The orders from three shops in July 2021 were as follows:
Tasty shop ordered 100 of type A, 50 of type B and 25 of type C.
Fragrant shop ordered 120 of type A, 60 of type B and 20 of type C.
Unusual shop ordered 80 of type A, 70 of type B and 30 of type C.

1. Find the combined filling rate and Tap A's filling rate.

Combined rate of Tap A and Tap B:
$\frac{1}{2.4}=\frac{5}{12}$
tank per hour.

Rate of Tap A:
$\frac{1}{6}$
tank per hour.

2. Subtract rates to find Tap B's rate, then convert it to time taken.

Rate of Tap B:
$\frac{5}{12}-\frac{1}{6}=\frac{5}{12}-\frac{2}{12}=\frac{3}{12}=\frac{1}{4}$
tank per hour.

So time taken by Tap B alone:
$\frac{1}{\frac{1}{4}}=4$
hours.

1. Write the weights of type A, type B and type C as a $3\times1$ column matrix.

Type A, B and C have weights $2$ kg, $5$ kg and $10$ kg respectively.

Since $P$ is a $3\times1$ matrix,
$P=\begin{pmatrix}2\\5\\10\end{pmatrix}$

1. Write the prices of type A, type B and type C as a $1\times3$ row matrix.

The respective prices are type A: $6.20, type B: $13.50, type C: $24.00.

Since $Q$ is a $1\times3$ matrix,
$Q=\begin{pmatrix}6.20&13.50&24.00\end{pmatrix}$

1. Write the quantities ordered as a $3\times3$ matrix with rows representing shops and columns representing types A, B and C.

Using the order Tasty, Fragrant, Unusual for the rows, and type A, type B, type C for the columns,
$R=\begin{pmatrix}100&50&25\\120&60&20\\80&70&30\end{pmatrix}$

1. Set up the matrix product $RP$ using the order matrix and the weight matrix.

From earlier parts,
$R=\begin{pmatrix}100&50&25\\120&60&20\\80&70&30\end{pmatrix},\qquad P=\begin{pmatrix}2\\5\\10\end{pmatrix}$

So
$RP=\begin{pmatrix}100&50&25\\120&60&20\\80&70&30\end{pmatrix}\begin{pmatrix}2\\5\\10\end{pmatrix}$

2. Multiply each row of $R$ by the column matrix $P$ to find each shop's total weight.

$RP=\begin{pmatrix}100(2)+50(5)+25(10)\\120(2)+60(5)+20(10)\\80(2)+70(5)+30(10)\end{pmatrix}$

$=\begin{pmatrix}200+250+250\\240+300+200\\160+350+300\end{pmatrix}$

$=\begin{pmatrix}700\\740\\810\end{pmatrix}$

1. Choose a row matrix that adds the three shop totals in $RP$.

Since $RP$ is a $3\times1$ matrix, $S$ must be a $1\times3$ matrix so that $SRP$ is $1\times1$.

To add the three entries of $RP$,
$S=\begin{pmatrix}1&1&1\end{pmatrix}.$

1. Set up the matrix product for total revenue.

Since
$S=\begin{pmatrix}1&1&1\end{pmatrix},\quad R=\begin{pmatrix}100&50&25\\120&60&20\\80&70&30\end{pmatrix},\quad Q=\begin{pmatrix}6.20&13.50&24.00\end{pmatrix},$
the total revenue is given by
$SRQ^T=\begin{pmatrix}1&1&1\end{pmatrix}\begin{pmatrix}100&50&25\\120&60&20\\80&70&30\end{pmatrix}\begin{pmatrix}6.20\\13.50\\24.00\end{pmatrix}.$

2. Carry out the multiplication to find the total revenue.

First find the total number of packages of each type:
$\begin{pmatrix}1&1&1\end{pmatrix}\begin{pmatrix}100&50&25\\120&60&20\\80&70&30\end{pmatrix}=\begin{pmatrix}300&180&75\end{pmatrix}.$
Then multiply by the price column matrix:
$\begin{pmatrix}300&180&75\end{pmatrix}\begin{pmatrix}6.20\\13.50\\24.00\end{pmatrix}$
$=300(6.20)+180(13.50)+75(24.00)$
$=1860+2430+1800=6090.$