How do you add fractions with different denominators?

Rewrite both fractions so they are counting the same size of piece, then add the counts. For 1/3 + 1/4, twelfths work for both: 1/3 is 4/12 and 1/4 is 3/12, so the answer is 7/12. You cannot add the numbers as they stand because thirds and quarters are different-sized pieces - the common denominator is the step that makes them comparable.

Why do you not add the denominators?

Because the denominator is not a quantity being added - it names the size of the piece. In 2/5 the 5 says 'fifths' and the 2 says 'two of them'. Adding 1/2 + 1/2 by adding both rows would give 2/4, which is a half, when the true answer is 1. Once the denominators match you are adding counts of identical pieces, and only the numerator moves.

How do you simplify a fraction?

Divide the top and the bottom by the same number until nothing divides both. 18/24 divides by 6 to give 3/4. Dividing both by the same number is dividing by 1 in disguise, so the value never changes - only the description does. The fastest route is the highest common factor, but repeated halving gets there too.

How do you compare two fractions?

Give them a common denominator and compare the numerators, or cross-multiply: for 3/5 against 5/8, compare 3 times 8 with 5 times 5, so 24 against 25, and 5/8 is larger. Cross-multiplying is the common-denominator method with the shared bottom left unwritten, which is why it works.

What is an equivalent fraction?

A different description of the same amount: 1/2, 2/4, 3/6 and 50/100 are all one half. Multiplying top and bottom by the same number cuts every piece into the same number of smaller pieces and takes proportionally more of them, so the amount you are holding does not change.

Fractions

Almost everything people find hard about fractions comes from one idea being skipped: the bottom number is not a quantity, it is a unit. In 3/5, the 5 says what size of piece you are counting — fifths — and the 3 says how many you have.

You can only add things measured in the same unit, which is the whole story of common denominators.

What a fraction actually says

Cut something into equal pieces. The bottom number (the denominator) says how many pieces the whole was cut into. The top number (the numerator) says how many of those you are holding.

3/4 is: cut into 4 equal parts, take 3 of them.

Three of four pieces is only 3/4 if the four pieces are the same size. This is the difference between a fraction and a rough share, and it is what makes fractions arithmetic rather than description.

A bigger denominator means smaller pieces

1/8 is smaller than 1/3, even though 8 is bigger than 3. Cutting a cake into eight gives you less per slice than cutting it into three.

This is the single most common source of confusion, and it comes from reading the bottom number as a size instead of a count of cuts. More cuts, smaller pieces.

Equivalent fractions

Multiply the top and the bottom by the same number and the value does not change.

1/2 = 2/4 = 3/6 = 50/100. 2/3 = 8/12 (both × 4).

You are cutting every existing piece into the same number of smaller pieces, and then taking proportionally more of them. Twice as many pieces, each half the size — you are holding exactly what you were holding before. What changed is the description, not the amount.

Adding, subtracting and comparing all start by rewriting fractions until they describe the same size of piece.

Simplifying

Run equivalence backwards: divide the top and the bottom by the same number until nothing divides both.

18/24 → divide both by 6 → 3/4. 8/12 → divide both by 4 → 2/3.

If you cannot spot the highest common factor, halve repeatedly, or take out an obvious 2, 3 or 5 and look again. 40/100 → 20/50 → 10/25 → 2/5. You get to the same place, just in more steps.

Dividing top and bottom by the same number is multiplying by 1 in disguise — 6/6 is 1, and multiplying by 1 changes nothing. That is why simplifying is always allowed and never changes the answer.

Comparing fractions

Same bottom: compare the tops

5/8 > 3/8, because five eighths is more eighths than three.

Same top: the bigger bottom is smaller

3/5 > 3/8 — the same number of pieces, but fifths are bigger than eighths.

Neither: cross-multiply

To compare 3/5 and 5/8, multiply each top by the other bottom.

3 × 8 = 24 against 5 × 5 = 25, so 5/8 is the larger.

This is the common-denominator method with the shared bottom left unwritten. Both fractions would be rewritten over 40 — as 24/40 and 25/40 — and since the bottoms are then identical, only the tops decide. Skipping the step you would not have used is the whole trick.

