Two farmers,Ramkishan and Gurcharan Singh,cultivate only three varieties of rice: Basmati,Permal,and Naura. The sales (in Rupees) of these varieties of rice by both farmers in the months of September and October are given by the following matrices $A$ and $B$.
September Sales (in Rupees)
$A = \begin{bmatrix} \text{Basmati} & \text{Permal} & \text{Naura} \\ 10,000 & 20,000 & 30,000 \\ 50,000 & 30,000 & 10,000 \end{bmatrix} \begin{matrix} \\ \text{Ramkishan} \\ \text{Gurcharan Singh} \end{matrix}$
October Sales (in Rupees)
$B = \begin{bmatrix} \text{Basmati} & \text{Permal} & \text{Naura} \\ 5,000 & 10,000 & 6,000 \\ 20,000 & 10,000 & 10,000 \end{bmatrix} \begin{matrix} \\ \text{Ramkishan} \\ \text{Gurcharan Singh} \end{matrix}$
$(i)$ Find the combined sales in September and October for each farmer in each variety.
$(ii)$ Find the decrease in sales from September to October.
$(iii)$ If both farmers receive $2\%$ profit on gross sales,compute the profit for each farmer and for each variety sold in October.
September Sales (in Rupees)
$A = \begin{bmatrix} \text{Basmati} & \text{Permal} & \text{Naura} \\ 10,000 & 20,000 & 30,000 \\ 50,000 & 30,000 & 10,000 \end{bmatrix} \begin{matrix} \\ \text{Ramkishan} \\ \text{Gurcharan Singh} \end{matrix}$
October Sales (in Rupees)
$B = \begin{bmatrix} \text{Basmati} & \text{Permal} & \text{Naura} \\ 5,000 & 10,000 & 6,000 \\ 20,000 & 10,000 & 10,000 \end{bmatrix} \begin{matrix} \\ \text{Ramkishan} \\ \text{Gurcharan Singh} \end{matrix}$
$(i)$ Find the combined sales in September and October for each farmer in each variety.
$(ii)$ Find the decrease in sales from September to October.
$(iii)$ If both farmers receive $2\%$ profit on gross sales,compute the profit for each farmer and for each variety sold in October.
(N/A) $(i)$ The combined sales in September and October for each farmer in each variety is given by the sum of matrices $A$ and $B$:
$A+B = \begin{bmatrix} \text{Basmati} & \text{Permal} & \text{Naura} \\ 15,000 & 30,000 & 36,000 \\ 70,000 & 40,000 & 20,000 \end{bmatrix} \begin{matrix} \\ \text{Ramkishan} \\ \text{Gurcharan Singh} \end{matrix}$
$(ii)$ The decrease in sales from September to October is given by the difference of matrices $A$ and $B$:
$A-B = \begin{bmatrix} \text{Basmati} & \text{Permal} & \text{Naura} \\ 5,000 & 10,000 & 24,000 \\ 30,000 & 20,000 & 0 \end{bmatrix} \begin{matrix} \\ \text{Ramkishan} \\ \text{Gurcharan Singh} \end{matrix}$
$(iii)$ The profit is $2\%$ of the October sales $(B)$:
$0.02 \times B = 0.02 \times \begin{bmatrix} 5,000 & 10,000 & 6,000 \\ 20,000 & 10,000 & 10,000 \end{bmatrix} = \begin{bmatrix} 100 & 200 & 120 \\ 400 & 200 & 200 \end{bmatrix}$
Thus,in October,Ramkishan receives Rs. $100$,Rs. $200$,and Rs. $120$ as profit for the three varieties,and Gurcharan Singh receives Rs. $400$,Rs. $200$,and Rs. $200$ as profit for the three varieties,respectively.
$A+B = \begin{bmatrix} \text{Basmati} & \text{Permal} & \text{Naura} \\ 15,000 & 30,000 & 36,000 \\ 70,000 & 40,000 & 20,000 \end{bmatrix} \begin{matrix} \\ \text{Ramkishan} \\ \text{Gurcharan Singh} \end{matrix}$
$(ii)$ The decrease in sales from September to October is given by the difference of matrices $A$ and $B$:
$A-B = \begin{bmatrix} \text{Basmati} & \text{Permal} & \text{Naura} \\ 5,000 & 10,000 & 24,000 \\ 30,000 & 20,000 & 0 \end{bmatrix} \begin{matrix} \\ \text{Ramkishan} \\ \text{Gurcharan Singh} \end{matrix}$
$(iii)$ The profit is $2\%$ of the October sales $(B)$:
$0.02 \times B = 0.02 \times \begin{bmatrix} 5,000 & 10,000 & 6,000 \\ 20,000 & 10,000 & 10,000 \end{bmatrix} = \begin{bmatrix} 100 & 200 & 120 \\ 400 & 200 & 200 \end{bmatrix}$
Thus,in October,Ramkishan receives Rs. $100$,Rs. $200$,and Rs. $120$ as profit for the three varieties,and Gurcharan Singh receives Rs. $400$,Rs. $200$,and Rs. $200$ as profit for the three varieties,respectively.
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