Fraction Calculator
Add, subtract, multiply, and divide fractions with instant simplified fraction results, mixed numbers, and step-by-step explanations.
Calculate Fraction Arithmetic
Enter numerators, denominators, and choose an arithmetic operation.
Fraction Calculation Output
Quick Summary
Our fraction calculator with steps performs addition, subtraction, multiplication, and division on proper, improper, and mixed fractions. It converts unsimplified results into their simplest reduced form, mixed number equivalent, and exact decimal format with step-by-step mathematical reasoning.
How It Works: Fraction Arithmetic Rules
Fraction arithmetic follows fundamental algebraic rules depending on the operation:
1. **Addition & Subtraction:** Find a Least Common Denominator (LCD), adjust numerators proportionately, add/subtract numerators while keeping the common denominator.
2. **Multiplication:** Multiply top numerators together and bottom denominators together ($\a / b \times \c / d = \ac / bd$).
3. **Division:** Multiply the first fraction by the reciprocal (flipped version) of the second fraction ($\a / b \div \c / d = \a / b \times \d / c$).
4. **Simplification:** Divide numerator and denominator by their Greatest Common Divisor (GCD).
Formula Explanation
The standard fraction arithmetic equations for two fractions $\a / b$ and $\c / d$ are:
Step-by-Step Worked Example
Here is a detailed 5-step breakdown for adding 3/4 + 1/6:
- Step 1 (Find Common Denominator): $\text{LCD}(4, 6) = 12$.
- Step 2 (Convert Fractions to LCD): $\3 / 4 = \3 \times 3 / 4 \times 3 = \mathbf{\9 / 12}$; $\1 / 6 = \1 \times 2 / 6 \times 2 = \mathbf{\2 / 12}$.
- Step 3 (Add Numerators): $\9 + 2 / 12 = \mathbf{\11 / 12}$.
- Step 4 (Check Simplification GCD): $\text{GCD}(11, 12) = 1$, so $\mathbf{\11 / 12}$ is already in simplest reduced form.
- Step 5 (Convert to Decimal): $11 \div 12 = \mathbf{0.9167}$!
Calculation Examples: Real-World Scenario Comparison
Compare fraction arithmetic results across operations:
| Operation | Fraction 1 | Fraction 2 | Simplified Fraction Result | Mixed Number | Decimal |
|---|---|---|---|---|---|
| Addition (+) | 3/4 | 1/6 | 11/12 | 11/12 | 0.9167 |
| Subtraction (-) | 5/8 | 1/4 | 3/8 | 3/8 | 0.3750 |
| Multiplication (×) | 2/3 | 3/5 | 2/5 | 2/5 | 0.4000 |
| Division (÷) | 3/4 | 1/2 | 3/2 | 1 1/2 | 1.5000 |
Benefits of Using the Fraction Calculator
Utilizing this calculator provides essential mathematical clarity and speed:
- Eliminates Manual Common Denominator Errors: Automatically finds LCD and converts numerators without manual cross-multiplication mistakes.
- Provides Multiple Formats: Displays simplest proper/improper fraction, mixed number, and decimal representation simultaneously.
- Supports All 4 Basic Arithmetic Operations: Easily toggle between addition, subtraction, multiplication, and division.
- Automatic GCD Reduction: Reduces huge fractions like 24/36 down to 2/3 in a fraction of a second.
Frequently Asked Questions (FAQ)
What is a proper fraction?
A proper fraction is a fraction where the numerator (top number) is smaller than the denominator (bottom number), such as 3/4 or 2/5.
What is an improper fraction?
An improper fraction has a numerator that is greater than or equal to its denominator, such as 7/4 or 11/5.
How do you convert an improper fraction to a mixed number?
Divide numerator by denominator. The whole number quotient becomes the whole part, and the remainder becomes the new numerator (e.g. 7 ÷ 4 = 1 remainder 3, so 1 3/4).
Why must denominators be non-zero?
Division by zero is mathematically undefined. A denominator of 0 represents dividing a quantity into zero parts, which has no valid numerical value.
How do you multiply two fractions?
Multiply top numbers together to get the new numerator, and bottom numbers together to get the new denominator, then simplify.
How do you divide two fractions?
Keep the first fraction, change division to multiplication, and flip the second fraction upside down (multiply by reciprocal).
What is a reciprocal of a fraction?
The reciprocal of a fraction is created by swapping its numerator and denominator (e.g. reciprocal of 3/4 is 4/3).
How do you simplify a fraction to lowest terms?
Find the Greatest Common Divisor (GCD) of numerator and denominator, then divide both numbers by the GCD.
Can fractions be negative?
Yes! A fraction is negative if either the numerator or the denominator is negative, but not both.
How do you add fractions with different denominators?
Find the Least Common Denominator (LCD), convert each fraction so they share the LCD, then add the numerators.