Dividing Fractions Calculator

What this calculator does

So, you need to divide one fraction by another. How many quarter-cups fit in three cups. Splitting 3/4 of a tank between two trips. This tool divides the first fraction by the second and gives the answer as a fraction, with any whole number brought to the front, and as a decimal.

Each fraction has three boxes: a whole number, a numerator, and a denominator, so mixed numbers like 2 1/2 go straight in. Dividing fractions has a reputation for being the confusing one, but it rests on a single trick, and once you see it, it is no harder than multiplying.

How to use it

1. Enter the fraction being divided in the first row: whole number if any, then numerator over denominator.
2. Enter the fraction you are dividing by in the second row.
3. Press Calculate.

A zero denominator is refused, and so, in effect, is dividing by zero, since a second fraction equal to zero has no meaning to divide by.

Flip the second one and multiply

Here is the whole trick. To divide by a fraction, turn it upside down and multiply instead. Keep the first fraction as it is, flip the second so its top and bottom swap, and then multiply straight across the way you would any two fractions. Dividing by 2/3 is the same as multiplying by 3/2.

That is genuinely all of it. The hard-sounding part of dividing fractions is really just a multiplication with one fraction flipped first, and the tool does exactly that.

The method it uses

For a/b divided by c/d, the tool computes

(a × d) / (b × c)

which is a/b times the flipped d/c, then reduces. A mixed number is folded into a single top-heavy fraction first, as whole × denominator + numerator. The arithmetic runs on math.js using exact whole-number ratios, so the answer stays exact, and it reads the sign so a negative result keeps its minus.

A worked example, step by step

Take 3/4 divided by 2/3.

• Flip the second fraction: 2/3 becomes 3/2.
• Now multiply: 3/4 times 3/2 is (3 × 3) over (4 × 2), which is 9/8.
• 9/8 is more than a whole, so pull it out: 9/8 is 1 1/8.

As a decimal that is 1.125. The tool shows 1 1/8 and the decimal together.

Why dividing can make the answer bigger

With whole numbers, dividing makes things smaller, so it surprises people that 6 divided by 1/2 is 12. But read it as a question: how many halves fit into 6? Each whole holds two halves, and there are six wholes, so twelve halves fit. Dividing by a number smaller than 1 gives an answer larger than what you started with, because you are asking how many small pieces fit inside, and small pieces fit many times over.

This is the mirror image of multiplying. Multiplying by a proper fraction shrinks a number; dividing by one grows it. The flip is what carries you between the two.

What the reciprocal really is

The flipped fraction has a name: the reciprocal. The reciprocal of 2/3 is 3/2, and the reciprocal of 5 is 1/5. Its defining feature is that a number times its reciprocal is always 1, since 2/3 times 3/2 is 6/6. Dividing by a fraction asks how many of that fraction go into your amount, and multiplying by its reciprocal is the clean way to answer that. To divide by a whole number, use the same idea in reverse: dividing by 4 is multiplying by 1/4, because 4 is 4/1 and its reciprocal is 1/4.

Questions people ask

Which fraction do I flip, the first or the second?

Only the second, the one you are dividing by. The first fraction stays exactly as it is. Flipping the wrong one gives the wrong answer, so the order matters.

Can dividing really make the answer larger?

Yes, whenever you divide by a fraction less than 1. Six divided by a half is twelve, because twelve halves fit into six. Divide by something more than 1 and the answer shrinks, as you would expect.

What is a reciprocal?

It is the fraction turned upside down, top and bottom swapped. The reciprocal of 3/4 is 4/3. A number multiplied by its reciprocal always equals 1.

How do I divide by a whole number?

Treat the whole number as itself over 1 and flip it. Dividing by 4 is the same as multiplying by 1/4, so 3/4 divided by 4 is 3/16.

Does it reduce the answer?

Yes, to lowest terms, with any whole number split off into a mixed number.

References

On the reducing step and the notation. Once the flip and multiply are done, the answer is cut to lowest terms by dividing through by the greatest common divisor, found with the algorithm Euclid recorded around 300 BC. The bar we write between numerator and denominator is credited to the Moroccan mathematician al-Hassar in the twelfth century and was carried into Europe by Fibonacci's Liber Abaci in 1202.

1. Euclid, Elements, Book VII, Propositions 1 and 2 (c. 300 BC), the greatest common divisor algorithm used to reduce the answer to lowest terms.
2. al-Hassar (12th century) and Leonardo of Pisa, Fibonacci, Liber Abaci (1202), for the horizontal fraction bar in the form used today.