4. To convert a positive mixed number into an improper fraction, multiply the whole number by the denominator and add the numerator.
Why simplify a fraction?
Simplifying a fraction makes it easier to read, compare, communicate, and use in later calculations. For example, 12/18 and 2/3 have the same value, but 2/3 is the reduced form.
The calculator uses the greatest common divisor (GCD) of the numerator and denominator to reduce the result. Dividing both by the same non-zero factor preserves the value of the fraction.
Fractions and decimal values
A fraction can be converted to a decimal by dividing its numerator by its denominator. Some fractions produce terminating decimals, while others produce repeating decimals.
1/4 = 0.25
1/2 = 0.5
1/3 = 0.333333…
5/4 = 1.25
FreeCalz displays the decimal result to six decimal places. The simplified fraction remains the exact form of the result, while the decimal is a rounded display representation.
Common fraction mistakes
- Adding denominators directly: In general, 1/2 + 1/3 is not 2/5. Different denominators need a common denominator.
- Forgetting to simplify: A result such as 6/15 can be reduced to 2/5.
- Using zero as a denominator: A denominator of zero is not valid.
- Forgetting the reciprocal during division: Dividing by c/d means multiplying by d/c.
- Dropping negative signs: Negative values must be carried through the arithmetic consistently.
Where are fractions used?
Fractions are part of everyday arithmetic as well as more advanced mathematics. They are commonly used in measurements, recipes, proportions, probability, ratios, construction dimensions, science, engineering, finance, and algebra.
Fractions are especially useful when an exact value is important. Keeping a result as a fraction can avoid rounding during intermediate steps and can make mathematical relationships easier to see.
Understanding your fraction result
The simplified result is the exact fractional answer after common factors have been removed. The decimal result expresses the same value as a decimal and is displayed to six decimal places.
The step-by-step section shows the original inputs, the operation, the intermediate numerator and denominator, the GCD used for simplification, and the final answer. This makes it easier to check the arithmetic rather than relying only on the final number.
Frequently asked questions
How do you add two fractions?
Find a common denominator, convert both fractions to equivalent fractions with that denominator, add the numerators, and simplify the result.
Can fractions with different denominators be added?
Yes. They normally need to be rewritten with a common denominator before addition or subtraction.
How do you multiply fractions?
Multiply the numerators together and the denominators together, then reduce the resulting fraction if a common factor exists.
How do you divide fractions?
Multiply the first fraction by the reciprocal of the second fraction. The divisor must not be zero.
What does simplifying a fraction mean?
It means dividing the numerator and denominator by their greatest common divisor so the fraction has no common factor greater than 1.
Why can’t a denominator be zero?
Division by zero is undefined in ordinary arithmetic, so a fraction with zero as its denominator is not a valid numerical fraction.
Does 2/3 equal 4/6?
Yes. Multiplying both the numerator and denominator of 2/3 by 2 gives 4/6, so the two fractions represent the same value.
Does this fraction calculator show decimals?
Yes. After calculating and simplifying the fraction, the calculator displays its decimal value to six decimal places.