How do you add fractions?

If the bottom numbers already match, add the tops and leave the bottom alone: 2/7 + 3/7 = 5/7. If they do not match, rewrite both fractions over a common denominator first, then add. 1/2 + 1/3 becomes 3/6 + 2/6 = 5/6.

Why do the denominators have to be the same?

Because the bottom number says how big a piece is, and pieces of different sizes cannot be counted together. Two halves and three thirds is not five of anything. Rewriting both fractions over one denominator re-cuts them into pieces of a single size, and then counting works.

Why do you not add the bottoms?

Because the bottom is not a quantity, it is a label saying what kind of piece you are counting. 1/4 + 1/4 = 2/4, which is a half — and adding the bottoms would give 2/8, a quarter, which is what you started with. The label stays; only the count goes up.

How do I find the least common denominator?

It is the least common multiple of the two bottom numbers: the smallest number both divide into. For 4 and 6 it is 12. Multiplying the two bottoms always works too (24 here), and gives the right answer with bigger numbers that reduce at the end, so it is a fair fallback if the LCM is not obvious.

Is subtracting fractions the same?

Identical, right up to the last step. Get a common denominator, then subtract the tops instead of adding them: 5/6 - 1/4 becomes 10/12 - 3/12 = 7/12. Everything that is hard about adding fractions is hard about subtracting them in exactly the same way.

Adding Fractions

Adding fractions is the one operation with an extra step in it, and the extra step is not arbitrary: pieces of different sizes cannot be counted together. Once you have seen the bars re-cut, the rule is obvious rather than remembered.

By Nethanel Bar, Co-founder & CEO

Last updated August 25, 2026

Adding fractions has a reputation, and it is deserved in one narrow sense: it is the only one of the four operations with an extra step. Multiplying and dividing go straight through. Adding stops and asks you to rewrite both fractions first.

The extra step is not a convention someone invented. It is forced, and the bars below are why.

Fraction 1 top1

Fraction 1 bottom2

Fraction 2 top1

Fraction 2 bottom3

1/2 + 1/3 = 3/6 + 2/6 = 5/6common denominator6

The top two bars are cut differently, so their pieces cannot be counted together. The third bar re-cuts both into pieces of one size - then they simply add.

Different-sized pieces cannot be counted together

The top two bars are the two fractions. At the starting values one is cut into halves and the other into thirds — different cuts, so their pieces are different sizes. There is no honest way to say how many pieces you have in total, because a half and a third are not the same thing. "One piece plus one piece is two pieces" would give 2/5, which is smaller than the half you started with.

The third bar fixes it. Both fractions are re-cut into sixths: the half becomes 3 of them, the third becomes 2 of them, and now they are pieces of one size that can simply be pushed together.

1/2 + 1/3 = 3/6 + 2/6 = 5/6

Notice what the bottom number did during the addition: nothing. It is not a quantity being added, it is a label saying what kind of piece is being counted. Three sixths plus two sixths is five sixths for the same reason three apples plus two apples is five apples.

That is also why adding the bottoms is wrong. 1/4 + 1/4 is 2/4, a half — and adding the bottoms would give 2/8, which is a quarter, the amount you had before you added anything.

When the bottoms already match, there is nothing to do

Set both denominators to the same number in the figure and the third bar is cut like the other two, because the work has already been done:

2/7 + 3/7 = 5/7

Add the tops, keep the bottom, reduce if you can. This is the whole rule, and everything else on this page exists to get you to this situation.

Finding the common denominator

You need a number that both bottoms divide into. Two ways to get one:

The least common denominator is the smallest such number — the least common multiple of the two bottoms. It keeps the arithmetic small.

Multiplying the two bottoms always works, needs no thought, and gives bigger numbers that reduce at the end.

the sum	least common denominator	multiply the bottoms
1/2 + 1/3	6	6
1/4 + 1/6	12	24
3/8 + 1/6	24	48
2/5 + 1/10	10	50

Both columns reach the same answer. The right-hand one just carries larger numbers on the way, so 1/4 + 1/6 arrives as 10/24 and needs reducing to 5/12, where the left-hand route lands on 5/12 directly.

Watch the readout in the figure as you move the denominators: when one bottom already divides into the other, the common denominator is just the larger one. 2/5 + 1/10 needs 10, not 50, because fifths are already made of tenths.

Rewriting each fraction

Once you have the denominator, each fraction is rewritten by multiplying its top and bottom by whatever gets it there — an equivalent fraction, nothing more:

3/8 = (3*3)/(8*3) = 9/24

1/6 = (1*4)/(6*4) = 4/24

9/24 + 4/24 = 13/24

The multiplier is different for each fraction, and that is normal: 8 needs a 3 to reach 24 and 6 needs a 4. What has to be the same for both is the number you end up over.

Subtracting is the same page

Everything above holds, with one sign changed at the very end:

5/6 - 1/4 = 10/12 - 3/12 = 7/12

Common denominator, then subtract the tops. If the answer comes out negative, that is a real answer and not a mistake — 1/4 - 5/6 is -7/12.

Try one

If you got 4/14, the bottoms were added. If you got 26/48, you multiplied the bottoms rather than taking their least common multiple — the amount is right, so reduce it and you will land on 13/24 anyway.

Common questions

How do you add fractions?: If the bottom numbers already match, add the tops and leave the bottom alone: 2/7 + 3/7 = 5/7. If they do not match, rewrite both fractions over a common denominator first, then add. 1/2 + 1/3 becomes 3/6 + 2/6 = 5/6.
Why do the denominators have to be the same?: Because the bottom number says how big a piece is, and pieces of different sizes cannot be counted together. Two halves and three thirds is not five of anything. Rewriting both fractions over one denominator re-cuts them into pieces of a single size, and then counting works.
Why do you not add the bottoms?: Because the bottom is not a quantity, it is a label saying what kind of piece you are counting. 1/4 + 1/4 = 2/4, which is a half — and adding the bottoms would give 2/8, a quarter, which is what you started with. The label stays; only the count goes up.
How do I find the least common denominator?: It is the least common multiple of the two bottom numbers: the smallest number both divide into. For 4 and 6 it is 12. Multiplying the two bottoms always works too (24 here), and gives the right answer with bigger numbers that reduce at the end, so it is a fair fallback if the LCM is not obvious.
Is subtracting fractions the same?: Identical, right up to the last step. Get a common denominator, then subtract the tops instead of adding them: 5/6 - 1/4 becomes 10/12 - 3/12 = 7/12. Everything that is hard about adding fractions is hard about subtracting them in exactly the same way.