Fractions trip up students at every level — from elementary arithmetic to university calculus. Yet the rules are surprisingly simple once you see them applied clearly. Whether you're struggling with adding fractions with different denominators or need a quick refresher on dividing mixed numbers, this guide walks you through all four operations with step-by-step examples.
By the end of this guide, you'll be able to add, subtract, multiply, and divide any fraction with confidence — and you'll have a free embedded calculator to check your work instantly.
A fraction represents a part of a whole. It has two parts separated by a line:
For example, in ¾, the numerator 3 means you have 3 parts, and the denominator 4 means the whole is divided into 4 equal parts.
| Type | Definition | Example |
|---|---|---|
| Proper fraction | Numerator < Denominator | 3/4, 1/2, 5/8 |
| Improper fraction | Numerator ≥ Denominator | 7/4, 9/5, 11/3 |
| Mixed number | Whole number + proper fraction | 1¾, 2½, 3⅓ |
| Equivalent fractions | Different notation, same value | 1/2 = 2/4 = 4/8 |
Adding fractions has two cases depending on whether the denominators are the same or different.
The LCD is the smallest number both denominators divide into evenly. For 1/3 + 1/4, the LCD is 12 (since 3×4=12 and no smaller number works).
Multiply numerator and denominator of each fraction to get the LCD as denominator: 1/3 = 4/12 and 1/4 = 3/12.
4/12 + 3/12 = 7/12. Keep the denominator. Simplify if possible.
Subtraction follows the exact same rules as addition — find a common denominator, then subtract the numerators.
Multiplication is actually the easiest fraction operation — no common denominator needed!
In 2/3 × 3/4, the 3 in the numerator of the second fraction and the 3 in the denominator of the first can cancel before you multiply: (2/1) × (1/4) = 2/4 = 1/2. This avoids large numbers and simplification steps later.
Dividing fractions uses a simple trick: multiply by the reciprocal. The reciprocal of a fraction is just the fraction flipped upside down.
A fraction is in its simplest form when the numerator and denominator share no common factors other than 1. To simplify, find the Greatest Common Divisor (GCD) and divide both parts by it.
List factors of both numbers and find the largest one they share. For 12/18: factors of 12 are 1,2,3,4,6,12; factors of 18 are 1,2,3,6,9,18. GCD = 6.
12 ÷ 6 = 2 and 18 ÷ 6 = 3. So 12/18 simplifies to 2/3.
For large numbers, use Euclid's method: GCD(48, 18) → 48 = 2×18+12 → GCD(18,12) → 18=1×12+6 → GCD(12,6) = 6. The GCD is the last non-zero remainder.
Mixed numbers combine a whole number with a fraction (like 2¾). To perform any arithmetic with them, first convert to improper fractions.
Enter any two fractions below and choose the operation. The calculator shows the result in both fraction and decimal form.
Enter numerators and denominators for each fraction, then choose your operation.
For more tools including fraction simplification, mixed number conversion, and fraction-to-decimal conversion, visit our full Math Calculator suite.
Fractions aren't just a classroom exercise — they appear constantly in daily life:
Find the Least Common Denominator (LCD), convert each fraction to have that denominator, then add the numerators. For 1/3 + 1/4: LCD = 12, so 4/12 + 3/12 = 7/12.
Multiply numerators together and denominators together. No common denominator needed. For 2/3 × 3/4: (2×3)/(3×4) = 6/12 = 1/2. Cross-canceling before multiplying simplifies the work.
Use "Keep, Change, Flip": keep the first fraction, change division to multiplication, and flip the second fraction. For 2/3 ÷ 4/5: 2/3 × 5/4 = 10/12 = 5/6.
Find the GCD of numerator and denominator, then divide both by that number. For 12/16: GCD = 4, so 12÷4 = 3 and 16÷4 = 4, giving 3/4.
Convert to improper fractions first, find the LCD, add, and convert back. For 1½ + 2⅓: 3/2 + 7/3 = 9/6 + 14/6 = 23/6 = 3⅚.
Fraction arithmetic follows four clear rules. Adding and subtracting require a common denominator. Multiplying is straightforward — numerator times numerator, denominator times denominator. Dividing means multiplying by the reciprocal. Always simplify your answer using the GCD.
Practice these four operations with our free calculator and you'll handle any fraction problem with confidence — in school, the kitchen, or the workshop.