A baking school offers beginner, intermediate and advanced baking classes on weekdays and weekends in August this year. The matrix B shows the number of participants attending the baking classes in August.

1. Represent the three class fees as a column matrix in the required order.

The order is beginner, intermediate, advanced.

So the $3 \times 1$ matrix is
$F=\begin{pmatrix}30\\40\\50\end{pmatrix}.$

1. Set up the row-by-column multiplication for $BF$.

Using $B=\begin{pmatrix}22&20&9\\8&10&11\end{pmatrix},\quad F=\begin{pmatrix}30\\40\\50\end{pmatrix},$
$C=BF=\begin{pmatrix}22&20&9\\8&10&11\end{pmatrix}\begin{pmatrix}30\\40\\50\end{pmatrix}.$

2. Calculate each element of the product matrix.

$C=\begin{pmatrix}22(30)+20(40)+9(50)\\8(30)+10(40)+11(50)\end{pmatrix}$
$=\begin{pmatrix}660+800+450\\240+400+550\end{pmatrix}$
$=\begin{pmatrix}1910\\1190\end{pmatrix}.$

1. Interpret each element of the product matrix in context.

Since $B$ gives the numbers of participants for weekdays and weekends, and $F$ gives the class fees, the product $C=BF=\begin{pmatrix}1910\\1190\end{pmatrix}$ gives the total fees collected.

The first element, $1910$, represents the total fees from weekday classes.

The second element, $1190$, represents the total fees from weekend classes.

1. Find the total amount of fees collected in August.

From part (b), total August fees
$=1910+1190=3100$

2. Calculate the total September fees after applying the changes to each class type.

Total participants in August:

Beginner: $22+8=30$

Intermediate: $20+10=30$

Advanced: $9+11=20$

September participants:

Beginner: $80\%\times30=24$

Intermediate: $110\%\times30=33$

Advanced: $3\times20=60$

September fees
$=24(30)+33(40)+60(50)$
$=720+1320+3000=5040$

3. Calculate the percentage increase from August fees to September fees.

Increase in fees
$=5040-3100=1940$

Percentage increase
$=\frac{1940}{3100}\times100\%=62.5806\ldots\%$
$\approx62.6\%$