A fraction and a mixed number can represent the same value. The fraction form is useful for calculation,
while the mixed-number form is often easier to interpret visually.
\[
\begin{aligned}
\frac{p}{q},
\qquad q\ne0.
\end{aligned}
\]
The number \(p\) is the numerator, and the number \(q\) is the denominator.
The denominator cannot be zero because division by zero is undefined.
A fraction is proper if the absolute value of the numerator is smaller than the denominator:
A fraction is improper if the numerator is at least as large as the denominator in absolute value:
\[
\begin{aligned}
|p|\ge q.
\end{aligned}
\]
For example, \(2/7\) is proper, while \(23/7\) is improper.
3. Mixed number form
A mixed number writes a value as a whole part plus a proper fraction:
\[
\begin{aligned}
3\frac{2}{7}
&=
3+\frac{2}{7}.
\end{aligned}
\]
The mixed number \(3\frac{2}{7}\) means 3 whole units and 2 more seventh-parts.
4. Simplifying first
Before converting, it is often best to simplify the fraction. Simplification uses the greatest common divisor:
\[
\begin{aligned}
\frac{p}{q}
&=
\frac{p/\gcd(p,q)}{q/\gcd(p,q)}.
\end{aligned}
\]
Example:
\[
\begin{aligned}
\frac{8}{12}
&=
\frac{8\div4}{12\div4}\\
&=
\frac{2}{3}.
\end{aligned}
\]
5. Converting an improper fraction to a mixed number
To convert \(p/q\) to a mixed number, divide the absolute value of the numerator by the denominator:
\[
\begin{aligned}
|p| &= q\cdot k+r,
\qquad 0\le r
The quotient \(k\) becomes the whole part, and the remainder \(r\) becomes the numerator of the fractional part.
Therefore:
\[
\begin{aligned}
\frac{p}{q}
&=
\text{sign}(p)\left(k+\frac{r}{q}\right).
\end{aligned}
\]
6. Worked example: \(23/7\)
Divide 23 by 7:
\[
\begin{aligned}
23 &= 7\cdot3+2.
\end{aligned}
\]
So the quotient is 3 and the remainder is 2. Therefore:
\[
\begin{aligned}
\frac{23}{7}
&=
3\frac{2}{7}.
\end{aligned}
\]
7. Negative improper fractions
For a negative fraction, convert the absolute value first and then attach the negative sign to the whole mixed value.
\[
\begin{aligned}
-\frac{17}{5}
&=
-\left(\frac{17}{5}\right).
\end{aligned}
\]
Since
\[
\begin{aligned}
17 &= 5\cdot3+2,
\end{aligned}
\]
we get:
\[
\begin{aligned}
-\frac{17}{5}
&=
-3\frac{2}{5}.
\end{aligned}
\]
8. Converting a mixed number to an improper fraction
To convert \(k\frac{m}{n}\) to an improper fraction, multiply the whole part by the denominator and add the fractional numerator:
\[
\begin{aligned}
k\frac{m}{n}
&=
\frac{k\cdot n+m}{n}.
\end{aligned}
\]
Then simplify the fraction if possible.
9. Worked example: \(3\frac{2}{7}\)
\[
\begin{aligned}
3\frac{2}{7}
&=
\frac{3\cdot7+2}{7}\\
&=
\frac{21+2}{7}\\
&=
\frac{23}{7}.
\end{aligned}
\]
10. Negative mixed numbers
A negative mixed number means the negative sign applies to the entire value:
\[
\begin{aligned}
-2\frac{1}{3}
&=
-\left(2+\frac{1}{3}\right).
\end{aligned}
\]
Convert the positive mixed number first:
\[
\begin{aligned}
2\frac{1}{3}
&=
\frac{2\cdot3+1}{3}\\
&=
\frac{7}{3}.
\end{aligned}
\]
Then attach the negative sign:
\[
\begin{aligned}
-2\frac{1}{3}
&=
-\frac{7}{3}.
\end{aligned}
\]
11. Decimal approximation
A fraction can also be written approximately as a decimal:
\[
\begin{aligned}
\frac{23}{7}
&\approx
3.285714.
\end{aligned}
\]
The decimal is helpful for checking size, but the exact answer should usually remain as a simplified fraction or mixed number.
12. Formula summary
The table below uses plain text formulas in table cells to avoid raw LaTeX rendering problems.
13. Common mistakes
- Forgetting to simplify the fraction before or after converting.
- Putting the quotient in the numerator instead of using it as the whole part.
- Forgetting that the remainder must be smaller than the denominator.
- Forgetting that a negative mixed number applies the negative sign to the whole value.
- Using a denominator of zero, which is undefined.
Key idea: a mixed number and an improper fraction can describe exactly the same value. The conversion only changes the form, not the number.
