Adding Fractions Calculator

What this calculator does

So, you have two fractions and you need their sum. Maybe half a cup and a third of a cup. Maybe 2 3/4 hours of work and another 1 1/2 hours. This calculator adds them and hands the answer back two ways: as a fraction, with any whole number pulled out to the front, and as a decimal.

Each fraction gets three boxes: a whole number, a numerator, and a denominator. The middle pair, numerator over denominator, is the fraction itself. The whole-number box is there so you can type a mixed number like 2 3/4 straight in, without turning it into a top-heavy fraction first. Leave that box empty and you simply have a plain fraction.

How to use it

1. Fill in the first fraction: a whole number if you have one, then the numerator on top and the denominator on the bottom.
2. Do the same for the second fraction.
3. Press Calculate.

The one rule the tool holds you to is that a denominator cannot be zero, because a fraction with nothing underneath has no value to work with. Everything else it will take, negative numbers included.

Why you cannot just add across

Here is the thing that trips almost everyone at the start. You cannot add 1/2 and 1/3 by adding the tops and adding the bottoms. 1/2 + 1/3 is not 2/5. If it were, adding a third onto a half would leave you with less than the half you began with, which cannot be right.

The reason is that a half and a third are different-sized pieces, and you cannot count different-sized pieces together until you cut them to a common size. Halves and thirds both divide evenly into sixths, so rewrite both over 6: a half becomes 3/6, a third becomes 2/6. Now the pieces match, and adding is easy again, 3 sixths plus 2 sixths is 5 sixths. That shared bottom number is the common denominator, and finding one is really the whole job of adding fractions.

The method it uses

The tool does not go hunting for the smallest possible common denominator. It uses the one that always works: the two denominators multiplied together. For a/b + c/d that gives

(a × d + c × b) / (b × d)

Multiply each numerator by the other fraction's denominator, add those two results for the new top, and put it all over the product of the denominators. Then reduce. If you typed a mixed number, it is folded into a single top-heavy fraction first, as whole × denominator + numerator, so 2 3/4 becomes 11/4 before anything else happens. The arithmetic runs on math.js, a JavaScript maths library whose fraction type keeps everything as exact whole-number ratios instead of drifting into rounding, so the fraction you get back is exact.

A worked example, step by step

Take 3/4 + 2/3.

• New top, multiplying across: 3 × 3 + 2 × 4 = 9 + 8 = 17.
• New bottom, multiplying the denominators: 4 × 3 = 12.
• So the raw answer is 17/12. That is more than one whole, so pull the whole out: 17/12 is 1 with 5 left over, which is 1 5/12.

As a decimal that is about 1.417. The tool shows both forms, the mixed number 1 5/12 and the decimal rounded to three places.

Adding mixed numbers without converting first

Because every fraction here has its own whole-number box, you can add mixed numbers directly, which is where a lot of by-hand slips creep in. Say 2 1/2 + 1 3/4. Behind the scenes each is folded into a single fraction: 2 1/2 is 5/2, and 1 3/4 is 7/4. Then it is the same routine, 5/2 + 7/4 = (5 × 4 + 7 × 2) / (2 × 4) = (20 + 14) / 8 = 34/8, which reduces to 17/4, or 4 1/4. You do none of that yourself. You type 2, 1, 2 across the first row and 1, 3, 4 across the second, and read off 4 1/4.

Why the answer comes back already reduced

You may notice the result often arrives in smaller numbers than the working above would suggest. That is on purpose. 34/8 and 17/4 are the same value, but 17/4 is the tidy form, and the tool always gives you the tidy form. It gets there by finding the largest number that divides both the top and the bottom, the greatest common divisor, and dividing both by it. 34 and 8 share a factor of 2, so both are halved to 17/4. That same reducing step is why 2/4 comes back as 1/2 and 6/8 comes back as 3/4.

Questions people ask

Can I add more than two fractions at once?

Two at a time. To add three, add the first two, then type that answer back into the first row and add the third to it. The result is the same as adding all three together.

What if both fractions already share a denominator?

Then it is the easy case: keep the denominator and add the tops. The tool still runs the full method, and you land on the same answer, reduced.

Does it simplify the answer for me?

Yes, always, down to lowest terms, and it splits off any whole number so you get a clean mixed number rather than a top-heavy fraction.

Can I add a negative fraction?

Yes. Put the minus sign on the numerator or on the whole number. Adding a negative fraction is the same as subtracting a positive one, and the tool handles the sign for you.

Why does the decimal stop at three places?

Some sums, like anything ending in thirds, run on forever as a decimal. The tool rounds the decimal to three places so it reads cleanly, but the fraction it gives you alongside is exact, with nothing lost.

References

On the reducing step and the notation. Cutting a fraction to lowest terms means dividing top and bottom by their greatest common divisor, and the method for finding that divisor is one of the oldest algorithms we have, set down by Euclid around 300 BC. The stacked way we write fractions, one number over another with a bar between, is younger than the arithmetic itself: the horizontal bar is generally credited to the Moroccan mathematician al-Hassar in the twelfth century and reached Europe through Fibonacci's Liber Abaci in 1202.

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