Fraction Calculator

How to Use the Fraction Calculator

Fraction arithmetic is one of the few places a calculator can be more accurate than a spreadsheet. Everything here is done in whole numbers from start to finish, so one third stays exactly one third rather than becoming 0.33333333333333331.

1. Enter the first fraction as a numerator and a denominator. Negative numerators are fine; a negative denominator is normalised automatically so the sign always sits on top.
2. Choose the operation with the toggle. Addition and subtraction need a common denominator; multiplication and division do not, which is why the second pair is usually easier by hand.
3. Enter the second fraction. For division, this is the one that gets flipped — dividing by a fraction is multiplying by its reciprocal.
4. Read the result in lowest terms. The calculator divides both parts by their greatest common factor, found with Euclid's algorithm rather than by trial and error.
5. Check the working table. It shows the raw result before simplification and the common factor that was removed, so you can follow the steps rather than trusting the answer.

Improper fractions are shown as mixed numbers in the tiles, so 21/10 also appears as 2 1/10. Both forms are correct; which one is preferable depends entirely on what you are doing next.

Fraction Formulas

Four operations. Two of them need a common denominator and two of them do not.

Euclid's algorithm finds the greatest common divisor by repeatedly replacing the larger number with the remainder of dividing it by the smaller one, until nothing is left over. It is over two thousand years old and still the fastest practical method, which is a reasonable claim to the most durable algorithm ever written down.

Example

7/12 + 5/18

1. Multiply the denominators for a common denominator: 12 × 18 = 216.
2. Rewrite both: 7/12 becomes 126/216, and 5/18 becomes 60/216.
3. Add the numerators: 126 + 60 = 186/216.
4. Find the greatest common divisor of 186 and 216: it is 6.
5. Divide both parts: 186 ÷ 6 = 31, and 216 ÷ 6 = 36. The answer is 31/36.
6. As a decimal that is 0.861111, and as a percentage 86.11%.

The shorter route

Using the lowest common denominator instead saves a step. The lowest common denominator of 12 and 18 is 36, so 7/12 = 21/36 and 5/18 = 10/36, giving 31/36 immediately with no simplification needed. Both routes reach the same answer; the second is neater by hand and the first is easier to program, which is why the calculator does it the first way and then reduces.

The other three operations

On the same two fractions: subtracting gives 11/36 (0.305556), multiplying gives 35/216 (0.162037), and dividing gives 21/10 — which is 2 1/10 as a mixed number, or 2.1 exactly. Notice that dividing by a fraction smaller than one produces a result larger than what you started with, which is the part that feels wrong until you remember that dividing by a half means asking how many halves fit inside.

Why Exact Arithmetic Matters

Every step of the worked example, from the inputs to lowest terms.

The decimal is identical before and after simplification, which is the point of simplifying: 186/216 and 31/36 are the same number written two ways. Lowest terms is a convention for readability rather than a mathematical requirement.

Fractions and decimals are not interchangeable in every situation. One third is exactly representable as a fraction and not as a decimal — 0.333333 is an approximation however many digits you add. In any calculation involving thirds, sevenths or ninths, staying in fractions until the final step avoids an error that compounds quietly through everything downstream.

That is also why this page does its arithmetic in integers. A spreadsheet adding 7/12 and 5/18 as decimals returns 0.8611111111111112 — correct to fifteen places and not exactly 31/36. For most purposes that is irrelevant; for anything involving repeated multiplication or an exact comparison, it is not, and the ratio calculator has the same property for the same reason.

Four Things Worth Knowing About Fractions

Four things worth knowing about working with fractions.

A negative denominator is not wrong, just untidy. The fraction 3/−4 equals −3/4, and this calculator normalises the sign onto the numerator. Doing so makes comparisons and simplification behave predictably.

Improper fractions are often more useful than mixed numbers. 21/10 is easier to multiply than 2 1/10, which is why algebra stays in improper form and recipes do not. Convert at the end, not in the middle.

Comparing fractions needs a common denominator too. Whether 7/12 or 5/8 is larger is not obvious by inspection; over a common denominator of 24 they become 14/24 and 15/24, and the answer is immediate. Cross-multiplying works just as well: 7 × 8 = 56 against 5 × 12 = 60.

Not every decimal is a fraction. Any terminating or repeating decimal can be written as a fraction, but irrational numbers such as π and the square root of two cannot be — that is precisely what irrational means. A calculator that offers you a fraction for π is offering an approximation.

There is a practical reason fractions survive in an age of decimals, and it is not nostalgia. Some quantities divide naturally into thirds, sixths or twelfths — time, angles, musical intervals, imperial measurements — and every one of those becomes a repeating decimal the moment you convert it. A third of an hour is exactly twenty minutes and approximately 0.333 hours, and only one of those two statements is usable for anything.

The same applies in trades that work in imperial units. A measurement of 3 5/8 inches is exact, reads instantly off a tape measure, and adds to another fraction without any loss. Written as 3.625 it is also exact, and written as 3.63 it is not — which is the sort of quiet rounding that turns into a visible gap across twelve joists.

For converting between fractions, decimals and percentages in the other direction, the percentage calculator handles the same arithmetic from the percentage end, and the ratio calculator does it for the two-part comparisons that look like fractions and behave slightly differently.