Fraction Calculator

Four operations, one rule that flips

A fraction is a division written down: 3/4 is three divided by four. Arithmetic on fractions follows from that, and the four operations do not behave the same way. Adding and subtracting need a shared denominator, multiplying goes straight across, and dividing flips the second fraction before multiplying.

The formulas in the table are the general ones, and they work without finding a least common denominator first. They also produce larger numbers that then need simplifying, which is why the school method finds the smallest shared denominator instead. Both routes land on the same answer. The calculator shows the simplified result and the steps, so you can check either one.

Adding and subtracting: the least common denominator

Halves and thirds cannot be counted together until they are the same size. Rewrite both with a denominator of 6: 1/2 = 3/6 and 1/3 = 2/6, so 3/6 + 2/6 = 5/6. The answer is about 0.8333, and it is less than 1, which is a useful check because both addends are below 1 and at least one is well below it.

Adding the tops and the bottoms is the classic error: 1/2 + 1/3 is not 2/5. That answer, 0.4, is smaller than 1/2 alone, so a sum came out below one of its parts. Any time addition makes a number smaller than a part, the arithmetic is wrong.

The shared denominator does not have to be the smallest one. Twelfths work for halves, thirds, quarters and sixths; twenty-fourths cover eighths as well. Using the product of the two denominators always works and just costs an extra simplification step. The least common multiple is the smallest number both denominators divide, and the grade 5 standard on adding fractions with unlike denominators is where this method is taught.

Multiplying: straight across, then cancel

Multiplication is the easy one. Multiply the tops, multiply the bottoms, then simplify: 3/4 x 2/3 = 6/12, and 6/12 = 1/2. No common denominator is needed, because a product of fractions is a product of two divisions, and divisions combine without any special treatment.

Cancelling first keeps the numbers small. In 3/4 x 2/3, the 3 on top and the 3 on the bottom cancel, and the 2 and the 4 reduce to 1 and 2, which leaves 1/2 without any multiplication of large numbers. The result is identical; only the work changes.

Mixed numbers have to become improper fractions before multiplying. Two and a half times one and a third is 5/2 x 4/3 = 20/6, and 20/6 simplifies to 10/3, which is 3 1/3. Multiplying the whole numbers and the fractions separately, 2 x 1 and 1/2 x 1/3, gives 2 1/6, which is smaller than either factor and obviously wrong: both factors exceed one, so the product must exceed one.

That last check is worth keeping. A product of two numbers above 1 is above both of them, and a product involving a fraction below 1 comes out smaller than the other factor. Multiplication that makes a number bigger when the other factor is 1/2 has gone wrong.

Dividing: invert the second fraction

Division asks how many of one thing fit inside another. Three quarters divided by two fifths is the question of how many 2/5 portions fit in 3/4, and the answer is 15/8, or 1 7/8. The mechanical rule is to flip the divisor and multiply: 3/4 x 5/2 = 15/8.

The rule is not arbitrary. Dividing by 2/5 is the same as multiplying by 5/2, because 2/5 times 5/2 is 1, and dividing by 1 changes nothing. That identity is written a/b / (c/d) = (a x d) / (b x c), and the grade 6 standard on dividing fractions builds the rule from exactly that reasoning.

The second row is the one to keep in mind: half divided by a quarter is 2, because two quarters fit in a half. The third row scales that up, since 2 1/2 holds ten quarters. A whole number divisor behaves the same way; dividing by 3 is multiplying by 1/3, which is why 2/3 divided by 3 gives 2/9 rather than anything involving ninths of the original denominator.

Simplifying, improper fractions and mixed numbers

A fraction is in simplest form when the only number that divides both parts is 1. Divide the top and bottom by their greatest common divisor: 18/24 has a GCD of 6, so 18/24 becomes 3/4, and 3 and 4 share nothing else. Taking one factor at a time gets there too, it just takes longer.

An improper fraction has a top larger than its bottom. Seventeen fifths is 3 2/5, because 17 / 5 = 3 with 2 left over. Converting back is whole x denominator + numerator: 3 x 5 + 2 = 17, over 5. A mixed number is easier to picture and an improper fraction is easier to compute with, so the calculator shows both.

Two simplifications look alike and are not. Simplifying divides a factor out of both parts and leaves the value unchanged. Converting to a mixed number rewrites the form and also leaves the value unchanged. Multiplying the denominator without touching the numerator, which happens when someone adds the bottoms while adding fractions, does change the value, and that is the error covered above.

Comparing fractions without a calculator

Which is larger, 3/7 or 4/9? Cross multiply: 3 x 9 = 27 and 4 x 7 = 28, so 4/9 is larger, because 28/63 beats 27/63. Both denominators multiply out to 63, and the numerators decide it.

Benchmarks catch most of these faster than cross multiplying. A fraction above 1/2 has a top larger than half its bottom, so 4/7 is above a half and 3/8 is below it, and the comparison is settled without arithmetic. For fractions that cluster near a known value, compare each to that value: 5/8 is 1/8 above 1/2 and 7/12 is 1/12 above it, so 7/12 is closer to a half and is the smaller of the two.

Same numerator is the third shortcut. With the same top, the smaller bottom is the larger fraction, because the same amount is being split into fewer pieces: 3/5 beats 3/8 for the same reason a fifth of a pizza is more than an eighth.

When the exact fraction matters more than the decimal

One third has no exact decimal form. 1/3 is 0.3333 and the threes never stop, so any decimal answer is a rounded one, and rounding three fractions and then adding them can miss the exact total by a hair. Working in fractions keeps the value exact until the last step, which is why construction, machining and recipe scaling all run on fractions.

A decimal is easier for money and for anything a calculator has to compare. The rule of thumb: keep the fraction while you compute, convert at the end. A recipe doubled three times is 3 x 4 1/2 = 13 1/2 cups exactly; the same sum in rounded decimals is 13.4997 if you used 4.4999 somewhere along the way.

Exact conversions between units are a related but separate problem. The fraction to decimal calculator converts one value with a chosen number of decimal places, and the fraction to cups converter turns a fraction into measuring cups and spoons, which is a volume question with a kitchen-sized answer. This page does the arithmetic itself: two fractions in, one simplified fraction out, with the steps shown.

For measurements, the exact size of a unit is fixed by definition, not by a measuring cup. The NIST guide to SI units lists the exact values, and 1 inch is exactly 25.4 millimeters, so 7/8 inch is 22.225 mm with no rounding at all.