Add, subtract, multiply or divide two fractions and see the step that gets marked \u2014 not just the answer.
Two fractions, one operation
Exact throughout, with the true decimal.
Takes 7, 3/4, 1 1/2, 0.75, or 0.1(6) for a repeating decimal.
1/4 + 1/6
5/12
0.41(6) as a decimal.
Improper
5/12
numerator over denominator, lowest terms
Mixed number
5/12
already proper — no whole part
Decimal
0.41(6)
repeats, block of 1
Reduced by
nothing
already in lowest terms
Working, one line at a time
1Find the common denominatorLCM(4, 6) = 12The product 4 × 6 = 24 would also work, but it is 2× larger and the answer would need reducing afterwards.
2Rewrite both over it1/4 = 3/12
1/6 = 2/12Multiply top and bottom of the first by 3, of the second by 2.
3Add the numerators3 + 2 = 5, over 12The denominator does not change — it is the size of the pieces, and adding pieces does not resize them.
The denominator names the size of the piece. Adding pieces does not change their size, which is why it does not change.
Why that denominator
The least common multiple against the product of the denominators
Least common multiple
12
Multiplying them instead
larger, and needs reducing afterwards
5 and 12 share no factor above 1, so the answer is already in lowest terms.
The decimal repeats because the reduced denominator 12 has a prime factor other than 2 and 5 (3 after the twos and fives come out). The block 6 is 1 digit long, and that length is fixed by the denominator, not by how far you divide.
Every step is exact: the numbers here are whole-number fractions throughout, never floating point.
What this tool shows
Fractions are held as whole-number pairs the whole way through, so nothing is ever rounded and1/3 stays a third instead of becoming 0.3333333333333333. The decimal shown beside each answer is the exact one, with its repeating block in brackets.
Adding and subtracting with the least common denominator
Multiplying with cross-cancellation done first
Dividing by turning the divisor over
Mixed numbers and decimals as input
The exact decimal, with its repeating block
Whether the answer was already in lowest terms
Four operations, one exact engine Least common denominator, not the product Cross-cancelling shown before multiplying Exact decimals, repeat marked
Exact rational arithmetic, verified on every change.
Updated 7 September 2026 · Works in any browser, no installation
To add or subtract fractions, rewrite both over the least common multiple of their denominators, then add or subtract only the numerators. To multiply, cancel any factor shared between either numerator and either denominator, then multiply straight across. To divide, turn the second fraction over and multiply. 1/4 + 1/6 is 5/12, over a denominator of 12 rather than 24 — and the answer needs no reducing afterwards because 12 was the least one.
At a glance
Formula shown
a/b + c/d = (a\u00b7(L/b) + c\u00b7(L/d)) / L, where L = lcm(b, d). For multiplication a/b \u00d7 c/d = ac/bd, cancelled first. For division a/b \u00f7 c/d = a/b \u00d7 d/c.
Scenario support
Homework that is marked on the working; halving or trebling a recipe; adding measurements written in eighths and sixteenths; turning a fraction into a decimal a spreadsheet will take.
Educational estimate
Planning support from the values you enter — not professional advice.
Which denominator, and why
Adding fractions needs the pieces to be the same size, and the denominator is what names that size. Any common multiple of the two denominators works — but the LEAST one is the answer a marker wants, and it is the one this page uses.
For 1/4 + 1/6 the product is 24 and the least common multiple is 12. Both give the right value. Over 24 the answer arrives as 10/24, which then has to be reduced to 5/12 — and a reduction step that was never necessary looks, on a marked page, exactly like a mistake that was caught. Finding the LCM first avoids it.
The tool shows the multiple it chose and, when the product would have been larger, how much larger. When the two denominators share no factor the two are the same number and it says so rather than pretending it did something clever.
Cancel before you multiply
Multiplication is the operation with no common-denominator step, which makes it the easy one — and the one where people still build unnecessarily large numbers. 4/9 × 3/8 multiplied straight across is 12/72, which then reduces to 1/6.
Cancelling first avoids the large pair entirely. The 4 and the 8 share a factor of 4; the 3 and the 9 share a factor of 3. Take both out and the product is 1/3 × 1/2, which is 1/6 with nothing left to reduce. It works because the whole thing is a single product: a factor on either numerator cancels against either denominator.
This page always cancels first and shows what it cancelled, because that is the technique — the answer would be the same either way, and the point is not the answer.
Why dividing flips the second one
“Keep, change, flip” is a reliable mnemonic and a poor explanation. Here is the explanation. Dividing by 2/3 asks how many two-thirds fit into the thing. Two-thirds fit one-and-a-half times into a whole, so dividing by 2/3 makes anything one-and-a-half times bigger — and one-and-a-half is 3/2, the flipped fraction.
