Fractions

A fraction writes part of a whole as two numbers separated by a fraction bar: $\frac{a}{b}$, read "a over b". The number $a$ above the bar is the numerator, and $b$ below it is the denominator. The denominator says into how many equal parts the whole was split, and the numerator says how many of those parts you take.

For example $\frac{3}{4}$ is three of four equal parts — split a chocolate bar into 4 pieces and eat 3, and you have eaten $\frac{3}{4}$ of it.

Equivalent fractions

The same part of a whole can be written many ways. If you multiply the numerator and denominator by the same number $k$, the fraction's value does not change:

For example:

Three ways of writing the same half. Doing this in reverse — dividing the numerator and denominator by the same number — is called reducing, covered below.

Adding fractions

Fractions with the same denominator add by just adding the numerators:

When the denominators differ, first bring both fractions to a common denominator (e.g. the product of the two denominators), then add:

For example:

Subtracting fractions

Subtraction works exactly the same way — only the sign between the numerators changes:

Fractions with the same denominator subtract straight away, $\frac{5}{9} - \frac{2}{9} = \frac{3}{9} = \frac{1}{3}$, and different denominators are brought to a common one first:

The last step reduces by $2$ — and a difference is worth checking by adding it back: $\frac{7}{12} + \frac{1}{6} = \frac{7}{12} + \frac{2}{12} = \frac{9}{12} = \frac{3}{4}$. It agrees.

Multiplying fractions

Multiplying fractions is simpler than adding — no common denominator needed. Multiply the numerators together and the denominators together:

For example:

The last step reduces the result by $3$.

Dividing fractions

Dividing by a fraction comes down to multiplying — just flip the divisor, swapping its numerator and denominator:

The fraction $\frac{d}{c}$ is called the reciprocal of $\frac{c}{d}$; a number times its reciprocal is always $1$, since $\frac{c}{d} \cdot \frac{d}{c} = \frac{cd}{dc} = 1$. That is why dividing by $\frac{c}{d}$ and multiplying by $\frac{d}{c}$ are the same operation.

It is always the second fraction — the one you divide by — that gets flipped. Unlike dividing whole numbers, dividing by a fraction can make the result larger: $\frac{3}{4}$ divided by $\frac{1}{2}$ is $\frac{3}{2}$, because the question is "how many halves fit into three quarters".

Reducing fractions

A fraction is in lowest terms when the numerator and denominator no longer share a common divisor bigger than $1$. To get there, divide both by their greatest common divisor (gcd):

For example $\mathrm{gcd}(18, 24) = 6$, so:

And $\frac{3}{4}$ reduces no further.

Mixed numbers

A fraction whose numerator is smaller than its denominator is called proper — it is less than one whole. When the numerator is greater than or equal to the denominator, the fraction is improper and holds at least one whole inside it. Such a fraction can be split into a whole part and a remainder, that is, written as a mixed number:

The notation $2\frac{3}{4}$ reads "two and three quarters" — it is a sum, even though no plus sign is written. Going the other way follows one formula:

Multiply the whole part by the denominator and add the numerator: $2\frac{3}{4} = \frac{2 \cdot 4 + 3}{4} = \frac{11}{4}$.

The improper form is usually the handier one for calculating — mixed numbers get converted before they are multiplied or divided. The mixed form is the clearer one in an answer: "$2\frac{3}{4}$ hours" says at once that it is under three hours, which $\frac{11}{4}$ does not.

Comparing fractions

Which is larger, $\frac{3}{5}$ or $\frac{5}{8}$? The numerators and denominators settle nothing on their own — only equal parts can be compared. Hence two methods.

The first: bring both fractions to a common denominator and compare the numerators.

The second, quicker one is cross-multiplication — the same arithmetic without writing the denominator down:

Here $3 \cdot 8 = 24$ against $5 \cdot 5 = 25$, so again $\frac{3}{5} < \frac{5}{8}$. The condition $b, d > 0$ matters: with a negative denominator the comparison flips, exactly as multiplying an inequality by a negative number does.

Two cases need no arithmetic at all: with equal denominators the larger fraction is the one with the larger numerator ($\frac{5}{7} > \frac{3}{7}$), and with equal numerators it is the one with the smaller denominator ($\frac{3}{5} > \frac{3}{8}$, because fifths are bigger pieces than eighths).

Common mistakes

• Adding numerators and denominators separately — $\frac{1}{2} + \frac{1}{3}$ is not $\frac{2}{5}$. You must find a common denominator first.
• Finding a common denominator to multiply — multiplying fractions never needs one; that step is unnecessary.
• Flipping the first fraction when dividing — in $\frac{3}{4} \div \frac{2}{5}$ it is the divisor, $\frac{2}{5}$, that gets flipped; flipping $\frac{3}{4}$ gives an entirely different number.
• Comparing fractions by their denominators — $\frac{3}{8}$ is not larger than $\frac{3}{5}$ just because $8 > 5$; with equal numerators, the larger denominator means smaller pieces.
• Multiplying mixed numbers part by part — $2\frac{1}{2} \cdot 3$ is not $6\frac{1}{2}$; convert to $\frac{5}{2}$ first and only then multiply, which gives $\frac{15}{2} = 7\frac{1}{2}$.
• Leaving a fraction unreduced — $\frac{6}{8}$ and $\frac{3}{4}$ are the same number, but a "reduce" question expects the lowest-terms form.