Fractions

Updated 4 October 2026

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 $\frac{3}{5}$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.

$\frac{3}{4}$3/4 is: cut into 4 equal parts, take 3 of them.

A bigger denominator means smaller pieces

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

What are equivalent fractions?

Equivalent fractions are fractions that name the same amount, and you make one by multiplying or dividing the top and bottom by the same number. Take $\frac{3}{4}$3/4 and multiply both parts by 2: you get $\frac{6}{8}$6/8, and six eighths of a bar covers exactly as much as three fourths. Each fourth has been cut into 2 eighths, so there are twice as many pieces and each is half the size. That is why the amount stays the same: multiplying the top and bottom by 2 is multiplying by $\frac{2}{2}$2/2, which is 1. It goes wrong if you change only one number, or add instead of multiplying. Adding 1 to both parts of $\frac{1}{2}$1/2 gives $\frac{2}{3}$2/3, which is more than a half, so the two are not equivalent. To check two fractions, cross-multiply: for $\frac{3}{4}$3/4 and $\frac{6}{8}$6/8, 3 × 8 = 24 and 4 × 6 = 24, and equal products mean equal fractions.

The full explanation, with its figures and questions: What makes two fractions equivalent?.

How do you simplify a fraction?

To simplify a fraction, divide the top and bottom by the same number, and keep going until no number bigger than 1 divides both. For $\frac{12}{30}$12/30, both numbers divide by 6, because 12 = 2 × 6 and 30 = 5 × 6, so $\frac{12}{30}$12/30 simplifies to $\frac{2}{5}$2/5. Dividing both parts by 6 is dividing by $\frac{6}{6}$6/6, which is 1, so the amount does not change; the pieces are just bigger. The fastest route is to divide once by the highest common factor. If you do not spot it, divide by any common factor and look again: $\frac{12}{30}$12/30 halves to $\frac{6}{15}$6/15, and $\frac{6}{15}$6/15 divides by 3 to give $\frac{2}{5}$2/5, the same answer in two steps. The case to watch is a fraction already in simplest form, such as $\frac{7}{12}$7/12, where no number but 1 divides both. To check, confirm the new top and bottom share no factor, then multiply back: 2 × 6 = 12 and 5 × 6 = 30.

$\frac{18}{24}\to$18/24 → divide both by $6\to \frac{3}{4}$6 → 3/4. $\frac{8}{12}\to$8/12 → divide both by $4\to \frac{2}{3}$4 → 2/3.

Halving twice and then dividing by 5 reaches the same place: $\frac{40}{100}\to \frac{20}{50}\to \frac{10}{25}\to \frac{2}{5}$40/100 → 20/50 → 10/25 → 2/5.

How do you compare two fractions?

To compare two fractions, give them the same bottom number, and then the bigger top is the bigger fraction. To compare $\frac{2}{3}$2/3 and $\frac{3}{4}$3/4, rewrite both in twelfths: $\frac{2}{3}=\frac{8}{12}$2/3 = 8/12 and $\frac{3}{4}=\frac{9}{12}$3/4 = 9/12, so $\frac{3}{4}$3/4 is larger. This works because once the pieces are the same size, you only have to count them. Two shortcuts skip the rewriting. If the bottoms already match, compare the tops. If the tops match, the fraction with the smaller bottom is bigger, because fewer cuts make bigger pieces: $\frac{2}{5}$2/5 is more than $\frac{2}{7}$2/7. When neither matches, cross-multiply, which does the rewriting in one step: for $\frac{2}{3}$2/3 and $\frac{3}{4}$3/4, 2 × 4 = 8 and 3 × 3 = 9, and the bigger product sits with the bigger fraction, $\frac{3}{4}$3/4. To check, subtract the smaller from the larger: $\frac{3}{4}-\frac{2}{3}=\frac{1}{12}$3/4 − 2/3 = 1/12, and a difference above zero confirms the order.

Same bottom: compare the tops

$\frac{5}{8}>\frac{3}{8}$5/8 > 3/8, because five eighths is more eighths than three.

Same top: the bigger bottom is smaller

$\frac{3}{5}>\frac{3}{8}$3/5 > 3/8 — the same number of pieces, but fifths are bigger than eighths.

Neither: cross-multiply

To compare $\frac{3}{5}$3/5 and $\frac{5}{8}$5/8, multiply each top by the other bottom.

3 × 8 = 24 against 5 × 5 = 25, so $\frac{5}{8}$5/8 is the larger.

Compare to a half

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

How do you add and subtract fractions?

