Fraction Calculator

Fraction Calculator

Fractions done exactly, the way they are taught and the way they should be: add, subtract, multiply and divide fractions and mixed numbers with the working shown, simplify with the common factor named, convert between fractions, decimals and percentages (including repeating decimals shown honestly, not chopped), compare two fractions, and work out a fraction of a number. Every answer is a whole-number fraction reduced by its greatest common divisor, so 23/6 means exactly 23/6, never a rounded-off imitation.

First fraction
Second fraction

Disclaimer: Fractions are calculated exactly with whole-number arithmetic, in your browser, for general information only. A decimal typed into the converter may have no exact fraction; when that happens the closest fraction within the denominator you allow is shown, with its error stated.

What this calculator does

Fraction arithmetic is exact or it is nothing. A decimal answer like 0.3333333 is a fraction that has been rounded and quietly changed; 1/3 is the thing itself. This calculator works the way fractions are meant to be worked: everything in whole numbers, answers reduced by the greatest common divisor, mixed numbers and improper fractions shown side by side, and the working displayed so a student can copy the method and a cook can trust the scaling. It adds, subtracts, multiplies and divides; simplifies with the common factor named; converts fractions to decimals and percentages, flagging the ones that repeat forever instead of pretending they end; converts decimals back to the closest honest fraction with the approximation error stated; compares two fractions by cross-multiplication; and answers the two everyday questions, what is a fraction of this number, and what fraction is this of that.

How to use it

  1. Entering fractions: each fraction has three boxes. Whole part is for mixed numbers like 2 1/2; leave it empty for a plain fraction. A whole number on its own goes in the whole box with the other two empty. For a negative mixed number, put the minus on the whole part and keep the numerator positive, which is the standard way of writing it; the calculator says so if you do it the other way round.
  2. Arithmetic shows its working in the standard form: for addition and subtraction, the cross-multiplied numerator over the common denominator, then the simplification. For division, the working shows the flip: dividing by a fraction is multiplying by its reciprocal, which is the step most often recited and least often understood, so it is written out in full.
  3. Converting a decimal to a fraction asks how large a denominator you will accept, because that choice is the whole trade-off. 0.333 has no exact fraction; with denominators up to 100 the closest is 1/3, out by 0.000333, and the page shows that error rather than hiding it. With 10,000 allowed, 3333/10000 is closer still but useless in a kitchen. Pick the size of answer you can actually use.
  4. Repeating decimals are shown with the repeating part in brackets: 1/6 is 0.1(6), meaning the 6 goes on forever. That notation is the honest one; a chopped decimal like 0.1666666667 is a rounding wearing a costume.
  5. Comparing fractions uses cross-multiplication, the method that needs no common denominator and no decimals: multiply each numerator by the other denominator, and the bigger product belongs to the bigger fraction. The gap between the two is given as an exact fraction too.

How it is calculated

All arithmetic is integer arithmetic on numerators and denominators: a/b + c/d = (ad + cb)/bd, a/b – c/d = (ad – cb)/bd, a/b x c/d = ac/bd, a/b / c/d = ad/bc. Every result is reduced by the greatest common divisor of its numerator and denominator, computed by the Euclidean algorithm, and signs are normalized so the denominator is always positive. Decimals are produced by long division, which is also how repeating cycles are detected: a remainder that appears twice means the digits between its appearances repeat forever. Nothing is ever rounded inside the fraction work; only decimal displays are rounded, and they are labelled as displays.

Worked examples

Adding mixed numbers: 1 1/2 + 2 1/3. As improper fractions that is 3/2 + 7/3, the working gives (9 + 14) / 6 = 23/6, which is 3 5/6, the decimal 3.8333333333, or 383.33%. Notice the answer is bigger than both inputs by the right amount and the mixed form is the one you would say out loud.

The four operations on plain fractions: 3/4 – 1/2 = 1/4. 2/3 x 3/5 = 6/15 = 2/5. 1/2 / 1/4 = 1/2 x 4/1 = 2, which is the answer that surprises people until it is phrased as a question: how many quarters are there in a half.

