What this calculator does
Fractions are exact in a way decimals often are not. One third is precisely one third, while 0.333 is an approximation that accumulates error through a calculation. That precision is why fractions persist in cooking, construction, music and mathematics itself.
The four operations follow different rules. Multiplication and division are the easy pair: multiply straight across, or invert and multiply. Addition and subtraction need a common denominator first, which is where most of the difficulty lies. This calculator handles all four and simplifies the result.
The formula
For addition and subtraction, both fractions are converted to a common denominator by cross-multiplying, then the numerators are combined. Multiplication multiplies numerators and denominators separately. Division inverts the second fraction and multiplies. The result is then reduced by dividing both parts by their greatest common divisor.
| Term | Meaning |
|---|---|
| Numerator | The top number: how many parts. |
| Denominator | The bottom number: how many parts make a whole. |
| Simplify | Divide both parts by their greatest common divisor. |
| Mixed number | A whole number alongside a proper fraction, such as 1 ½. |
The inputs explained
| Field | What to enter |
|---|---|
| Numerator 1 | Numerator of the first fraction. |
| Denominator 1 | Denominator of the first fraction. |
| Operation | Which operation to perform. |
| Numerator 2 | Numerator of the second fraction. |
| Denominator 2 | Denominator of the second fraction. |
When to use it
Scaling a recipe
Two-thirds of three-quarters of a cup is the kind of arithmetic that recipes demand constantly. Multiply the two fractions and read the simplified result.
Measuring in construction
Imperial measurement runs on halves, quarters, eighths and sixteenths. Adding and subtracting them is exactly what this handles, and the mixed-number output matches how a tape measure is read.
Checking homework
The simplified fraction, mixed number, decimal and percentage are all shown, which covers whatever form an answer is expected in.
Converting for comparison
The decimal output makes two awkward fractions instantly comparable: 7/16 against 4/9 is far easier to judge as 0.4375 against 0.4444.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
Adding 3/4 to different fractions
Three-quarters added to a range of second fractions, with the denominator varying.
| Adding 1/d | Result | Mixed number | Decimal |
|---|---|---|---|
| 1/2 | 5/4 | 1 1/4 | 1.250 |
| 1/3 | 13/12 | 1 1/12 | 1.083 |
| 1/4 | 1 | 1 | 1.000 |
| 1/6 | 11/12 | 11/12 | 0.9167 |
| 1/8 | 7/8 | 7/8 | 0.8750 |
| 1/12 | 5/6 | 5/6 | 0.8333 |
The four operations on the same pair
Three-quarters and five-sixths, combined every way.
| Operation | Result | Mixed number | Decimal | Percentage |
|---|---|---|---|---|
| Add | 19/12 | 1 7/12 | 1.583 | 158.3% |
| Subtract | -1/12 | 1/12 | -0.0833 | -8.33% |
| Multiply | 5/8 | 5/8 | 0.6250 | 62.5% |
| Divide | 9/10 | 9/10 | 0.9000 | 90.0% |
Questions
How do I add fractions with different denominators?
Convert both to a common denominator first. Multiplying the two denominators always works, though using the lowest common multiple keeps the numbers smaller. Then add the numerators and simplify.
Why does dividing by a fraction make the number bigger?
Dividing asks how many of the second fit into the first. Since 1/2 fits into 3 six times, dividing 3 by 1/2 gives 6. Whenever you divide by something less than one, the result grows.
How do I simplify a fraction?
Divide both the numerator and denominator by their greatest common divisor. For 18/24 the GCD is 6, giving 3/4. The GCD calculator will find it for larger numbers.
What is an improper fraction?
One where the numerator is at least as large as the denominator, such as 7/4. It is perfectly valid and often easier to calculate with: the mixed number 1 ¾ is the same value written differently.
Can I enter negative fractions?
Yes. Put the minus sign on the numerator. The calculator keeps the sign on the numerator in its output, which is the standard convention.
To convert a decimal into a fraction, see the decimal to fraction calculator. For simplifying, use GCD and LCM.