Grafolyn

Fraction Calculator

Add, subtract, multiply and divide fractions and mixed numbers with the common denominator shown, put them in order, and convert between fractions, decimals and percentages — exactly, including the repeating ones.

Add, subtract, multiply or divide, with the common denominator shown

Type a fraction as 3/4, a mixed number as 2 3/4, a decimal as 0.75, or a repeating decimal as 0.1(6) — the brackets mark the part that repeats.

5/6

as a decimal
0.8(3)
as a percentage
83.(3)%

The working

  1. Find the lowest common denominatorThe denominators are 2, 3. Their lowest common multiple is 6, so every fraction is rewritten over 6.LCD = 6
  2. Rewrite each fractionMultiply the top and bottom of each fraction by whatever turns its denominator into the common one. The value is unchanged — only how it is written.1/2 = 3/6, 1/3 = 2/6
  3. Add the numeratorsWith a common denominator the denominators stay put and only the tops combine.3 + 2 = 5, so the answer is 5/6
  4. Already in lowest termsThe numerator and denominator share no factor above 1, so nothing cancels.5/6

The working is the answer

A fraction question is almost never marked on the final number. It is marked on the common denominator you found, the fractions you rewrote over it, and the cancelling you did at the end. A calculator that prints 7/12 and nothing else has answered a question nobody asked.

So every operation here shows those steps. Add 3/4 and 1/6 and you get the lowest common denominator of 12, then 9/12 and 2/12 written out, then the numerators added, then the check for whether anything cancels. That is the four lines a marker wants to see.

How each operation works

Adding and subtracting

Fractions can only be added when the pieces are the same size, which is what a common denominator means. Find the lowest common multiple of the denominators, rewrite each fraction over it by multiplying top and bottom by the same amount, then add the numerators and leave the denominator alone.

Multiplying the denominators together always gives a common denominator, and for 1/4 and 1/6 that gives 24 rather than 12. The answer comes out the same after simplifying, but the numbers along the way are twice the size.

Multiplying

Straight across: numerators multiply, denominators multiply. No common denominator is needed, which surprises people who have just spent a week on addition. 2/3 × 3/4 is 6/12, which simplifies to 1/2.

Dividing

Flip the second fraction and multiply. Dividing by 1/4 is the same as multiplying by 4, which is worth thinking about for a moment: dividing by something smaller than one makes the answer bigger. That is the step people forget, and it is the one the question is testing.

Repeating decimals, exactly

One third is 0.333… forever. Most calculators show 0.3333333333 and move on, which is close but wrong, and converting that back gives 3333333333/10000000000 rather than 1/3.

This tool writes the repeating part in brackets instead — 0.(3) for a third, 0.(142857) for a seventh, 0.1(6) for a sixth — and that form is exact. It converts straight back to the fraction it came from, because the brackets carry the information that an endless string of digits cannot.

The conversion back uses the standard trick: subtract the number from a shifted copy of itself so the infinite tails cancel, leaving an ordinary equation. For 0.(3), ten times the number minus the number is 3, so nine times it is 3 and the number is 1/3. The tool shows that working.

A decimal terminates exactly when its denominator, in lowest terms, has no prime factors other than 2 and 5 — which is why halves, quarters, fifths and eighths come out neatly and thirds, sixths and sevenths never do. That has nothing to do with the fractions and everything to do with base ten.

Four mistakes this will catch

  1. Adding the denominators. 1/2 + 1/3 is not 2/5. The denominators name the size of the pieces, and they do not combine.
  2. Forgetting to flip when dividing. Dividing by 3/4 and multiplying by 3/4 give different answers, and only one of them is right.
  3. Losing the sign of a mixed number. −2 3/4 is −11/4. Treating it as −2 + 3/4 gives −5/4, which is a different number entirely.
  4. Stopping before lowest terms. 6/12 is correct and usually not accepted. The tool says explicitly whether anything cancels, so there is no guessing.

What it does not do

Questions

How do I type a mixed number?

With a space: 2 3/4 means two and three quarters. A negative mixed number puts the sign on the whole thing, so −2 3/4 is −11/4, not −2 + 3/4. That is how it is written on paper and how it is read here.

What do the brackets in 0.1(6) mean?

They mark the digits that repeat forever. 0.1(6) is 0.16666… which is exactly 1/6. Writing it that way is the only way to type a repeating decimal accurately, since the usual bar over the digits is not something a keyboard offers. Type it that way and the conversion is exact; type 0.16666 instead and you get 16666/100000, which is a different number.

Why does 1/3 show as 0.(3) rather than 0.3333333333?

Because 0.(3) is exactly one third and 0.3333333333 is not. The repeating form carries the whole value in a finite amount of writing, which is the point of the notation. It also converts straight back, so nothing is lost on the round trip.

What is the lowest common denominator, and why does it matter?

It is the smallest number every denominator divides into — for 1/4 and 1/6 it is 12. You can add fractions over any common denominator, and multiplying the two denominators together always gives one, but it leaves you with larger numbers to simplify at the end. Using the lowest one keeps the arithmetic small, and it is what a question means when it asks you to show your working.

Why compare fractions over a common denominator instead of as decimals?

Because it is the method the question is testing, and it is exact. Once 5/8 and 7/11 are written as 55/88 and 56/88, the comparison is between 55 and 56 and needs no calculator at all. Converting to decimals works too, but it hides the reasoning and can mislead when two values differ only in the eighth decimal place.

Can it handle very large numerators?

Yes. The arithmetic runs on arbitrary-precision integers, so a fraction with a hundred-digit numerator adds and simplifies exactly. Nothing here ever converts to a floating-point number along the way, which is what makes the results exact rather than nearly right.