This fraction calculator adds, subtracts, multiplies, and divides any two fractions and automatically simplifies the answer to lowest terms. It also shows the result as a decimal and, when the answer is an improper fraction, as a mixed number — the three forms teachers most often ask for.
Enter the numerator and denominator of each fraction, choose an operation, and calculate. Negative fractions are supported: just enter a negative numerator. The note under the results shows the unsimplified fraction and the greatest common divisor (GCD) used to reduce it, so you can follow every step.
The Rules for Each Fraction Operation
Each operation follows a fixed rule:
- Addition: a/b + c/d = (ad + cb) / bd. Example: 1/2 + 3/4 = (4 + 6)/8 = 10/8 = 5/4.
- Subtraction: a/b − c/d = (ad − cb) / bd. Example: 3/4 − 1/6 = (18 − 4)/24 = 14/24 = 7/12.
- Multiplication: a/b × c/d = ac / bd. Example: 2/3 × 3/5 = 6/15 = 2/5.
- Division: a/b ÷ c/d = a/b × d/c = ad / bc. Example: 1/2 ÷ 3/4 = 4/6 = 2/3.
Addition and subtraction require a common denominator, which is why both numerators get cross-multiplied first. Multiplication is the simplest rule — straight across. Division works by flipping the second fraction (taking its reciprocal) and multiplying.
Simplifying with the Greatest Common Divisor
A fraction is in lowest terms when its numerator and denominator share no common factor other than 1. To simplify, divide both by their greatest common divisor (GCD).
- Find the GCD with the Euclidean algorithm: repeatedly replace the larger number with the remainder of dividing it by the smaller, until the remainder is 0.
- Example: simplify 18/24. GCD(18, 24): 24 mod 18 = 6, 18 mod 6 = 0, so GCD = 6. Then 18 ÷ 6 = 3 and 24 ÷ 6 = 4, giving 3/4.
- If the simplified numerator is larger than the denominator (an improper fraction such as 7/4), it can also be written as the mixed number 1 3/4.
This calculator performs both steps automatically after every operation.
Worked Example: 1/2 + 3/4
Add 1/2 and 3/4 using the cross-multiplication rule (ad + cb)/bd.
Step 1: Multiply diagonally and add: (1 × 4) + (3 × 2) = 4 + 6 = 10. That is the new numerator. Step 2: Multiply the denominators: 2 × 4 = 8. Result so far: 10/8. Step 3: Simplify. GCD(10, 8) = 2, so 10/8 = 5/4. Step 4: Convert to a mixed number: 5 ÷ 4 = 1 remainder 1, so 5/4 = 1 1/4.
As a decimal, 5/4 = 1.25. A quick check: 1/2 is 0.5 and 3/4 is 0.75, and 0.5 + 0.75 = 1.25, confirming the answer.
Frequently Asked Questions
How do you add fractions with different denominators?
Cross-multiply to get a common denominator: a/b + c/d = (ad + cb)/bd. For 1/3 + 1/4, compute (1×4 + 1×3)/(3×4) = 7/12. Then simplify if the numerator and denominator share a common factor.
How do you divide fractions?
Keep the first fraction, change division to multiplication, and flip the second fraction — often remembered as "keep, change, flip." So 1/2 ÷ 3/4 becomes 1/2 × 4/3 = 4/6 = 2/3. Dividing by a fraction less than 1 always produces a larger result.
How do I convert a fraction to a decimal?
Divide the numerator by the denominator. For 3/8, compute 3 ÷ 8 = 0.375. Some fractions produce repeating decimals — 1/3 = 0.333… — because the denominator has prime factors other than 2 and 5.
What is an improper fraction and a mixed number?
An improper fraction has a numerator larger than (or equal to) its denominator, like 7/4. A mixed number writes the same value as a whole number plus a proper fraction: 7/4 = 1 3/4. Divide numerator by denominator; the quotient is the whole part and the remainder is the new numerator.
Why can a denominator never be zero?
A fraction represents division, and division by zero is undefined in mathematics. There is no number that, multiplied by 0, gives a nonzero numerator, so expressions like 5/0 have no value. This calculator returns an error whenever a denominator is zero.