A mixed number combines a whole-number part and a fractional part. Multiplication and division are usually easiest
after converting each mixed number to an improper fraction.
1. Mixed number to improper fraction
A mixed number can be converted using:
\[
\begin{aligned}
w\frac{a}{b}
&=
\frac{wb+a}{b}.
\end{aligned}
\]
For example:
\[
\begin{aligned}
2\frac{1}{2}
&=
\frac{2\cdot2+1}{2}\\
&=
\frac{5}{2}.
\end{aligned}
\]
2. Multiplying mixed numbers
To multiply mixed numbers:
- Convert each mixed number to an improper fraction.
- Cancel common factors if possible.
- Multiply numerators and denominators.
- Convert the result back to a mixed number if needed.
The basic multiplication rule is:
\[
\begin{aligned}
\frac{a}{b}\times\frac{c}{d}
&=
\frac{ac}{bd}.
\end{aligned}
\]
3. Worked example: \(2\frac{1}{2}\times3\frac{3}{4}\)
Convert both mixed numbers:
\[
\begin{aligned}
2\frac{1}{2}
&=
\frac{5}{2},\\
3\frac{3}{4}
&=
\frac{15}{4}.
\end{aligned}
\]
Multiply:
\[
\begin{aligned}
\frac{5}{2}\times\frac{15}{4}
&=
\frac{5\cdot15}{2\cdot4}\\
&=
\frac{75}{8}.
\end{aligned}
\]
Convert to a mixed number:
\[
\begin{aligned}
\frac{75}{8}
&=
9\frac{3}{8}.
\end{aligned}
\]
Therefore:
\[
\begin{aligned}
\boxed{2\frac{1}{2}\times3\frac{3}{4}=9\frac{3}{8}}.
\end{aligned}
\]
4. Cross-cancellation
Before multiplying, you can cancel common factors between any numerator and the opposite denominator:
\[
\begin{aligned}
\frac{a}{b}\times\frac{c}{d}.
\end{aligned}
\]
You may cancel a common factor between \(a\) and \(d\), or between \(c\) and \(b\).
This works because it divides numerator and denominator by the same factor.
For example:
\[
\begin{aligned}
\frac{4}{9}\times\frac{3}{10}
&=
\frac{4}{3}\times\frac{1}{10}\\
&=
\frac{4}{30}\\
&=
\frac{2}{15}.
\end{aligned}
\]
5. Dividing mixed numbers
Dividing by a fraction means multiplying by its reciprocal:
\[
\begin{aligned}
\frac{a}{b}\div\frac{c}{d}
&=
\frac{a}{b}\times\frac{d}{c},
\qquad c\ne0.
\end{aligned}
\]
The second fraction cannot be zero, because division by zero is undefined.
6. Worked division example
Solve:
\[
\begin{aligned}
4\frac{2}{3}\div1\frac{1}{6}.
\end{aligned}
\]
Convert to improper fractions:
\[
\begin{aligned}
4\frac{2}{3}
&=
\frac{14}{3},\\
1\frac{1}{6}
&=
\frac{7}{6}.
\end{aligned}
\]
Multiply by the reciprocal:
\[
\begin{aligned}
\frac{14}{3}\div\frac{7}{6}
&=
\frac{14}{3}\times\frac{6}{7}.
\end{aligned}
\]
Cross-cancel:
\[
\begin{aligned}
\frac{14}{3}\times\frac{6}{7}
&=
\frac{2}{1}\times\frac{2}{1}\\
&=
4.
\end{aligned}
\]
Therefore:
\[
\begin{aligned}
\boxed{4\frac{2}{3}\div1\frac{1}{6}=4}.
\end{aligned}
\]
7. Why division uses the reciprocal
Dividing by \(c/d\) asks how many groups of size \(c/d\) fit into the first amount.
Algebraically, dividing by \(c/d\) is the same as multiplying by \(d/c\):
\[
\begin{aligned}
x\div\frac{c}{d}
&=
x\times\frac{d}{c}.
\end{aligned}
\]
8. Zero cases
Multiplying by zero gives zero:
\[
\begin{aligned}
0\times a&=0.
\end{aligned}
\]
But division by zero is undefined:
\[
\begin{aligned}
a\div0
\quad\text{is undefined.}
\end{aligned}
\]
So the second mixed number cannot equal zero in a division problem.
9. Converting the result back to a mixed number
An improper fraction can be converted to a mixed number by division:
\[
\begin{aligned}
\frac{75}{8}
&=
9\frac{3}{8},
\end{aligned}
\]
because \(75\div8=9\) remainder \(3\).
10. Formula summary
The table below uses plain text formulas in table cells to avoid raw LaTeX rendering problems.
11. Common mistakes
- Multiplying mixed numbers without converting them to improper fractions first.
- For division, forgetting to flip the second fraction.
- Flipping both fractions instead of only the divisor.
- Canceling across addition or subtraction instead of multiplication.
- Forgetting to simplify the final answer.
- Trying to divide by a mixed number equal to zero.
Key idea: convert mixed numbers to improper fractions first; for division, multiply by the reciprocal of the second fraction.
