What the six tabs do
Fraction problems come in a few distinct shapes, and cramming them into one form makes every one of them worse. The tabs across the top split them:
- Fractions — the ordinary case, two fractions and one operation, entered as numerator over denominator.
- Mixed Numbers — for anything written as a whole number beside a fraction, like
2 3/4. - Simplify — reduce a single fraction to its lowest terms and see the factor that was cancelled.
- Decimal to Fraction — including repeating decimals, which most calculators cannot do exactly.
- Fraction to Decimal — with the repeating block marked rather than cut off.
- Big Numbers — the same arithmetic when the numbers are far too large for ordinary decimal arithmetic to hold.
Every tab shows the working, not just the answer, because in almost every case the working is the thing being marked.
Adding and subtracting: the common denominator
You cannot add sevenths to eighths any more than you can add metres to feet. The pieces have to be the same size first, and that is all a common denominator is — a re-cutting of both fractions into pieces of a matching size.
The reliable way is to multiply the two denominators together. For 2/7 + 3/8 that gives 56, so each fraction is rewritten over 56:
2/7becomes16/56, multiplying top and bottom by 8.3/8becomes21/56, multiplying top and bottom by 7.- Now the numerators add:
16 + 21 = 37, giving37/56.
Multiplying top and bottom by the same number does not change a fraction's value — it is multiplication by one, wearing a disguise. That is the whole justification for the step.
A shortcut worth knowing. The product of the denominators always works but is often bigger than it needs to be. The lowest common multiple usually leaves less to simplify at the end: for 1/4 + 1/6, the product is 24 but the LCM is 12, and starting from 12 gets you to 5/12 with no cancelling afterwards. Our LCM calculator finds it for any set of numbers.
Subtraction is identical — same common denominator, then subtract the numerators instead of adding them.
Multiplying and dividing
Multiplication is the easy one, and it surprises people who have just fought through addition. There is no common denominator to find: multiply the numerators, multiply the denominators, simplify. 3/4 × 1/6 = 3/24 = 1/8.
Division looks stranger. You turn the second fraction upside down and multiply: 3/4 ÷ 1/6 becomes 3/4 × 6/1 = 18/4 = 9/2. That is not a trick to memorise; it follows from what division asks. How many sixths fit inside three quarters? Each whole contains six sixths, so the question naturally multiplies by six.
One consequence catches people out: dividing by a fraction smaller than one makes the answer bigger. Nine halves is more than three quarters, and that is correct, not an error.
Mixed numbers and improper fractions
A mixed number like 2 3/4 is two wholes plus three quarters. It reads well but calculates badly, so the first move is always to convert it into a single improper fraction:
2 3/4 → (2 × 4 + 3) / 4 → 11/4
Multiply the whole part by the denominator, add the numerator, keep the denominator. Then do the arithmetic normally and convert back at the end if the answer is wanted in mixed form.
Negatives are where this goes wrong. In -2 3/4, the minus sign applies to the whole quantity, not just the 2. It is negative two and negative three quarters, so the improper form is -11/4, not -8/4 + 3/4. Getting this backwards flips the sign of the fractional part and produces an answer that is wrong by one and a half. The Mixed Numbers tab handles the sign for you and shows the converted form so you can check it.
Simplifying, and why it matters
A fraction is in lowest terms when the numerator and denominator share no factor except one. To get there, divide both by their greatest common factor. For 36/120 that factor is 12, giving 3/10.
You can chip away in stages — halve it, halve it again — and arrive at the same place. Finding the greatest common factor in one go just saves rounds. Our GCF calculator does that for any pair.
It matters for three reasons. Marks are usually only awarded for the simplified form. Simplified fractions are far easier to compare against each other. And in longer calculations, simplifying at each step keeps the numbers small enough to work with by hand.
Fractions and decimals in both directions
Decimal to fraction
Count the digits after the point and put the whole thing over that power of ten, then simplify. 1.375 has three digits, so it starts as 1375/1000; the greatest common factor is 125, leaving 11/8.
Repeating decimals need different handling and most tools quietly refuse. 0.333… is not 333/1000; it is exactly one third. Type the repeating digits in brackets — 0.(3) — and the calculator returns 1/3. It uses the standard algebraic identity: for a block of n repeating digits, the denominator is n nines.
