Question

A manufacturer sells the products x,y,z in two markets, annual sales are indicated below.
$\begin{array}{cccc}Market& Products& & \\ I& 10000& 2000& 18000\\ II& 6000& 20000& 8000\end{array}$
(a)If unit sale prices of x,y and z are Rs 2.50, Rs 1.50 and Rs 1.00 respectively. Find the total revenue in each market with the help of matrix algebra.

(b)If the unit costs of the above three commodities are Rs 2.00m Rs 1.00 and 50 paise respectively. Find the gross profit.

Answer 1

Matrix representing the sales is $A=\left[\begin{array}{ccc}10000& 2000& 18000\\ 6000& 20000& 8000\end{array}\right]$
Matrix representing the sale price per unit is $B=\left[\begin{array}{c}2.50\\ 1.50\\ 1.00\end{array}\right]$
Total revenue in each market is given by the product.
$AB=\left[\begin{array}{ccc}10000& 2000& 18000\\ 6000& 20000& 8000\end{array}\right]\left[\begin{array}{c}\frac{5}{2}\\ \frac{3}{2}\\ 1\end{array}\right]$
$=\left[\begin{array}{c}10000\times \frac{5}{2}+2000\times \frac{3}{2}+18000\times 1\\ 6000\times \frac{5}{2}+20000\times \frac{3}{2}+8000\times 1\end{array}\right]=\left[\begin{array}{c}46000\\ 53000\end{array}\right]$
Hence, total revenue in market I is Rs 46000 and that in in market II is Rs 53000.

The matrix representing the cost price per unit is $C=\left[\begin{array}{c}2.00\\ 1.00\\ 0.50\end{array}\right]$
$\therefore$ Total cost in the two markets is given by the product
$AC=\left[\begin{array}{ccc}10000& 2000& 18000\\ 6000& 20000& 8000\end{array}\right]\left[\begin{array}{c}2\\ 1\\ \frac{1}{2}\end{array}\right]AC=\left[\begin{array}{c}10000\times 2+2000\times 1+18000\times \frac{1}{2}\\ 6000\times 2+20000\times 1+8000\times \frac{1}{2}\end{array}\right]=\left[\begin{array}{c}31000\\ 36000\end{array}\right]$
Profit in market I=Rs(46000-31000)=Rs 15000
and profit in market II=Rs(53000-36000)=Rs 17000
$\therefore$ The gross profit =Rs (15000+17000)=Rs 32000.