Fraction Calculator
Last updated: 27 June 2026
Reviewed by Gavin Meiring, Lead research and primary author ยท Doctoral Candidate (Corporate Governance) ยท Research and drafting assisted by AI
- Ancient Egyptians only used unit fractions (fractions with 1 in the numerator, like 1/3 or 1/5). To write 3/4, they'd write 1/2 + 1/4.
- Continued fractions โ fractions within fractions โ can represent any real number exactly. The golden ratio ฯ is the 'simplest' continued fraction: [1; 1, 1, 1, 1, ...].
- The fraction โ cannot be written exactly as a decimal โ the 3s repeat forever. This is because 10 is not divisible by 3. In base 12, โ is exactly 0.4.
Fraction Calculator
A fraction calculator performs addition, subtraction, multiplication, and division with fractions and mixed numbers, showing both the working and the simplified result. It is used by students, teachers, and anyone who needs to work accurately with fractions without manual calculation.
How to Use the Fraction Calculator
- Enter the numerator and denominator of your first fraction in the two input fields.
- Select the operation: add, subtract, multiply, or divide.
- Enter the numerator and denominator of your second fraction.
- For mixed numbers, enter the whole number part in the separate field provided.
- Click Calculate to see the result as a simplified fraction and as a decimal.
The Formula
Addition and subtraction require a common denominator. Find the Lowest Common Denominator (LCD) of the two fractions, convert each fraction, then add or subtract the numerators. For multiplication: multiply the numerators together and the denominators together, then simplify. For division: multiply by the reciprocal of the second fraction (flip numerator and denominator, then multiply). Always simplify the result by dividing numerator and denominator by their Greatest Common Factor (GCF).
Real-World Example
Calculate 3/4 plus 2/5. The LCD of 4 and 5 is 20. Convert: 3/4 = 15/20 and 2/5 = 8/20. Add numerators: 15 plus 8 = 23. Result: 23/20. As a mixed number: 1 and 3/20. As a decimal: 23 divided by 20 = 1.15. To verify the simplification: the GCF of 23 and 20 is 1, so 23/20 is already fully simplified.
Simplifying Fractions
A fraction is in its simplest form when the numerator and denominator share no common factors other than 1. To simplify, find the GCF of the numerator and denominator and divide both by it. For example, 18/24 has a GCF of 6: divide both by 6 to get 3/4. You can also simplify progressively by dividing by any common factor repeatedly until no further simplification is possible. Equivalent fractions represent the same value with different numbers: 1/2, 2/4, 3/6, and 50/100 are all equivalent.
Frequently Asked Questions
What is an improper fraction and how do I convert it to a mixed number? An improper fraction has a numerator larger than its denominator, such as 11/4. To convert to a mixed number, divide the numerator by the denominator: 11 divided by 4 = 2 remainder 3, giving the mixed number 2 and 3/4.
Why do fractions need a common denominator for addition but not for multiplication? Addition and subtraction of fractions require equal-sized parts (common denominators) to combine meaningfully. Multiplication works differently: you are finding a fraction of a fraction, which is done by multiplying numerators and denominators directly without needing them to match.
How do I divide by a fraction? Dividing by a fraction is the same as multiplying by its reciprocal. To divide 3/4 by 2/3: flip the second fraction to get 3/2, then multiply: 3/4 multiplied by 3/2 = 9/8 = 1 and 1/8.
How are fractions used in everyday life? Fractions appear in cooking (half a cup, three-quarters of a teaspoon), construction (measurements in inches), finance (interest rates expressed as fractions of a percent), and time (a quarter of an hour). Understanding fractions is a foundational arithmetic skill with wide practical applications.
All four operations on the same pair
The example on this page works through 3/4 plus 2/5. Running the other three operations on the same pair shows how differently they behave, and how each result simplifies.
| Operation | Working | Exact result | Simplified | Decimal |
|---|---|---|---|---|
| Addition | 15/20 plus 8/20 | 23/20 | 1 and 3/20 | 1.1500 |
| Subtraction | 15/20 minus 8/20 | 7/20 | 7/20 | 0.3500 |
| Multiplication | 3 multiplied by 2 over 4 multiplied by 5 | 6/20 | 3/10 | 0.3000 |
| Division | 3/4 multiplied by 5/2 | 15/8 | 1 and 7/8 | 1.8750 |
Addition and subtraction share the common denominator step, and the result keeps the denominator of 20 because neither numerator shares a factor with it. Multiplication skips the common denominator entirely and then needs simplifying: 6/20 reduces by a greatest common factor of 2. Division flips the second fraction and multiplies, which is why the answer is larger than both inputs.
A second pair exercises the simplify step harder. 9/12 multiplied by 8/15 gives 72/180, and the greatest common factor of 72 and 180 is 36, so the answer reduces to 2/5, a decimal of 0.4000. Reducing before multiplying gives the same answer faster: 9/12 reduces to 3/4, 8/15 stays as it is, and 3/4 multiplied by 8/15 gives 24/60, which is 2/5. Both routes reach the same place, and the second involves smaller numbers.
