Karen is selling pancakes for a one-day charity event. She specialises in three different types of pancakes which are plain, chocolate and cheese at the cost of $2, $3 and $4 each respectively. The matrix A shows the average number of pancakes sold in the morning and afternoon of the day.

1. Evaluate each element of the product matrix

$T=(1\quad 1)\begin{pmatrix}83&86&72\\46&67&47\end{pmatrix}$

$=\begin{pmatrix}1(83)+1(46)&1(86)+1(67)&1(72)+1(47)\end{pmatrix}$

$=\begin{pmatrix}129&153&119\end{pmatrix}$

1. Interpret each element of matrix $T$ in context

Since matrix $A$ gives the numbers sold in the morning and afternoon, multiplying by $(1\ 1)$ adds the two rows.

Therefore
$T=\begin{pmatrix}129&153&119\end{pmatrix}$
represents the total numbers sold for the day for the three columns: plain, chocolate and cheese respectively.

1. Set up the matrix multiplication for total revenue

From part (a)(i),
$T=\begin{pmatrix}129&153&119\end{pmatrix}.$
The selling prices are $2, $3 and $4 respectively, so

2. Evaluate the matrix product

$=258+459+476=1193.$

1. Find the total number of pancakes requiring the basic recipe

From earlier parts,
$T=\begin{pmatrix}129&153&119\end{pmatrix}$
So the total number of pancakes sold is
$129+153+119=401.$

2. Calculate the required quantities and cost of the basic ingredients

The basic recipe is for $12$ servings.

Eggs needed:
$\frac{2}{12}\times 401=66.83\ldots$
So she needs $3$ trays of eggs.
$3\times \$6=\$18$

Milk needed:
$\frac{500}{12}\times 401=16708.33\ldots\text{ ml}=16.708\ldots\text{ litres}$
So she needs $17$ one-litre cartons.

Using the bundle price, $16$ cartons cost
$8\times \$5.95=\$47.60$
and one more carton costs $3.50.
$\text{Milk cost}=\$47.60+\$3.50=\$51.10$

Flour needed:
$\frac{255}{12}\times 401=8521.25\text{ g}=8.52125\text{ kg}$
So she needs $9$ packets of flour.
$9\times \$2.50=\$22.50$

3. Calculate the required quantities and cost of the add-on ingredients

For chocolate pancakes, cocoa powder needed is
$\frac{60}{12}\times 153=765\text{ g}.$
So she needs $1$ tin of cocoa powder.
$\text{Cocoa powder cost}=\$13.50$

For cheese pancakes, cream cheese needed is
$\frac{100}{12}\times 119=991.67\ldots\text{ g}.$
So she needs $1$ block of cream cheese.
$\text{Cream cheese cost}=\$15.50$

4. Find the total cost price and profit

Total cost price is
$18+51.10+22.50+13.50+15.50=120.60.$
So the cost price is $120.60.

From part (a)(iii), total revenue is $1095.

Profit is
$1095-120.60=974.40.$

5. Compare the donation with 900% of the cost price

All profit is donated, so donation is $974.40.

Now,
$900\%\text{ of cost price}=9\times 120.60=1085.40.$
Since
$974.40<1085.40,$
her donation is not more than $900\%$ of her cost price.