Fraction Calculator

Add or subtract: a/b ± c/d = (a·d ± c·b) / (b·d) Multiply: a/b × c/d = (a·c) / (b·d) Divide: a/b ÷ c/d = (a·d) / (b·c) Then divide both parts by their greatest common divisor

First fraction

A numerator and a denominator, both whole numbers. The numerator may be negative or zero; the denominator may not be zero and may not exceed a hundred thousand

Operator

One of the four operations, chosen from a list because only four exist — adding, subtracting, multiplying or dividing the first fraction by the second. Division is the one that is performed by turning the second fraction upside down, so it is also the one where a numerator of zero in the second fraction is refused

Second fraction

The other operand, with the same rules: any whole numerator, a nonzero denominator up to a hundred thousand. Its numerator becomes the denominator of the reciprocal when the operation is division

Before reducing

The fraction straight out of the rule, with no cancelling done yet: five sixths over six for a half plus a third. Its denominator is the product of the two denominators, which is why this form is sometimes called cross-multiplication — and why it can be larger than the smallest common denominator would be

Lowest terms

The same value with the greatest common divisor divided out of both parts: five sixths over six cancels to five sixths. This is the answer, and the step between the two outputs is the whole of fraction reduction

Decimal

The answer as a decimal, to six places: five sixths is 0.833333. A value that does not divide evenly is rounded rather than truncated, and the six digits are a display choice rather than a statement about the fraction's precision

Four operations, four occasions, and one habit worth having. Adding is what you do with quantities that come in different parts of a whole — a third of a cup plus a quarter of a cup, and the page tells you it is seven twelfths. Subtracting is the same with the sign reversed, and it is the operation where the order of the two fractions matters: a half minus a third is a sixth while a third minus a half is minus a sixth, so the fraction you enter first is the one being subtracted from. Multiplying is the simplest of the four and the one most likely to be overthought — multiply across, top times top and bottom times bottom, and there is no common denominator to find; a half of two thirds is a third. Dividing is multiplying by the reciprocal, and it is the one people reach for a calculator to do: three quarters divided by a half is three quarters times two, which is six quarters, which is one and a half. Beneath all four sits a habit: check the answer against a rough decimal. Five sixths over six is a bit more than eight tenths, and so is five sixths, so the reduction did not change the value — which is the only thing the reduction is allowed to leave alone. If the two decimals disagree, the error is in the arithmetic rather than in the cancelling, and no amount of re-reading the lowest-terms output will find it.

1. A half plus a third

   1. Common denominator by cross-multiplication: 2 × 3 = 6
   2. First numerator: 1 × 3 = 3
   3. Second numerator: 1 × 2 = 2
   4. Add the numerators over the common denominator: (3 + 2) / 6 = 5/6
   5. Nothing cancels — 5 and 6 share no factor — so the unreduced fraction and the answer are the same, and the decimal is 0.833333

   The default case, and one where the two fraction outputs coincide. That happens whenever the cross-multiplied form is already in lowest terms, which for a sum of two unit fractions means the denominators were coprime. It is worth seeing once before the cases where they differ, so that a page showing one fraction twice does not look like a bug.

2. A quarter plus a sixth, in both forms

   1. Common denominator by cross-multiplication: 4 × 6 = 24
   2. First numerator: 1 × 6 = 6; second numerator: 1 × 4 = 4
   3. Add them over the common denominator: (6 + 4) / 24 = 10/24
   4. Find the greatest common divisor of 10 and 24: 2
   5. Divide both parts by 2: the answer is 5/12, and as a decimal 0.416667

   The example that explains the two outputs. Twenty-four is not the smallest number four and six both divide — twelve is — so the unreduced fraction is larger than it needs to be, and the cancelling step does real work. Going through the product rather than the smallest common multiple is not laziness: it is a rule that works for every pair of denominators without any thought about their factors, and the reduction afterwards brings it back to the tidy form. Ten twenty-fourths and five twelfths are the same number, which the matching decimals at the end of the table confirm.

3. Two thirds times three quarters

   1. Multiply across: 2 × 3 = 6 over 3 × 4 = 12
   2. No common denominator is needed — that step belongs to adding and subtracting only
   3. Cancel the six: 6 ÷ 6 = 1 and 12 ÷ 6 = 2
   4. The answer is 1/2, which is 0.5

   Multiplication, where the denominators are simply multiplied and nothing has to be matched first. The cancelling here is worth noticing: two thirds times three quarters is a half, smaller than either operand because both are less than one — a result that surprises people who expect multiplying to make things bigger, and which a check against 0.5 confirms in one step.

4. A half divided by three quarters

   1. Turn the second fraction upside down: 3/4 becomes 4/3
   2. Multiply instead: (1 × 4) / (2 × 3) = 4/6
   3. Cancel the two: 4 ÷ 2 = 2 and 6 ÷ 2 = 3
   4. The answer is 2/3, which is 0.666667

   Division, which is multiplication by the reciprocal — the rule that makes the second fraction's numerator land in the answer's denominator. Note the direction: a half divided by three quarters is two thirds, larger than the half you started with, because the divisor is less than one. Dividing by a number below one always grows the result, and that is the quickest sanity check available here.

