The short answer
To add or subtract fractions, first convert them to a common denominator, then add or subtract the numerators. To multiply, multiply straight across — numerator × numerator, denominator × denominator. To divide, flip the second fraction and multiply instead. Every result should end up simplified by dividing both parts by their greatest common factor.
Key takeaways
- Addition and subtraction need a common denominator first; multiplication and division don't — trying to find one anyway just adds unnecessary steps.
- "Keep, change, flip" for division works because dividing by a number is the same as multiplying by its reciprocal.
- A fraction is only in simplest form when its numerator and denominator share no common factor besides 1 — always divide by the GCF at the end.
- An improper fraction (numerator ≥ denominator) and its equivalent mixed number represent the same value — converting between them never changes what the fraction is worth.
The four operations, side by side
| Operation | Rule | Example |
|---|---|---|
| Addition | a/b + c/d = (ad+bc)/bd | 1/4 + 1/6 = 5/12 |
| Subtraction | a/b − c/d = (ad−bc)/bd | 3/4 − 1/6 = 7/12 |
| Multiplication | a/b × c/d = ac/bd | 2/3 × 3/5 = 2/5 |
| Division | a/b ÷ c/d = a/b × d/c | 1/2 ÷ 3/4 = 2/3 |
Worked example: adding fractions with different denominators
1/4 + 1/6 — LCM of 4 and 6 is 12
1/4 = 3/12
1/6 = 2/12
3/12 + 2/12 = 5/12 (already in simplest form)
Using the least common multiple (12, not just multiplying the denominators together to get 24) keeps the numbers as small as possible, which makes the arithmetic easier and often means less simplifying at the end.
Simplifying and mixed numbers
Simplifying means dividing the numerator and denominator by their greatest common factor (GCF). For 8/12, the GCF is 4, so 8/12 = 2/3.
A mixed number combines a whole number with a fraction, like 2¾. To convert an improper fraction to a mixed number, divide the numerator by the denominator: 11 ÷ 4 = 2 remainder 3, so 11/4 = 2¾. To go the other way, multiply the whole number by the denominator, add the numerator, and keep the same denominator: 2¾ = (2×4+3)/4 = 11/4.
Common mistakes to avoid
- Adding numerators and denominators separately — 1/2 + 1/3 is not 2/5. Fractions must share a denominator before their numerators can be added.
- Leaving the final answer unsimplified — 8/12 is numerically correct but 2/3 is the expected simplest form.
- Finding a common denominator before multiplying or dividing — that step only applies to addition and subtraction.
- Converting a mixed number to a decimal before doing fraction arithmetic on it — convert to an improper fraction instead to keep the exact value.
Related calculators
- Ratio Calculator — simplify and compare ratios using the same GCF logic.
- Percentage Calculator — convert a fraction result into a percentage.
- Greatest Common Factor Calculator — double-check the GCF used to simplify a fraction.
- Least Common Multiple Calculator — find the common denominator for adding or subtracting.