Put formally: dividing by a number is multiplying by whatever undoes it, and what undoes 2/3 is 3/2, because their product is 1. The flip is not a trick with the notation, it is the reciprocal doing its job.
Dividing by zero is the one case with no answer. Not a very large answer, not infinity: there is no number that multiplied by zero gives anything but zero, so the question has no solution. This tool says that rather than returning an error code.
The decimal beside the answer
Most calculators print a decimal that has been rounded and do not say so. This one prints the exact decimal, with any repeating part in brackets: 11/12 is 0.91(6), meaning the 6 goes on forever.
Whether it repeats at all is decided entirely by the denominator once the fraction is in lowest terms. If the only prime factors are 2s and 5s the decimal stops; anything else and it repeats forever. That is why eighths and twentieths terminate, and why thirds, sixths, sevenths and twelfths do not.
The length of the repeat is fixed too, and the Fraction to Decimal Calculator works it out from the denominator before doing any dividing at all.
Mixed numbers and decimals as input
All four of these mean a number and all four are accepted: 7,3/4, 1 1/2, 0.75. So is0.1(6), where the brackets mark the repeating block — that one is read as exactly 1/6, not as 0.16666 rounded.
A mixed number must have a proper fraction after the whole: 1 1/2 is fine and 1 5/4 is refused, because it is really 9/4 and writing it that way hides a whole. The tool says so rather than silently guessing.
Whatever goes in, the answer comes out in all three forms at once, so nothing has to be converted afterwards.
Where fractions go wrong
Adding the denominators. 1/2 + 1/3 is not 2/5. The denominator is the size of the piece, and adding two halves-and-thirds does not produce fifths of anything. The common-denominator step exists precisely to stop this.
Finding a common denominator for a multiplication. Harmless but wasted: multiplication does not need one, and the extra step introduces bigger numbers for nothing.
Flipping the wrong one. In a ÷, it is the SECOND fraction that turns over. Flipping the first gives the reciprocal of the answer, which is often close enough to look plausible.
Stopping before lowest terms. 10/24 is the right value and usually the wrong answer. This page tells you what it divided by at the end, or says the fraction was already reduced — which is itself worth knowing.
Sources and methodology
Fraction arithmetic is settled mathematics; what follows is where the conventions this page follows are written down.
Method. Every value is parsed into an exact fraction of BigInts and stays one; no step passes through a floating-point number, which is why 1/3 + 1/3 is exactly 2/3 here and not 0.6666666666666666. The least common multiple is computed by Euclid rather than by multiplying the denominators, cross-cancellation is applied before any multiplication, and the decimal shown is the exact expansion with its repeating block derived from the denominator. That engine is verified on every change against 58 hand-written assertions, including that all four operations agree with independently cross-multiplied BigInt arithmetic across twenty-five thousand generated pairs, and that every decimal expansion reads back as the identical fraction. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Simplify FractionsReduce a fraction to lowest terms with both routes to the divisor shown side by side — Euclid line by line and the shared primes — and a stated proof when nothing can be cancelled.
Mixed NumberAdd, subtract, multiply and divide mixed numbers with the borrow shown where it happens, plus a second independent check worked through improper fractions.
Fraction to DecimalConvert a fraction to its exact decimal, with the length of the repeat worked out from the denominator before dividing and the long division shown remainder by remainder.
Decimal to FractionTurn any decimal into its exact fraction, repeating ones included, with the four algebra lines that cancel the infinite tail instead of rounding it away.
Comparing FractionsOrder up to eight fractions with all three methods shown on your own numbers, and a figure for how many decimal places the closest pair need before they stop looking equal.
PercentageSolve X% of Y, what percent X is of Y, reverse percentage, increase/decrease, discounts, and tax, tip, or commission.
The working matters more than the answer on this page, so each section below takes one step and says what it is for. If you only need the reduction, the Simplify Fractions Calculator does that step alone; if you need to compare rather than combine, the Comparing Fractions Calculator puts several in order at once.
Educational use disclaimer
This is an educational tool. The arithmetic is exact and independently verified, but the working shown is one standard route among several — your course may present a different order of steps for the same answer.
Published the fraction calculator: all four operations on exact BigInt fractions, with the working shown as the product rather than the answer.
Addition uses the least common multiple rather than the product of the denominators, and says how much larger the product would have been — an unnecessary reduction afterwards reads like a caught mistake on a marked page.
Multiplication cross-cancels before multiplying, so 4/9 × 3/8 never builds 12/72; the decimal beside every answer is the exact expansion with its repeating block in brackets.
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