Mixed numbers and negative fractions
A mixed number is shorthand for a sum, so 2 and 1/3 means 2 plus 1/3, which is 7/3 as an improper fraction. Converting first keeps one set of rules in play for the whole calculation.
| Calculation | Converted | Exact result | Mixed number | Decimal |
|---|---|---|---|---|
| 2 and 1/3 plus 1 and 3/4 | 7/3 plus 7/4 | 49/12 | 4 and 1/12 | 4.0833 |
| 7/6 plus 5/4 | unchanged | 29/12 | 2 and 5/12 | 2.4167 |
| 5/6 minus 7/15 | unchanged | 11/30 | 11/30 | 0.3667 |
| minus 3/4 plus 5/8 | unchanged | minus 1/8 | minus 1/8 | minus 0.1250 |
Two things in that table are easy to get wrong by hand. The first is the sign on a negative fraction: the minus sign belongs to the numerator, so minus 3/4 is the same quantity as 3 over minus 4, and adding 5/8 to it gives a negative result because 6/8 is larger than 5/8. The second is the denominator in the third row: 6 and 15 share a factor of 3, so their lowest common multiple is 30 and not 90. Finding the lowest common multiple rather than multiplying the denominators together keeps the numbers small, and 5/6 minus 7/15 becomes 25/30 minus 14/30, which is 11/30.
Finding the divisor that does the work
Two functions sit underneath every row above. The greatest common factor of two numbers is the largest integer that divides both without a remainder, and it is what you divide by to simplify. The lowest common multiple is the smallest number that both divide into, and it is what you convert to before adding or subtracting.
They are linked: the greatest common factor of two numbers multiplied by their lowest common multiple equals the product of the two numbers. For 6 and 15, the greatest common factor is 3, the lowest common multiple is 30, and 3 multiplied by 30 is 90, which is 6 multiplied by 15. That relationship is a quick way to check the common multiple you have chosen, and to see why the two smallest denominators in the table above are 30 and 20 rather than 90 and 20.
Euclid's algorithm gets the greatest common factor without factoring either number. For 180 and 210, divide 210 by 180 to get a remainder of 30, then divide 180 by 30 to get a remainder of 0. The last non-zero remainder, 30, is the greatest common factor. Repeating that loop is what makes a calculator able to simplify a fraction with a nine digit numerator instantly, and it handles numbers that are far too large to factor by inspection.
Why the tool returns a fraction and not a decimal
Any fraction whose denominator has prime factors other than 2 and 5 produces a decimal that never terminates. One third is 0.3333 and so on forever, and one seventh repeats a six digit cycle. A calculator that shows only decimals is forced to round, and the rounding is where the value starts to drift.
Floating point arithmetic in a computer has the same problem in a more surprising form. In binary, 0.1 and 0.2 have no exact representation, and adding them returns 0.30000000000000004 rather than 0.3. The error is around 5.6 times ten to the power of minus 17, which is invisible in most work and decisive in some: a total that should come to exactly 0.3 fails an equality test, and a chained calculation can carry the residue forward. Keeping the value as a fraction of two integers avoids the problem entirely, because the denominator and numerator are whole numbers and the arithmetic on them is exact.
That is why the result appears as a simplified fraction first and a decimal second. Use the fraction when the answer feeds another calculation, and the decimal when you need to read the size of it.
Comparing two fractions without converting to decimals
Working out which of two fractions is larger by converting both to decimals loses precision, and the loss grows with awkward denominators. Cross multiplication keeps the comparison exact: multiply the numerator of each fraction by the denominator of the other.
| Comparison | Cross products | Result |
|---|---|---|
| 5/6 against 7/8 | 5 multiplied by 8 gives 40; 7 multiplied by 6 gives 42 | 7/8 is larger |
| 3/4 against 2/5 | 3 multiplied by 5 gives 15; 2 multiplied by 4 gives 8 | 3/4 is larger |
| 7/15 against 5/6 | 7 multiplied by 6 gives 42; 5 multiplied by 15 gives 75 | 5/6 is larger |
The larger cross product belongs to the larger fraction. The method needs no common denominator and no rounding, so it compares exact values however awkward the denominators are. Sorting a list of fractions uses the same comparison repeatedly, comparing pairs until the order settles.
Where the fraction method stops applying
Four boundaries of the method are worth stating.
- Division by zero has no answer. An input of 0 in the denominator of the second fraction, in a division, leaves the operation undefined rather than returning a large number. A calculator that returns something is guessing.
- The arithmetic is exact for rational numbers and only handles rational numbers. Roots, logarithms and constants such as pi cannot be written as a fraction of two integers, so a result that involves them will be a decimal approximation.
- Simplifying to lowest terms assumes integer numerator and denominator. A fraction written as 2.5/4 is a valid expression and not a fraction in the form this method simplifies.
- Whole numbers work through every operation. A whole number is a fraction with a denominator of 1, so 3 plus 2/5 converts to 15/5 plus 2/5 and gives 17/5, which is 3 and 2/5.
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