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# A manufacturer produces three products $x, y, z$ which he sells in two markets. Annual sales are indicated below: $\begin{array} { c c } \textbf{Market} & \textbf{Products} \\ I & 10,000 \quad 2,000 \quad 18,000 \\ II & 6,000 \quad 20,000 \quad 8,000 \end{array}$ If the unit costs of the above three commodities are Rs 2.00, Rs 1.00 and 50 paise respectively. Find the gross profit.

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• If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B: $\begin{bmatrix}AB\end{bmatrix}_{i,j} = A_{i,1}B_{1,j} + A_{i,2}B_{2,j} + A_{i,3}B_{3,j} ... A_{i,n}B_{n,j}$
Step 1: (b) Given
The unit cost price of commodities x,y and z are given respectively as Rs 2.00,Rs 1.00 and 0.50 paise. Thus cost price of each market can be obtained by
$\begin{bmatrix}10000 & 2000 & 18000\\6000 & 20000 & 8000\end{bmatrix}\begin{bmatrix}2.00\\1.00\\0.50\end{bmatrix}$
$\begin{bmatrix}10000\times 2+ 2000\times 1 + 18000\times 0.50\\6000\times 2+20000\times 1+8000\times 0.5 \end{bmatrix}$
$\begin{bmatrix}20000+2000+9000\\12000+20000+4000 \end{bmatrix}$
$\begin{bmatrix}31000\\36000\end{bmatrix}$
Total cost price =31000+36000=67000.
Step 2: From the subdivision (i) we have the revenue cost as 99000
Gross profit =Revenue cost - Total cost price
$\;\;\quad\qquad\;\;=$99000-67000
$\;\;\quad\qquad\;\;=$32000
Gross profit=Rs 32000.
edited Mar 19, 2013