Mr. Charle's class experienced the following changes in student numbers:
1. 2009-2010: A 5\% increase in the number of students.
2. 2010-2011: Another 5\% increase in the number of students.

We need to find the total percentage increase in the number of students from 2009 to 2011.

Step 1: Assume an Initial Number of Students
Let's assume the initial number of students in $\(\mathbf{2 0 0 9}\)$ is $\(\mathbf{1 0 0}\)$ (for simplicity in percentage calculations).

Step 2: Calculate the Number of Students After the First Increase (2009-2010)
A 5\% increase on the initial 100 students:

$$
\(\begin{gathered}
\text { Increase }=100 \times \frac{5}{100}=5 \\
\text { Students in } 2010=100+5=105
\end{gathered}\)
$$


Step 3: Calculate the Number of Students After the Second Increase (2010-2011)
Now, apply another 5\% increase on the new count of 105 students:

$$
\\(text { Increase }=105 \times \frac{5}{100}=5.25\)
$$


Students in $\(2011=105+5.25=110.25\)$


Step 4: Determine the Total Percentage Increase from 2009 to 2011
The number of students increased from $\(\mathbf{1 0 0}\)$ to $\(\mathbf{1 1 0 . 2 5}\)$ over two years.
To find the total percentage increase:

$$
\(\begin{aligned}
\text { Total Increase } & =110.25-100=10.25 \\
\text { Percentage Increase } & =\left(\frac{10.25}{100}\right) \times 100=10.25 \%
\end{aligned}\)
$$


Key Insight
When applying sequential percentage increases, the total increase is not simply the sum of the individual percentages ( $5 \%+5 \%=10 \%$ ). Instead, it includes the effect of compounding:

$$
\(\begin{gathered}
\text { Final Value }=\text { Initial Value } \times\left(1+\frac{5}{100}\right) \times\left(1+\frac{5}{100}\right)=100 \times 1.05 \times 1.05=110.25 \\
\text { Total Percentage Increase }=(1.05 \times 1.05-1) \times 100=10.25 \%
\end{gathered}\)
$$


Final Answer
The total percentage increase in the number of students from 2009 to 2011 is:

$$
\(10.25 \%\)
$$