Simplifying: 24/36 has a greatest common divisor of 12; dividing top and bottom by 12 gives 2/3. Simplifying is not cosmetic: 2/3 can be compared, added and pictured, and 24/36 mostly cannot.

Conversions: 3/8 is 0.375 exactly and 37.50%, a terminating decimal because 8 is made only of 2s and 5s. 1/6 is 0.1(6), repeating, because 6 contains a factor of 3. 0.625 backwards is exactly 5/8. 12.5% is 1/8, which is why 12.5% of anything is found by dividing by 8.

Comparing: 3/4 against 5/7. Cross-multiply: 3 x 7 = 21 against 5 x 4 = 20, so 3/4 is larger, by exactly 1/28, about 0.0357. Close fractions are where intuition fails and this method earns its keep.

Reading your result

The simplified fraction is the answer; everything else is a translation. The mixed number is for measuring and speaking, the decimal is for comparing sizes at a glance and for feeding into other calculations, and the percentage is for comparing against wholes. If the decimal line ends in a bracketed repeat, treat any decimal version of that fraction as an approximation wherever it is used next: three lots of 0.333 is 0.999, but three lots of 1/3 is exactly 1, and the gap matters in everything from medication to money-splitting.

Limitations

  • Numbers, not algebra. This page works with numeric fractions. Fractions containing letters and expressions belong to the Rational Expression Calculator; mixing the two is how students end up simplifying things that cannot be simplified.
  • Decimal to fraction is sometimes an approximation, and says so. A decimal with no exact fraction can only be matched closely. The error line is part of the answer, not fine print: if the error is too big for your purpose, allow a bigger denominator or use the decimal as it stands.
  • Very long repeating cycles are truncated in the display. Some fractions repeat after dozens of digits (1/97 repeats after 96). The cycle detection runs far enough for any fraction a person would type, and beyond that the page marks the decimal as continuing rather than inventing an ending.
  • Whole-number inputs. Numerators, denominators and whole parts are whole numbers, as fractions require. Decimal amounts inside a fraction question, like 1.5/2, should be scaled first (that one is 3/4), or entered through the decimal converter.

Frequently asked questions

How do I add fractions with different denominators?

Multiply each numerator by the other fraction’s denominator, add those products, and put the result over the product of the two denominators, then simplify. That is exactly the working this page shows. The schoolbook version finds the least common denominator first, which gives smaller numbers but the same answer after simplifying; for two fractions the cross method is shorter and never wrong.

Why does dividing by a fraction flip it?

Because division asks how many of the divisor fit into the number, and flipping converts that into a multiplication that can be done mechanically. 1/2 divided by 1/4 asks how many quarters fit in a half, which is 2, and 1/2 x 4/1 computes the same thing. The working line shows both forms so the step is visible rather than a magic rule about turning and multiplying.

What makes a decimal terminate or repeat?

The denominator, once the fraction is simplified. If its only prime factors are 2 and 5, the decimal terminates: eighths, tenths, twentieths. Any other prime factor in the denominator, a 3, a 7, an 11, makes it repeat forever. That is why thirds and sevenths are the familiar repeating ones, and why the converter can promise, for any fraction you enter, which kind it will be.

How do I simplify a fraction by hand?

Find the largest number that divides both the numerator and the denominator, the greatest common divisor, and divide both by it. If you cannot spot it, divide by any common factor you can see and repeat: 24/36 by 4 is 6/9, and 6/9 by 3 is 2/3. The simplify mode names the full GCD in one step, which is the check for your staged version.

Related calculators

Method: integer numerator and denominator arithmetic for all operations, reduction by greatest common divisor (Euclidean algorithm), long division with remainder tracking for decimal expansions and repeating cycles, and closest-fraction search over the allowed denominator range for decimal inputs, with approximation error always displayed. Last reviewed: 13 October 2026.