Question

Two farmers Ramkrishan and Gurucharan Singh cultivates only three varieties of rice namely Basmati, Permal and Naura. The sale (in rupees) of these varieties of rice by both the farmers in the month of September and October are given by the following matrices A and B.
September Sales (in Rupees)
$\begin{array}{ccccc}& & \text{Basmati}& \text{Permal}& \text{Naura}\end{array}$
$A=\left[\begin{array}{ccc}10,000& 20,000& 30,000\\ 50,000& 30,000& 10,000\end{array}\right]\begin{array}{c}\text{Ramkishan}\\ \text{Gurcharan Singh}\end{array}$
October Sales (in Rupees)
$\begin{array}{ccccc}& & \text{Basmati}& \text{Permal}& \text{Naura}\end{array}$
$B=\left[\begin{array}{ccc}5,000& 10,000& 6,000\\ 20,000& 10,000& 10,000\end{array}\right]\begin{array}{c}\text{Ramkishan}\\ \text{Gurucharan}\end{array}$
$\left(i\right)$ Find the combined sales in September and Octomber for each farmer in each variety.
$\left(ii\right)$ Find the decrease in sales from September to October.
$\left(iii\right)$ If both farmers receive $2$ profit on gross sales, compute the profit for each farmer and for each variety sold in October.

Answer 1

$\left(i\right)$ September Sales (in Rupees)
$\begin{array}{ccccc}& & \text{Basmati}& \text{Permal}& \text{Naura}\end{array}$
$A=\left[\begin{array}{ccc}10,000& 20,000& 30,000\\ 50,000& 30,000& 10,000\end{array}\right]\begin{array}{c}\text{Ramkishan}\\ \text{Gurucharan}\end{array}$
October Sales (in Rupees)
$\begin{array}{ccccc}& & \text{Basmati}& \text{Permal}& \text{Naura}\end{array}$
$B=\left[\begin{array}{ccc}5,000& 10,000& 6,000\\ 20,000& 10,000& 10,000\end{array}\right]\begin{array}{c}\text{Ramkishan}\\ \text{Gurucharan}\end{array}$
Combined sales in September and October for each farmer in each variety will be
$A+B=\left[\begin{array}{ccc}10,000& 20,000& 30,000\\ 50,000& 30,000& 10,000\end{array}\right]+\left[\begin{array}{ccc}5,000& 10,000& 6,000\\ 20,000& 10,000& 10,000\end{array}\right]$
$\Rightarrow A+B=\left[\begin{array}{ccc}15,000& 30,000& 36,000\\ 70,000& 40,000& 20,000\end{array}\right]$
$\therefore$ Combined sales in September and October $=$
$\begin{array}{ccc}\text{Basmati}& \text{Permal}& \text{Naura}\end{array}$
$\left[\begin{array}{ccc}15,000& 30,000& 36,000\\ 70,000& 40,000& 20,000\end{array}\right]\begin{array}{c}\text{Ramkishan}\\ \text{Gurucharan}\end{array}$

$\left(ii\right)$ Given:
September Sales (in Rupees)
$\begin{array}{ccccc}& & \text{Basmati}& \text{Permal}& \text{Naura}\end{array}$
$A=\left[\begin{array}{ccc}10,000& 20,000& 30,000\\ 50,000& 30,000& 10,000\end{array}\right]\begin{array}{c}\text{Ramkishan}\\ \text{Gurucharan}\end{array}$
October Sales (in Rupees)
$\begin{array}{ccccc}& & \text{Basmati}& \text{Permal}& \text{Naura}\end{array}$
$B=\left[\begin{array}{ccc}5,000& 10,000& 6,000\\ 20,000& 10,000& 10,000\end{array}\right]\begin{array}{c}\text{Ramkishan}\\ \text{Gurucharan}\end{array}$
Decrease in sales from September to October $=A-B$
$A-B=\left[\begin{array}{ccc}10,000& 20,000& 30,000\\ 50,000& 30,000& 10,000\end{array}\right]-\left[\begin{array}{ccc}5000& 10,000& 6000\\ 20,000& 10,000& 10,000\end{array}\right]$
$\Rightarrow A-B=\left[\begin{array}{ccc}5000& 10,000& 24,000\\ 30,000& 20,000& 0\end{array}\right]$
$\therefore$ Decrease in sales from September to October $=$
$\begin{array}{ccc}\text{Basmati}& \text{Permal}& \text{Naura}\end{array}$
$\left[\begin{array}{ccc}5000& 10,000& 24,000\\ 30,000& 20,000& 0\end{array}\right]\begin{array}{c}\text{Ramkishan}\\ \text{Gurucharan}\end{array}$

$\left(iii\right)$ Given:
October Sales (in Rupees)
$\begin{array}{ccccc}& & \text{Basmati}& \text{Permal}& \text{Naura}\end{array}$
$B=\left[\begin{array}{ccc}5000& 10,000& 6,000\\ 20,000& 10,000& 10,000\end{array}\right]\begin{array}{c}\text{Ramkishan}\\ \text{Gurucharan}\end{array}$
Both farmers receive $2$ profit on gross sales.
Profit $=2$ (Sales on October)
Profit $=\frac{2}{100}\times B$
Profit $=0.02\times B$
Profit $=0.02\times \left[\begin{array}{ccc}5000& 10,000& 6,000\\ 20,000& 10,000& 10,000\end{array}\right]$
Profit $=\left[\begin{array}{ccc}100& 200& 120\\ 400& 200& 200\end{array}\right]$
Hence,
Profit for each farmer and for each variety sold in october $=$
$\begin{array}{ccc}\text{Basmati}& \text{Permal}& \text{Naura}\end{array}$
$\left[\begin{array}{ccc}100& 200& 120\\ 400& 200& 200\end{array}\right]\begin{array}{c}\text{Ramkishan}\\ \text{Gurucharan Singh}\end{array}$