Fraction Calculator
Result is simplified to lowest terms
How Fraction Operations Work
A fraction represents a part of a whole, written as a numerator (top) over a denominator (bottom). The fraction 3/4 means three out of four equal parts. Fractions are essential in cooking, construction, music theory, and anywhere precise parts of a whole matter. This calculator performs the four basic operations and reduces every answer to lowest terms using the greatest common factor (GCF).
Addition and subtraction require a common denominator. To add 1/2 and 1/3, convert to equivalent fractions with denominator 6: 3/6 + 2/6 = 5/6. The general formula is: a/b + c/d = (ad + bc) / bd. Subtraction follows the same pattern with minus instead of plus. When denominators already match, simply add or subtract the numerators and keep the denominator.
Multiplication is more straightforward: multiply the numerators together and the denominators together. (2/3) × (4/5) = 8/15. Division is multiplication by the reciprocal: flip the second fraction and multiply. (2/3) ÷ (4/5) = (2/3) × (5/4) = 10/12, which simplifies to 5/6 by dividing both parts by their GCF of 2.
Simplifying (reducing) a fraction means dividing both numerator and denominator by their GCF until no common factor remains. The fraction 12/18 has GCF(12, 18) = 6, so 12/18 = 2/3. A fraction is in lowest terms when the GCF of its numerator and denominator is 1. Improper fractions (numerator larger than denominator) like 7/4 can be expressed as mixed numbers (1 3/4), though this calculator displays the simplified improper form.
Watch for special cases: dividing by zero is undefined, and any number over zero is undefined (not infinity in basic arithmetic). Zero over any nonzero denominator equals zero. When working with negative fractions, apply the sign to either the numerator or denominator — conventionally to the numerator — and simplify as usual. Fraction operations underpin ratio calculations, probability, and algebra, making this one of the most practical everyday math tools.
Examples
| Example | Result |
|---|---|
| 1/2 + 1/3 | 5/6 |
| 3/4 × 2/5 | 3/10 |
| 2/3 ÷ 4/5 | 5/6 |
| 5/8 + 3/8 | 1 |
| 7/12 − 1/4 | 1/3 |
| Simplify 12/18 | 2/3 |
| 2/5 + 1/5 | 3/5 |
Frequently asked questions
Find a common denominator (often the product of both denominators), convert each fraction, then add the numerators. Example: 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
Divide both the numerator and denominator by their greatest common factor (GCF). Example: 12/18 → divide both by 6 → 2/3.
The reciprocal of a/b is b/a. To divide fractions, multiply the first by the reciprocal of the second: 2/3 ÷ 4/5 = 2/3 × 5/4.