Fraction to decimal
Divide the top by the bottom. What comes out depends entirely on the denominator once the fraction is simplified. If it is built only from twos and fives — the factors of ten — the decimal ends. Otherwise it repeats forever.
1/8has denominator 2×2×2, so it terminates:0.125.1/20is 2×2×5, so it terminates:0.05.1/7has a 7 in it, so it cannot:0.(142857), repeating in a six-digit cycle forever.
This calculator shows the repeating block in brackets rather than rounding it off, so 2/7 comes back as 0.(285714) — the exact value, not an approximation of it.
Why this calculator does not round
Most calculators convert fractions to decimals internally, do the arithmetic, and convert back. That is fast and it is lossy: one third becomes 0.3333333333333333, and the error, though tiny, compounds through every subsequent step.
Everything here runs on exact whole-number arithmetic with no size limit. 1/3 stays one over three from beginning to end, and 1/3 + 1/6 returns exactly 1/2 rather than something that rounds to it. It is also why the Big Numbers tab is not a separate feature but the same code: a twenty-digit numerator is handled the same way as a single digit.
You can see the difference. In ordinary decimal arithmetic, 0.1 + 0.2 famously does not equal 0.3. As fractions, 1/10 + 2/10 is exactly 3/10, every time.
The mistakes that cost marks
- Adding the denominators.
1/2 + 1/3is not2/5. It is5/6. Only the numerators are added, and only after the denominators match. - Looking for a common denominator when multiplying. It is not needed, and it makes the arithmetic worse.
- Flipping the wrong fraction when dividing. The second one is inverted, never the first.
- Mishandling a negative mixed number. The sign covers the whole quantity, as above.
- Cancelling across an addition. You may cancel a common factor between a numerator and a denominator, but never a term that is being added —
(2+3)/2is not3. - Stopping before lowest terms.
6/8and3/4are the same number, but only one usually earns the mark.
Frequently asked questions
How do you add fractions with different denominators?
Give both fractions the same denominator, then add only the numerators. The quickest common denominator is the product of the two you have: for 2/7 + 3/8, multiply each fraction top and bottom so both sit over 56, giving 16/56 + 21/56 = 37/56. Using the lowest common multiple instead of the product usually leaves you with less simplifying to do at the end.
Why do you flip the second fraction when dividing?
Dividing by a number is the same as multiplying by its reciprocal, and the reciprocal of a fraction is that fraction turned upside down. So 3/4 divided by 1/6 becomes 3/4 multiplied by 6/1, which is 18/4, or 9/2. The rule is not arbitrary: dividing by one sixth asks how many sixths fit into three quarters, and that is naturally a multiplication by six.
What is a mixed number and how do I calculate with one?
A mixed number is a whole number written next to a fraction, such as 2 3/4. To calculate with it, convert it to an improper fraction first: multiply the whole part by the denominator and add the numerator, so 2 3/4 becomes 11/4. The Mixed Numbers tab does that conversion for you and gives the answer back in both forms.
How do I turn a decimal into a fraction?
Count the digits after the point and put the number over that power of ten, then simplify. For 1.375 there are three digits, so it starts as 1375/1000, and dividing both by 125 gives 11/8. Repeating decimals work differently and this calculator handles them too: type the repeating part in brackets, as 0.(3), and it returns 1/3 exactly rather than an approximation.
Does every fraction have an exact decimal?
No. A fraction terminates only when its simplified denominator is built from twos and fives, because those are the factors of ten. One eighth is 0.125 exactly, but one seventh never ends. This calculator marks the repeating block in brackets, showing 2/7 as 0.(285714), so you can see the exact pattern instead of a rounded value.
Can it handle very large numerators and denominators?
Yes. Every tab runs on exact whole-number arithmetic with no size limit, so a numerator with twenty digits is handled the same way as a single digit and nothing is rounded on the way through. That is why the answers here match a computer algebra system rather than drifting the way floating-point arithmetic does.
This calculator is provided for education and general reference. Results are computed in exact integer arithmetic and are not rounded, but you should still check that the values you entered are the ones you meant. Learn more about how we build and check these tools, or read our privacy policy.