Question

A manufacturer produces three products $x,y,z$ which he sells in two markets. Annual sales are indicated below:
Market Products

$I$	$10,000$	$2,000$	$18,000$
$II$	$6,000$	$20,000$	$8,000$

(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 $Rs.2.00,Rs.1.00$ and $50$ paise respectively. Find the gross profit.

Answer 1

(a) The unit sale prices of $x,y$ and $z$ are respectively$Rs.2.50,Rs.1.50$ and $Rs.1.00$.

Total revenue in market $I$ can be represented as:

$\left[\begin{array}{ccc}10000& 2000& 18000\end{array}\right]\left[\begin{array}{c}2.50\\ 1.50\\ 1.00\end{array}\right]$
$=10000\times 2.50+2000\times 1.50+18000\times 1.00$
$=25000+3000+18000$
$=46000$

Total revenue in market $II$ can be represented as:

$\left[\begin{array}{ccc}6000& 20000& 8000\end{array}\right]\left[\begin{array}{c}2.50\\ 1.50\\ 1.00\end{array}\right]$
$=6000\times 2.50+20000\times 1.50+8000\times 1.00$
$=15000+30000+8000$
$=53000$

So, the total revenue in market $I$ is Rs 46000 and in market $II$ is $Rs.53000$.

(b) The unit cost prices of $x,y$ and $z$ are respectively given as $Rs.2.00$, $Rs.1.00$ and $50$ paise.

So, the total cost prices of all the products in market $I$ can be represented as:

$\left[\begin{array}{ccc}10000& 2000& 18000\end{array}\right]\left[\begin{array}{c}2.00\\ 1.00\\ 0.50\end{array}\right]$
$=10000\times 2.00+2000\times 1.00+18000\times 0.50$
$=20000+2000+9000$
$=31000$

Since, the total revenue in market $I$ is $Rs.46000$.

So, the gross profit in this market is $Rs46000-Rs31000=Rs15000$.

The total cost prices of all the products in market $II$ can be represented as:

$\left[\begin{array}{ccc}6000& 20000& 8000\end{array}\right]\left[\begin{array}{c}2.00\\ 1.00\\ 0.50\end{array}\right]$
$=6000\times 2.00+20000\times 1.00+8000\times 0.50$
$=12000+20000+4000$
$=Rs36000$
Since, the total revenue in market II is $Rs.53000$.

So, the gross profit in this market is $Rs.53000-Rs.36000=Rs.17000$.