To add or subtract fractions, first give them the same bottom number, then add or subtract the tops and keep the bottom. For $\frac{2}{3}+\frac{1}{5}$2/3 + 1/5, both 3 and 5 divide into 15, so $\frac{2}{3}=\frac{10}{15}$2/3 = 10/15 and $\frac{1}{5}=\frac{3}{15}$1/5 = 3/15, and $\frac{10}{15}+\frac{3}{15}=\frac{13}{15}$10/15 + 3/15 = 13/15. The bottom stays 15 because it names the size of the pieces, and adding pieces of one size does not change their size. That is also why $\frac{1}{2}+\frac{1}{3}$1/2 + 1/3 is not $\frac{2}{5}$2/5: halves and thirds are different pieces, and they cannot be counted together until both are sixths. Subtraction uses the same steps: $\frac{2}{3}-\frac{1}{5}=\frac{10}{15}-\frac{3}{15}=\frac{7}{15}$2/3 − 1/5 = 10/15 − 3/15 = 7/15. One more step comes when the answer's top is bigger than its bottom, such as $\frac{5}{4}$5/4, which you rewrite as the mixed number $1\frac{1}{4}$1 1/4. To check, estimate: $\frac{2}{3}$2/3 is a bit less than 1 and $\frac{1}{5}$1/5 is small, so $\frac{13}{15}$13/15, just under 1, is the right size.

Same denominator: add the tops, keep the bottom

$\frac{2}{7}+\frac{3}{7}=\frac{5}{7}$2/7 + 3/7 = 5/7.

Different denominators: make them match first

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

$\frac{1}{3}+\frac{1}{4}\to$1/3 + 1/4 → twelfths $\to \frac{4}{12}+\frac{3}{12}=\frac{7}{12}$→ 4/12 + 3/12 = 7/12.

$\frac{2}{5}+\frac{1}{2}\to$2/5 + 1/2 → tenths $\to \frac{4}{10}+\frac{5}{10}=\frac{9}{10}$→ 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 $\frac{1}{6}+\frac{1}{4}$1/6 + 1/4 you could use 24, but 12 works and leaves less to simplify.

Subtraction is the same move

$\frac{3}{4}-\frac{2}{3}\to$3/4 − 2/3 → twelfths $\to \frac{9}{12}-\frac{8}{12}=\frac{1}{12}$→ 9/12 − 8/12 = 1/12.

Mixed numbers: deal with the wholes separately

$2\frac{1}{2}+1\frac{3}{4}\to 2+1=3$2 1/2 + 1 3/4 → 2 + 1 = 3, and $\frac{1}{2}+\frac{3}{4}=\frac{5}{4}=1\frac{1}{4}$1/2 + 3/4 = 5/4 = 1 1/4, so $4\frac{1}{4}$4 1/4.

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

How do you multiply and divide fractions?

To multiply fractions, multiply the tops together and the bottoms together: $\frac{3}{4}\times \frac{2}{5}=\frac{6}{20}=\frac{3}{10}$3/4 × 2/5 = 6/20 = 3/10. No common bottom is needed, because multiplying takes a part of a part: three fourths of two fifths is a smaller piece than either. To divide by a fraction, flip the second fraction upside down and multiply. For $\frac{3}{4}÷\frac{1}{8}$3/4 ÷ 1/8, flip $\frac{1}{8}$1/8 to $\frac{8}{1}$8/1 and multiply: $\frac{3}{4}\times \frac{8}{1}=\frac{24}{4}=6$3/4 × 8/1 = 24/4 = 6. This works because dividing asks how many eighths fit into three fourths, and each fourth holds 2 eighths, so three fourths hold 6. Only the second fraction flips, never the first. A mixed number must become an improper fraction first: $1\frac{1}{2}\times \frac{2}{3}$1 1/2 × 2/3 becomes $\frac{3}{2}\times \frac{2}{3}=\frac{6}{6}=1$3/2 × 2/3 = 6/6 = 1. To check a division, multiply the answer by the fraction you divided by: $6\times \frac{1}{8}=\frac{6}{8}=\frac{3}{4}$6 × 1/8 = 6/8 = 3/4, the number you started with.

The full explanation, with its figures and questions: How do you multiply and divide fractions?.

Fractions, decimals and percentages

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

The full explanation, with its figures and questions: Which fractions are worth knowing as percentages?.

The four mistakes worth naming

• Adding the denominators. $\frac{1}{2}+\frac{1}{2}$1/2 + 1/2 is 1, not $\frac{2}{4}$2/4. The bottom is the unit; it does not accumulate.
• Reading a bigger bottom as a bigger fraction. $\frac{1}{8}<\frac{1}{3}$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.

For the full teaching ladder — improper fractions and mixed numbers, fractions of amounts and ordering three at once — see equivalent fractions and mixed numbers.