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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.


Solution

(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:

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

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

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

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:

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

Since, the total revenue in market $$I$$ is $$Rs. 46000$$.
So, the gross profit in this market is $$Rs 46000-Rs 31000 =Rs 15000$$.

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

$$\displaystyle \left[ \begin{matrix} 6000 & 20000 & 8000 \end{matrix} \right] \left[ \begin{matrix} 2.00 \\ 1.00 \\ 0.50 \end{matrix} \right] $$
$$\displaystyle =6000\times 2.00+20000\times 1.00+8000\times 0.50$$
$$\displaystyle =12000+20000+4000$$
$$\displaystyle =Rs36000$$
Since, the total revenue in market II is $$Rs. 53000$$.
So, the gross profit in this market is $$Rs.53000 - Rs.36000 =Rs.17000$$.

Mathematics
NCERT
Standard XII

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