Fraction Calculator
Add, subtract, multiply and divide fractions.
Understanding Fraction Calculator
A fraction is a mathematical expression that represents a part of a whole, a ratio between quantities, or an exact numerical value. A fraction is written using two numbers separated by a horizontal or diagonal division line. The number on top is called the numerator, while the number on the bottom is called the denominator. For example, in 3/4, the numerator is 3 and the denominator is 4. The denominator indicates how many equal parts the whole is divided into, while the numerator indicates how many of those parts are being considered.
Fractions are an important part of arithmetic, algebra, geometry, probability, ratios, measurements, and many other areas of mathematics. Unlike rounded decimal values, fractions can preserve an exact result. For example, 1/3 is exactly one third, while its decimal representation 0.333333... continues indefinitely. Keeping the answer as a fraction can therefore be useful when exact mathematical precision is required.
Using the result in context
Fraction Calculator is intended to show the mathematical relationship behind the answer, not just return a number. Enter the quantities requested by the selected method—such as First Fraction, Operation, Second Fraction—and keep their units or mathematical meaning consistent. Where this calculator offers different shapes, methods, or forms, choose the one that matches the problem before entering the values.
Why the inputs matter
For a useful check, look at the units and the scale of the answer before accepting it. A formula can be evaluated correctly while an input is still inappropriate for the problem. Repeating the calculation with a simple test value or checking the result against the expected direction of change can help identify an input mistake.
About Fraction Calculator
A fraction is a mathematical expression that represents a part of a whole, a ratio between quantities, or an exact numerical value. A fraction is written using two numbers separated by a horizontal or diagonal division line. The number on top is called the numerator, while the number on the bottom is called the denominator. For example, in 3/4, the numerator is 3 and the denominator is 4. The denominator indicates how many equal parts the whole is divided into, while the numerator indicates how many of those parts are being considered.
Fractions are an important part of arithmetic, algebra, geometry, probability, ratios, measurements, and many other areas of mathematics. Unlike rounded decimal values, fractions can preserve an exact result. For example, 1/3 is exactly one third, while its decimal representation 0.333333... continues indefinitely. Keeping the answer as a fraction can therefore be useful when exact mathematical precision is required.
Fractions can be classified in several ways. A proper fraction has a numerator whose absolute value is smaller than the denominator, such as 3/5. An improper fraction has a numerator whose absolute value is equal to or greater than the denominator, such as 7/4. An improper fraction can also be expressed as a mixed number, so 7/4 can be written as 1 3/4. Negative fractions are also valid and represent values below zero.
Adding and Subtracting Fractions
When adding or subtracting fractions, the denominators must represent the same-sized parts. If the denominators are different, the fractions are first converted to equivalent fractions with a common denominator. The least common multiple of the denominators can be used to find a convenient least common denominator.
Once the denominators are the same, the numerators can be added or subtracted while the common denominator remains unchanged. For example, 1/2 + 1/3 can be rewritten as 3/6 + 2/6, which gives 5/6.
a/b + c/d = (ad + bc) / bd
Subtraction:
a/b − c/d = (ad − bc) / bd
Multiplying Fractions
Multiplication is generally simpler than addition or subtraction because the denominators do not need to be made equal. The numerators are multiplied together, and the denominators are multiplied together.
a/b × c/d = ac/bd
The resulting fraction should then be reduced to lowest terms when possible. In some cases, factors can be cancelled before multiplication to make the arithmetic easier and reduce the size of the numbers involved.
Dividing Fractions
To divide one fraction by another, the second fraction is inverted and the operation is changed from division to multiplication. The inverted fraction is called the reciprocal. For example, the reciprocal of 2/3 is 3/2.
a/b ÷ c/d = a/b × d/c = ad/bc
The second fraction cannot have a value of zero when it is used as a divisor. Division by zero is undefined, so an expression such as 1/2 ÷ 0 has no finite mathematical result.
Simplifying Fractions
A fraction can often be simplified by dividing both its numerator and denominator by their Greatest Common Divisor (GCD). The GCD is the largest positive integer that divides both numbers without leaving a remainder.
For example, the fraction 12/18 can be simplified because the GCD of 12 and 18 is 6. Dividing both numbers by 6 gives 2/3. The fractions 12/18 and 2/3 have exactly the same value, but 2/3 is in lowest terms.
Simplified Numerator = Numerator ÷ GCD
Simplified Denominator = Denominator ÷ GCD
Equivalent Fractions
Equivalent fractions have different numerators and denominators but represent the same value. Multiplying or dividing both parts of a fraction by the same non-zero number produces an equivalent fraction. For example, 1/2, 2/4, 3/6, and 50/100 all represent the same numerical value.
This property is particularly important when adding and subtracting fractions because it allows fractions with different denominators to be rewritten using a common denominator without changing their values.
Fraction to Decimal and Percentage
A fraction can be converted to a decimal by dividing the numerator by the denominator. The same decimal can then be expressed as a percentage by multiplying it by 100. These representations are useful when comparing fractions with decimal measurements or percentage values.
Percentage: (Numerator ÷ Denominator) × 100
How This Fraction Calculator Works
Enter the first fraction using the format numerator/denominator, choose an arithmetic operation, and then enter the second fraction. The calculator performs the selected operation using exact integer arithmetic rather than relying on an intermediate rounded decimal value.
After calculating the result, the calculator reduces the fraction to its lowest terms using the Greatest Common Divisor. It also provides the intermediate calculation, the unsimplified fraction, the simplified fraction, the mixed-number representation when applicable, the decimal equivalent, and the percentage equivalent.
Addition: a/b + c/d = (ad + bc)/bd
Subtraction: a/b − c/d = (ad − bc)/bd
Multiplication: a/b × c/d = ac/bd
Division: a/b ÷ c/d = ad/bc
Example
Common denominator = 6
1/2 = 3/6
1/3 = 2/6
3/6 + 2/6 = 5/6
Final Result = 5/6
Decimal = 0.833333
Percentage = 83.3333%
Why Use Fractions?
Fractions are especially useful when an exact result is more important than a rounded decimal approximation. They are commonly encountered in mathematical equations, measurements, recipes, construction, engineering, probability, financial calculations, ratios, and educational problems. Keeping calculations in fractional form can also make algebraic manipulations easier because the exact numerator and denominator remain visible.
Disclaimer
This calculator is intended for informational purposes only. While we strive for accuracy, we cannot guarantee that every result is correct for every situation.