A mixed number has a whole-number part and a fractional part. For example:
\[
\begin{aligned}
3\frac{3}{4}
\end{aligned}
\]
means:
\[
\begin{aligned}
3+\frac{3}{4}.
\end{aligned}
\]
1. Mixed numbers and improper fractions
A mixed number can be converted to an improper fraction using:
\[
\begin{aligned}
w\frac{a}{b}
&=
\frac{wb+a}{b}.
\end{aligned}
\]
For example:
\[
\begin{aligned}
3\frac{3}{4}
&=
\frac{3\cdot4+3}{4}\\
&=
\frac{15}{4}.
\end{aligned}
\]
2. Why common denominators are needed
Fractional parts can only be added or subtracted directly when their denominators match.
For unlike denominators, use a common denominator, usually the least common multiple.
\[
\begin{aligned}
L&=\operatorname{LCM}(b,d).
\end{aligned}
\]
Then rewrite both fractional parts with denominator \(L\).
3. Worked example: \(3\frac{3}{4}+2\frac{5}{6}\)
Add the whole-number parts:
\[
\begin{aligned}
3+2&=5.
\end{aligned}
\]
Now add the fractional parts:
\[
\begin{aligned}
\frac{3}{4}+\frac{5}{6}.
\end{aligned}
\]
The least common denominator is:
\[
\begin{aligned}
\operatorname{LCM}(4,6)&=12.
\end{aligned}
\]
Rewrite the fractions:
\[
\begin{aligned}
\frac{3}{4}
&=
\frac{9}{12},\\
\frac{5}{6}
&=
\frac{10}{12}.
\end{aligned}
\]
Add:
\[
\begin{aligned}
\frac{9}{12}+\frac{10}{12}
&=
\frac{19}{12}\\
&=
1\frac{7}{12}.
\end{aligned}
\]
Carry the extra whole number:
\[
\begin{aligned}
5+1\frac{7}{12}
&=
6\frac{7}{12}.
\end{aligned}
\]
Therefore:
\[
\begin{aligned}
\boxed{
3\frac{3}{4}+2\frac{5}{6}
=
6\frac{7}{12}
}.
\end{aligned}
\]
4. Carrying in addition
Carrying is needed when the sum of the fractional parts is at least 1.
For example:
\[
\begin{aligned}
\frac{9}{12}+\frac{10}{12}
&=
\frac{19}{12}\\
&=
1\frac{7}{12}.
\end{aligned}
\]
The extra 1 is added to the whole-number part.
5. Subtracting mixed numbers
To subtract mixed numbers, subtract the whole parts and fractional parts.
If the first fractional part is too small, borrow 1 whole from the first whole number.
Example:
\[
\begin{aligned}
5\frac{1}{3}-2\frac{5}{6}.
\end{aligned}
\]
Use common denominator 6:
\[
\begin{aligned}
\frac{1}{3}&=\frac{2}{6}.
\end{aligned}
\]
Since \(2/6\) is smaller than \(5/6\), borrow 1 whole:
\[
\begin{aligned}
5\frac{2}{6}
&=
4\frac{8}{6}.
\end{aligned}
\]
Now subtract:
\[
\begin{aligned}
4\frac{8}{6}-2\frac{5}{6}
&=
2\frac{3}{6}\\
&=
2\frac{1}{2}.
\end{aligned}
\]
6. Borrowing in subtraction
Borrowing means converting 1 whole into a fraction with the common denominator.
If the common denominator is \(L\), then:
\[
\begin{aligned}
1&=\frac{L}{L}.
\end{aligned}
\]
For example, if \(L=6\), then:
\[
\begin{aligned}
1&=\frac{6}{6}.
\end{aligned}
\]
This borrowed fraction is added to the first fractional part.
7. Improper-fraction method
Another reliable method is to convert each mixed number to an improper fraction first.
\[
\begin{aligned}
3\frac{3}{4}+2\frac{5}{6}
&=
\frac{15}{4}+\frac{17}{6}.
\end{aligned}
\]
Use common denominator 12:
\[
\begin{aligned}
\frac{15}{4}+\frac{17}{6}
&=
\frac{45}{12}+\frac{34}{12}\\
&=
\frac{79}{12}\\
&=
6\frac{7}{12}.
\end{aligned}
\]
This method is especially useful when subtraction gives a negative result.
8. Negative results
If the first mixed number is smaller than the second mixed number, subtraction gives a negative result.
For example:
\[
\begin{aligned}
2\frac{1}{6}-4\frac{3}{8}
&<0.
\end{aligned}
\]
In this case, the improper-fraction method keeps the sign clear.
9. Simplifying the final fractional part
The final fractional part should be reduced to lowest terms:
\[
\begin{aligned}
\frac{3}{6}
&=
\frac{1}{2}.
\end{aligned}
\]
A mixed-number answer is usually written with a proper fractional part.
10. Formula summary
The table below uses plain text formulas in table cells to avoid raw LaTeX rendering problems.
11. Common mistakes
- Adding or subtracting numerators without first finding a common denominator.
- Forgetting to carry when the fractional sum is greater than or equal to 1.
- Forgetting to borrow when the first fractional part is smaller in subtraction.
- Borrowing 1 but not converting it to \(L/L\).
- Leaving the final fractional part unsimplified.
- Confusing the mixed-number method with the improper-fraction method.
Key idea: use a common denominator for the fractional parts, then carry or borrow when the fraction part crosses one whole.
