Fraction Calculator
This free fraction calculator handles all four fraction operations - adding fractions, subtracting, multiplying, and dividing - plus a dedicated fraction simplifier. All results are reduced to lowest terms automatically, with complete step-by-step working shown.
All calculations use standard published formulas. Results are for informational use only.
How to use this fraction calculator
- Select a mode: Add, Subtract, Multiply, Divide, or Simplify.
- Enter the numerator and denominator for each fraction (whole numbers only).
- The result appears instantly in its fully simplified form.
- Check the step-by-step section below to see the full working.
All results are automatically reduced to lowest terms. If the result is a whole number - for example, 6/3 = 2 - the denominator is removed and the whole number is shown.
What is a fraction?
A fraction represents a part of a whole or a ratio of two integers. Written as numerator/denominator, the numerator is how many parts you have, and the denominator is how many parts make up the whole.
For example, 3/4 means 3 out of 4 equal parts. A pizza cut into 4 slices with 3 remaining is 3/4 of the whole pizza. Fractions also represent division: 3/4 = 3 divided by 4 = 0.75.
- Proper fraction - numerator is less than denominator: 3/4, 1/5
- Improper fraction - numerator is greater than or equal to denominator: 7/4, 5/5
- Mixed number - whole number plus a fraction: 1 3/4
- Unit fraction - numerator is 1: 1/2, 1/7, 1/100
How to add and subtract fractions
Fractions can only be added or subtracted when they share the same denominator. If they do not, you must first find the least common denominator (LCD) and convert both fractions.
Example: 1/3 + 1/4
- LCD(3, 4) = 12
- Convert: 4/12 + 3/12
- Add numerators: 7/12
- GCD(7, 12) = 1 - already simplified: 7/12
How to multiply and divide fractions
Multiplying fractions
Multiply numerators together and denominators together. Simplify the result.
Example: (2/3) x (3/4) = 6/12 = 1/2
Dividing fractions
Invert the second fraction (find its reciprocal) and multiply. Often taught as "keep, change, flip".
Example: (2/3) / (4/5) = (2/3) x (5/4) = 10/12 = 5/6
How to simplify a fraction
A fraction is in its simplest form when the numerator and denominator share no common factor greater than 1. To simplify, divide both by their GCD (greatest common divisor).
Example: Simplify 24/36.
- GCD(24, 36) = 12
- 24 / 12 = 2
- 36 / 12 = 3
- Simplified: 2/3
Where fractions appear in everyday life
Cooking and baking
Recipes constantly use fractions: 3/4 cup of sugar, 1/2 teaspoon of salt, 2/3 cup of milk. Scaling a recipe up or down requires multiplying each ingredient by a fraction. Understanding fraction arithmetic directly translates to cooking accuracy, especially when working with imperial measurements.
Finance and interest rates
Interest rates are often expressed as fractions: a mortgage rate of 1/4% per month = 0.25%. Loan repayments, investment returns, and tax rates involve fraction-to-decimal conversion routinely. Even splitting a bill involves dividing a total by the number of people - a fraction operation.
Construction and measurement
Imperial measurements - feet, inches, yards - are deeply fraction-based. A board measuring 5 and 3/8 inches requires adding the fractional component to calculate total material needed. Tile layouts, pipe cutting, and woodworking all depend on accurate fraction arithmetic.
Understanding fraction results
Improper fractions vs. mixed numbers
This calculator returns improper fractions when the numerator exceeds the denominator - for example, 7/4 rather than 1 3/4. Both forms are correct. To convert: divide numerator by denominator to get the whole number, the remainder becomes the new numerator. 7/4 = 1 remainder 3 = 1 3/4.
Negative fractions
A negative fraction can have the negative sign on the numerator, denominator, or out front - all three are equivalent: -3/4 = 3/-4 = -(3/4). This calculator places the negative sign on the numerator for clarity. The denominator in the result is always positive.
When the result is a whole number
When the denominator equals 1 after simplification, the result is a whole number. For example, 3/4 x 4/3 = 12/12 = 1. The calculator displays just the integer in these cases - no denominator is shown because dividing by 1 is implicit.
Quick tips for fraction calculations
- Adding/subtracting: find the LCD first - never just add denominators.
- Multiplying: multiply straight across - no LCD needed.
- Dividing: flip the second fraction and multiply.
- Always simplify: divide both parts by their GCD to get the simplest form.
- Cross-cancel before multiplying to keep numbers small and avoid large products.
Common fraction mistakes
Adding denominators instead of finding the LCD
A common error: 1/3 + 1/4 = 2/7. This is wrong. The correct approach finds LCD(3,4) = 12, then adds 4/12 + 3/12 = 7/12. Never add or subtract denominators - they must be made equal first.
Forgetting to simplify
After performing any operation, always check whether the result can be simplified. 6/12 is a valid answer technically, but 1/2 is expected in most contexts. Divide both numerator and denominator by their GCD to fully reduce the fraction.
Dividing instead of finding the reciprocal
When dividing fractions, a common mistake is directly dividing numerator by numerator and denominator by denominator. Instead, multiply by the reciprocal of the second fraction. (2/3) / (4/5) is (2/3) x (5/4), not (2/4)/(3/5).
Frequently Asked Questions
How do you add fractions with different denominators?
To add fractions with different denominators, find the least common denominator (LCD), convert both fractions to equivalent fractions using the LCD, then add the numerators. Finally, simplify the result. For example: 1/3 + 1/4. LCD = 12. Convert: 4/12 + 3/12 = 7/12.
How do you multiply fractions?
To multiply fractions, multiply the numerators together and multiply the denominators together. Then simplify the result. For example: (2/3) x (3/4) = 6/12 = 1/2. You can also cross-simplify before multiplying to keep numbers small: cancel any numerator with any denominator that share a common factor.
How do you divide fractions?
To divide by a fraction, multiply by its reciprocal (flip the second fraction). For example: (2/3) / (4/5) = (2/3) x (5/4) = 10/12 = 5/6. The phrase 'keep, change, flip' helps: keep the first fraction, change the operation to multiplication, flip the second fraction.
How do you simplify a fraction?
To simplify a fraction, find the GCD (greatest common divisor) of the numerator and denominator, then divide both by it. For example: 18/24. GCD(18,24) = 6. 18/6 = 3, 24/6 = 4. Simplified: 3/4. A fraction is fully simplified when the GCD of numerator and denominator is 1.
What is the LCD and how do you find it?
The LCD (least common denominator) is the smallest number that both denominators divide into evenly. To find it, calculate the LCM (least common multiple) of the two denominators. LCM(a,b) = (a x b) / GCD(a,b). For example: LCD of 4 and 6 = (4 x 6) / GCD(4,6) = 24/2 = 12.
Can a fraction have a zero denominator?
No. Division by zero is undefined in mathematics, so a fraction with denominator 0 is meaningless. When you see a zero denominator in a calculation, it means the expression has no defined value. This calculator requires non-zero denominators for all inputs.