A manufacturer produces three products x, y, z which he sells in two markets.
Question:

A manufacturer produces three products xyz which he sells in two markets.

Annual sales are indicated below:

(a) If unit sale prices of xy and 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.

Solution:

(a) The unit sale prices of xy, and are respectively given as Rs 2.50, Rs 1.50, and Rs 1.00.

Consequently, the total revenue in market I can be represented in the form of a matrix as:

$\left[\begin{array}{lll}10000 & 2000 & 18000\end{array}\right]\left[\begin{array}{l}2.50 \\ 1.50 \\ 1.00\end{array}\right]$

$=10000 \times 2.50+2000 \times 1.50+18000 \times 1.00$

$=25000+3000+18000$

$=46000$

The total revenue in market II can be represented in the form of a matrix as:

$\left[\begin{array}{lll}6000 & 20000 & 8000\end{array}\right]\left[\begin{array}{l}2.50 \\ 1.50 \\ 1.00\end{array}\right]$

$=6000 \times 2.50+20000 \times 1.50+8000 \times 1.00$

$=15000+30000+8000$

$=53000$

Therefore, the total revenue in market isRs 46000 and the same in market II isRs 53000.

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

Consequently, the total cost prices of all the products in market I can be represented in the form of a matrix as:

$\left[\begin{array}{lll}10000 & 2000 & 18000\end{array}\right]$$\left[\begin{array}{l}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 isRs 46000, the gross profit in this marketis (Rs 46000 − Rs 31000) Rs 15000.

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

$\left[\begin{array}{lll}6000 & 20000 & 8000\end{array}\right]\left[\begin{array}{l}2.00 \\ 1.00 \\ 0.50\end{array}\right]$

$=6000 \times 2.00+20000 \times 1.00+8000 \times 0.50$

$=12000+20000+4000$

$=$ Rs 36000

Since the total revenue in market II isRs 53000, the gross profit in this market is (Rs 53000 − Rs 36000) Rs 17000.