Compare to a half

Fastest check of all: double the top and see how it lands against the bottom. 7/16 — double 7 is 14, which is less than 16, so it is under a half.

Adding and subtracting fractions

Same denominator: add the tops, keep the bottom

2/7 + 3/7 = 5/7.

Two sevenths plus three sevenths is five sevenths for the same reason two meters plus three meters is five meters. The unit is not being added — it is what makes the counts addable.

Different denominators: make them match first

Find a number both bottoms divide into, rewrite both fractions over it, then add the tops.

1/3 + 1/4 → twelfths → 4/12 + 3/12 = 7/12.

2/5 + 1/2 → tenths → 4/10 + 5/10 = 9/10.

Multiplying the two denominators always gives a common one, and it always works. It just sometimes gives a bigger number than you need — for 1/6 + 1/4 you could use 24, but 12 works and leaves less to simplify.

Thirds and quarters are different-sized pieces, so their counts cannot be added as they stand. Twelfths can be made out of both — four of them make a third, three of them make a quarter — so rewriting into twelfths puts both amounts in one unit. Then, and only then, the counts add.

Subtraction is the same move

3/4 − 2/3 → twelfths → 9/12 − 8/12 = 1/12.

Mixed numbers: deal with the wholes separately

2 1/2 + 1 3/4 → 2 + 1 = 3, and 1/2 + 3/4 = 5/4 = 1 1/4, so 4 1/4.

For subtraction where the fraction part does not go, borrow one whole and turn it into pieces: 3 1/5 − 1 3/5 → borrow: 2 6/5 − 1 3/5 = 1 3/5.

Multiplying and dividing

These are the ones people expect to be harder, and they are easier.

Multiplying: straight across

2/3 × 4/5 = 8/15. No common denominator needed.

"Of" is what multiplication means here. Two thirds of four fifths: take four fifths, cut each fifth into three, and the whole is now in fifteenths. That is where the 15 comes from — and taking two of every three of those four pieces gives 8. Nothing has to match up first, because you are not adding counts, you are re-cutting.

Dividing: flip the second one and multiply

1/2 ÷ 1/6 = 1/2 × 6/1 = 6/2 = 3.

Read it as a question: how many sixths fit into a half? Three. That is the answer, and no flipping was needed to see it — the flip is a shortcut for "how many of these fit into that". Dividing by 1/6 makes things six times bigger for the same reason dividing a cake into sixth-sized portions gives you six times as many portions as dividing it into whole ones.

Fractions, decimals and percentages

Three notations for one idea. A fraction is a division waiting to happen: 3/4 is 3 ÷ 4 = 0.75, which is 75%.

Fraction	Decimal	Percentage
1/2	0.5	50%
1/4	0.25	25%
3/4	0.75	75%
1/5	0.2	20%
1/3	0.333…	33.3%
1/8	0.125	12.5%

Knowing this handful by sight removes most of the arithmetic from everyday percentage questions — there is more on that in the guide to percentages.

The four mistakes worth naming

Adding the denominators. 1/2 + 1/2 is 1, not 2/4. The bottom is the unit; it does not accumulate.
Reading a bigger bottom as a bigger fraction. 1/8 < 1/3. More cuts, smaller pieces.
Simplifying only the top. Whatever you do to one row you do to the other, or the value moves.
Finding a common denominator before multiplying. It is not wrong, it is wasted work — multiplication does not need matching units.

Practice these in the game

Math Challenge has a Fractions topic that drills adding and subtracting fractions — the two that need common denominators, and the two people get wrong. It starts with matching or friendly denominators and widens as you get them right.

It sits alongside topics for decimals, percentages and the rest of the arithmetic ladder, plus a free Daily Challenge that needs no account.

This page is the short answer. For the full teaching ladder — improper fractions and mixed numbers, fractions of amounts, ordering three at once, and an illustrated lesson behind every rung — see equivalent fractions and mixed numbers.

Your turn

Three to try — tap what you get.

1/3 + 1/4

Which is larger?

2/3 × 4/5

Practice fractions free →