Fraction Calculator
Add, subtract, multiply, or divide any two fractions. Get the simplified result, mixed number form, and a step-by-step explanation.
What is a Fraction Calculator?
A fraction calculator performs arithmetic operations — addition, subtraction, multiplication, and division — on fractions and returns the result in its simplest form. Working with fractions by hand requires finding common denominators, applying the correct operation, then simplifying using the greatest common divisor. This calculator does all of that automatically and shows the intermediate steps.
Whether you are checking homework, solving a recipe conversion, or working through an engineering problem, this tool gives you the exact simplified fraction alongside the mixed number form and the decimal equivalent — three representations of the same answer.
How to Add and Subtract Fractions
Addition and subtraction require the fractions to share a common denominator. The cleanest approach uses the Least Common Multiple (LCM) of the two denominators as the shared denominator:
- Find the LCM of the two denominators.
- Convert each fraction to an equivalent fraction with the LCM as the denominator by multiplying numerator and denominator by the appropriate factor.
- Add or subtract the numerators; keep the denominator.
- Simplify the result by dividing both numerator and denominator by their GCD.
How to Multiply and Divide Fractions
Multiplication is simpler — no common denominator is needed. Multiply the numerators together and the denominators together, then simplify:
Division is multiplication by the reciprocal. Flip the second fraction (swap numerator and denominator), then multiply:
For example: 1/2 ÷ 1/4 = 1/2 × 4/1 = 4/2 = 2. Dividing by a fraction less than 1 always gives a result larger than the original fraction.
Simplifying Fractions: The GCD Method
A fraction is in its simplest form when the numerator and denominator share no common factor other than 1. To simplify, divide both by their Greatest Common Divisor (GCD), computed using the Euclidean algorithm:
The Euclidean algorithm is efficient even for large numbers — it runs in O(log min(a,b)) steps, meaning it handles hundred-digit numbers in milliseconds.
Example — Dynamic (from form inputs)
Enter a fraction operation above to see step-by-step working here.
Static Example — 3/4 ÷ 3/8
Dividing 3/4 by 3/8:
- Flip the divisor: 3/8 becomes 8/3
- Multiply: 3/4 × 8/3 = (3×8)/(4×3) = 24/12
- Simplify: GCD(24,12) = 12, so 24/12 = 2
Result: 2 (a whole number), decimal = 2.0
About These Parameters
- Numerator
- The top number of a fraction — how many parts you have. Can be any integer, including negative numbers (a negative numerator produces a negative fraction).
- Denominator
- The bottom number — how many equal parts the whole is divided into. Cannot be zero (division by zero is undefined). A denominator of 1 means the fraction equals its numerator as a whole number.
- Operation
- The arithmetic operation to apply. Addition and subtraction require finding a common denominator. Multiplication and division do not — they operate directly on numerators and denominators.
Frequently Asked Questions
Why do you need a common denominator to add fractions?
Fractions with different denominators represent parts of wholes divided into different numbers of pieces. Adding 1/2 and 1/3 directly would be like adding "1 half-pizza" to "1 third-pizza" — the pieces are different sizes and cannot simply be counted together. Converting to a common denominator (sixths in this case: 3/6 + 2/6) makes the pieces the same size so the numerators can be added meaningfully. The LCM of the denominators gives the smallest common denominator, keeping numbers as small as possible.
What is a mixed number?
A mixed number combines a whole number and a proper fraction — for example, 2 1/3 means "two and one third." It is just another way to write an improper fraction (where the numerator is larger than the denominator). 7/3 = 2 1/3 because 7 ÷ 3 = 2 remainder 1, so the whole part is 2 and the remaining fraction is 1/3. Mixed numbers are often easier to visualise; improper fractions are easier to use in calculations.
How do you divide a fraction by a fraction?
Dividing by a fraction is the same as multiplying by its reciprocal — flip the second fraction upside down, then multiply. For example, 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6. This works because dividing by 4/5 is asking "how many times does 4/5 fit into 2/3?" The multiplication by the reciprocal gives the exact answer. One practical check: if you divide by a fraction less than 1 (like 1/2), the result is always larger than the original — 6 ÷ 1/2 = 12 means "12 halves fit into 6."
What does "simplify" mean for fractions?
Simplifying (or reducing) a fraction means writing it with the smallest possible whole numbers for numerator and denominator while keeping the same value. 6/8 simplifies to 3/4 because both are divisible by 2. A fraction is fully simplified when the only common factor of numerator and denominator is 1 — mathematicians say the fraction is in "lowest terms." Simplification does not change the fraction's value; it just makes it easier to read and work with.
Can fractions have negative numerators or denominators?
Yes. A negative fraction means the value is less than zero. By convention, the negative sign is kept on the numerator (or the whole fraction), not the denominator — so -3/4 is preferred over 3/(-4) even though both mean the same thing. When adding or subtracting, negative numerators follow the same rules as negative integers. When dividing, a negative divided by a negative gives a positive result.