Fraction Calculator
Add, subtract, multiply, or divide any two fractions. The calculator simplifies results to lowest terms and shows the decimal equivalent.
Fraction Calculator — How to Add, Subtract, Multiply & Divide Fractions
A fraction represents a part of a whole — a numerator (the top number) divided by a denominator (the bottom number). Fractions are one of the most fundamental building blocks of mathematics, appearing in cooking recipes, financial ratios, engineering tolerances, and statistical probabilities. This fraction calculator handles all four arithmetic operations and automatically simplifies every result to lowest terms.
What Is a Fraction?
A fraction has two parts: the numerator (top) and the denominator (bottom). The denominator tells you how many equal parts make up the whole, and the numerator tells you how many of those parts you have.
- Proper fraction: numerator is smaller than denominator (e.g. 3/4)
- Improper fraction: numerator is larger than denominator (e.g. 7/4)
- Mixed number: a whole number plus a proper fraction (e.g. 1¾)
How to Add Fractions
To add fractions, both fractions must have the same denominator. If they don't, you find the Lowest Common Denominator (LCD).
Steps to add fractions with different denominators:
1. Find the LCD of the two denominators
2. Convert each fraction so both share the LCD
3. Add the numerators together
4. Keep the LCD as the denominator
5. Simplify by dividing by the GCD
Example: 3/4 + 1/3
- LCD of 4 and 3 = 12
- 3/4 = 9/12 and 1/3 = 4/12
- 9/12 + 4/12 = 13/12 (or 1 and 1/12 as a mixed number)
How to Subtract Fractions
Subtraction follows exactly the same process as addition — find the LCD, convert, then subtract the numerators.
Example: 5/6 − 1/4
- LCD of 6 and 4 = 12
- 5/6 = 10/12 and 1/4 = 3/12
- 10/12 − 3/12 = 7/12
How to Multiply Fractions
Multiplying fractions is simpler than addition or subtraction — you do not need a common denominator.
Steps to multiply fractions:
1. Multiply the two numerators together
2. Multiply the two denominators together
3. Simplify the resulting fraction
Example: 2/3 × 3/5
- 2 × 3 = 6 (new numerator)
- 3 × 5 = 15 (new denominator)
- 6/15 = 2/5 (simplified by dividing by GCD of 3)
A useful shortcut: simplify diagonally before multiplying (called "cross-cancelling"). In the example above, 3 in the numerator and 3 in the denominator cancel to give 2/1 × 1/5 = 2/5 directly.
How to Divide Fractions
To divide fractions, flip the second fraction (find its reciprocal) and then multiply.
Example: 3/4 ÷ 2/5
- Flip 2/5 to get 5/2
- Multiply: 3/4 × 5/2 = 15/8
- 15/8 is already in lowest terms (= 1 and 7/8)
The phrase "keep, change, flip" is a useful memory aid: keep the first fraction, change ÷ to ×, flip the second fraction.
Converting Fractions to Decimals
To convert any fraction to a decimal, simply divide the numerator by the denominator:
| Fraction | Division | Decimal |
|---|---|---|
| 1/2 | 1 ÷ 2 | 0.5 |
| 3/4 | 3 ÷ 4 | 0.75 |
| 1/3 | 1 ÷ 3 | 0.333... |
| 7/8 | 7 ÷ 8 | 0.875 |
Some fractions are terminating decimals (like 3/4 = 0.75) and others are repeating decimals (like 1/3 = 0.333...). This calculator shows the decimal equivalent rounded to 6 significant figures.
Simplifying Fractions (Lowest Terms)
A fraction is in its lowest terms (fully simplified) when the numerator and denominator share no common factor greater than 1. To simplify, divide both by their Greatest Common Divisor (GCD).
Example: Simplify 18/24
- GCD of 18 and 24 = 6
- 18/6 = 3, 24/6 = 4
- Simplified: 3/4
This calculator automatically simplifies every result.
Real-World Uses of Fractions
Fractions appear throughout everyday life:
- Cooking: halving or doubling recipes (3/4 cup × 2 = 3/2 cups = 1½ cups)
- Finance: interest rates, tax fractions, currency exchange
- Construction: measurements in inches (3/8 inch, 5/16 inch)
- Statistics: probability (1/6 chance of rolling a six)
- Music: time signatures (3/4, 4/4 waltz and march rhythms)
Related Resources
Related Calculators
- Percentage Calculator — Convert fractions to percentages.
- Exponent Calculator — Raise fractions to positive or negative powers.
External Authority Resources
- Wolfram MathWorld on Fractions — Mathematical definition and arithmetic properties.
- Khan Academy: Fraction Operations — Interactive learning resources for fractions.
Frequently Asked Questions
To add fractions with different denominators, find the Lowest Common Denominator (LCD), convert each fraction so both share the LCD, then add the numerators and keep the LCD as the denominator. Finally, simplify the result by dividing numerator and denominator by their GCD.
Subtracting fractions works like addition — find the LCD, convert both fractions, then subtract the second numerator from the first. Keep the LCD as the denominator and simplify.
To multiply fractions, multiply the two numerators together and the two denominators together. Then simplify the resulting fraction by dividing by the GCD of the new numerator and denominator.
To divide fractions, flip the second fraction (find its reciprocal) and then multiply. For example, (3/4) ÷ (2/5) = (3/4) × (5/2) = 15/8.
A fraction is in its lowest terms (fully simplified) when the numerator and denominator share no common factors other than 1. For example, 6/8 simplifies to 3/4 because both are divisible by 2.
Divide the numerator by the denominator. For example, 3/4 = 3 ÷ 4 = 0.75. This calculator shows the decimal equivalent automatically alongside the fraction result.