5. A third minus a third, and nine quarters divided by nine quarters

   1. Common denominator: 3 × 3 = 9
   2. First numerator 1 × 3 = 3, second numerator 1 × 3 = 3
   3. Subtract: (3 − 3) / 9 = 0/9
   4. Reduce: zero over anything is zero, and the answer prints as 0 rather than as 0/9

   Zero, which is a legitimate answer and not an empty result — the two fractions were equal, and equal numbers subtract to nothing. The unreduced output keeps the shape of the work that produced it, nine in the denominator, while the answer collapses to the single digit. The same collapse happens at the other end: dividing nine quarters by nine quarters gives the whole number 1, with 36 over 36 as the unreduced form, so a fraction whose numerator and denominator are equal is printed as an integer rather than as a fraction over itself.

The unreduced output is cross-multiplication, not the smallest common denominator. Adding a quarter and a sixth goes through twenty-four rather than twelve, so the intermediate fraction is larger than the tidy one and looks wrong to anyone expecting the least common multiple. Both fractions are the same number and the answer is correct — the two outputs differ only in how much cancelling has been done — but if what you wanted was the smallest shared denominator, the reduced output is the one to read. Results are improper fractions, never mixed numbers: nine quarters stays nine quarters rather than becoming two and a quarter, and a whole result is printed as an integer, so a half divided by a half is 1 and not 1/1. The decimal output is rounded to six places, which makes it a check on the answer rather than a way of stating it: a third is 0.333333 and the addition of a third and two thirds shows as 0.999999 rather than as 1, while the fraction outputs through the same calculation give exactly 1. Denominators are capped at a hundred thousand, and that is a deliberate ceiling on how much cancelling the page will do at once. Three inputs are refused rather than answered: a denominator of zero, because the fraction has no value; a division whose second numerator is zero, because the reciprocal of zero is not a number; and a numerator large enough that the cross-multiplication leaves the range where whole-number arithmetic is exact, which the page reports as an error instead of printing an approximate fraction. Non-whole numerators are refused too, so a value like one and a half in a numerator field has to be converted to an improper fraction by hand before it will be accepted.

Why do the two fraction outputs differ?

Because one is the answer and the other is the work. The unreduced fraction comes straight out of the rule — the denominators multiplied together, the numerators crossed over the other side's denominator — and the reduced fraction is that same value with the greatest common divisor cancelled out of both parts. A quarter plus a sixth is 10/24 before cancelling and 5/12 after; the decimals match, which is the proof that only the writing changed.

Why is the unreduced denominator 24 when 12 would do?

Because the page finds a common denominator by multiplying the two denominators, not by searching for the smallest one both divide. Twenty-four is four times six, and it is a valid common denominator even though twelve is smaller. The trade is deliberate: multiplying always works and needs no analysis of the denominators, and the cancelling step afterwards brings the answer back to its lowest terms anyway. If you want the smallest shared denominator, read the reduced output.

How do I divide fractions?

Multiply by the reciprocal: turn the second fraction upside down and multiply across. A half divided by three quarters is a half times four thirds, which is four sixths, which reduces to two thirds. The rule exists because dividing by a number is multiplying by its inverse, and it is the reason the second fraction's numerator ends up in the answer's denominator — 0.666667 against 0.5 confirms the answer grew, as dividing by something less than one must.

Can the answer be a mixed number?

No — results are improper fractions, so nine quarters comes back as 9/4 rather than as 2 1/4. An improper fraction is the form that keeps working: adding or multiplying a mixed number means converting it back to an improper fraction first, so printing one would only mean converting it again at the next step. Turning 9/4 into two and a quarter is a division, and reading the decimal output does the same job for a quick check.

What happens if a denominator is zero?

The page reports an error, and there is no answer to give: a fraction with a zero denominator has no value, because the denominator is the number of parts the whole is divided into and dividing by zero is undefined. Dividing by a fraction whose numerator is zero is refused for the same reason — the reciprocal of zero is not a number — while a numerator of zero in the first fraction is fine and gives zero for any multiplication or division.

How do I check the answer is right?

Convert the two operands to decimals in your head and do the operation on those rough numbers. A half plus a third should land between 0.8 and 0.9, and five sixths is 0.833333. Multiplying two fractions both below one should give something smaller than either, and dividing by something below one should give something bigger. When the rough answer and the fraction disagree, the problem is in the arithmetic rather than in the cancelling, since cancelling can only change how a number is written.

• Fraction — numerator, denominator, the solidus and the rules for adding, multiplying and reducing fractions, including the common denominator that addition and subtraction require — Wolfram MathWorld (United States)
• Reduced Fraction — a fraction in lowest terms, obtained by dividing the numerator and denominator by their greatest common divisor, which is the step between the two fraction outputs on this page — Wolfram MathWorld (United States)
• Decimal fraction — an arithmetical fraction with an integral power of ten as its denominator, which is why a fraction like five twelfths has no exact decimal form and the decimal output has to be rounded — Encyclopedia of Mathematics, European Mathematical Society (international)