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:

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 $3$.

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.

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.
• Leaving a fraction unreduced — $\frac{6}{8}$ and $\frac{3}{4}$ are the same number, but a "reduce" question expects the lowest-terms form.