Fraction Calculator
Add, subtract, multiply, or divide two fractions.
Your Details
First Fraction
Second Fraction
How It Works
- Enter the first fraction's numerator and denominator.
- Choose an operation.
- Enter the second fraction and press Calculate.
Formula Used
Multiplication and division follow their own standard fraction rules.
Good to Know
- • Results are automatically simplified to lowest terms.
- • Dividing by a fraction is the same as multiplying by its reciprocal.
Important Notes
- • Denominators cannot be zero.
Why a Common Denominator Is Needed
Adding or subtracting fractions requires a common denominator because you can only directly combine pieces of the same size — you can't add "1 of 4 equal pieces" to "1 of 2 equal pieces" without first converting them so the pieces are the same size. This calculator handles that conversion automatically using cross-multiplication: multiplying each fraction's numerator by the other fraction's denominator, then combining over the product of both denominators.
The reason this works is that multiplying both the numerator and denominator of a fraction by the same number doesn't change its value — 1/4 is exactly equal to 2/8, or 3/12, or any other equivalent form. Cross-multiplication is really just finding equivalent forms of both fractions that happen to share a denominator, then adding the numerators directly.
Multiplication and Division Work Differently
Unlike addition and subtraction, multiplying fractions doesn't need a common denominator at all — you simply multiply the numerators together and the denominators together: (a/b) × (c/d) = (a×c)/(b×d). This is more straightforward than addition, which is worth remembering when a multi-step problem mixes operations.
Dividing by a fraction is equivalent to multiplying by its reciprocal (flipping the numerator and denominator): (a/b) ÷ (c/d) = (a/b) × (d/c). This is why division and multiplication of fractions share such closely related mechanics — division is really just multiplication in disguise, once you flip the second fraction.
Breaking Down the Terminology
- Numerator — the top number of a fraction, representing how many parts are being counted.
- Denominator — the bottom number, representing how many equal parts the whole is divided into.
- Common denominator — a shared denominator between two or more fractions, required for addition and subtraction.
- Reciprocal — a fraction flipped upside down (numerator and denominator swapped); used when dividing by a fraction.
- Simplify (reduce) — dividing both numerator and denominator by their greatest common divisor to express a fraction in its lowest terms.
A Hands-On Example: All Four Operations
Using 1/4 and 1/2 throughout:
Subtract: 1/2 − 1/4 = 2/4 − 1/4 = 1/4
Multiply: 1/4 × 1/2 = 1/8
Divide: 1/4 ÷ 1/2 = 1/4 × 2/1 = 2/4 = 1/2
Notice that multiplying two fractions smaller than 1 always produces a smaller result (1/4 × 1/2 = 1/8, smaller than either starting fraction), while dividing by a fraction smaller than 1 always produces a larger result — a pattern that can feel counterintuitive at first but follows directly from the reciprocal rule.
Operation Rules Table
| Operation | Rule |
|---|---|
| Add / Subtract | Convert to a common denominator first |
| Multiply | Multiply numerators and denominators directly |
| Divide | Multiply by the second fraction's reciprocal |
Traps to Avoid With Fractions
- Adding numerators and denominators directly without a common denominator — 1/4 + 1/2 is not 2/6; it requires converting to a shared denominator first.
- Forgetting to simplify the final result — 4/8 is a correct answer, but it should be simplified to 1/2 for a clean final form.
- Flipping the wrong fraction when dividing — only the second fraction (the divisor) gets flipped into its reciprocal.