Question

There are two book shops known by Suresh and Ganesh book shops. Their sales for books in three subjects- Physics, Chemistry and Mathematics for two months July and August 2009 are given by two matrices A and B.
July Sales (in rupees)
$\begin{align}
  & \ \ \ \begin{matrix}
   Physics & \ \ Chemistry & Mathematics \\
\end{matrix} \\
 & A=\left[ \begin{matrix}
   5600 \\
   6650 \\
\end{matrix}\ \ \ \ \ \ \ \ \ \ \begin{matrix}
   6750 \\
   7055 \\
\end{matrix}\ \ \ \ \ \ \ \ \ \ \ \begin{matrix}
   8500 \\
   8905 \\
\end{matrix}\ \right]\begin{matrix}
   Suresh \\
   Ganesh \\
\end{matrix} \\
\end{align}$
August Sales (in rupees)
$\begin{align}
  & \ \ \ \ \ \begin{matrix}
   Physics & \ \ Chemistry & Mathematics \\
\end{matrix} \\
 & B=\left[ \begin{matrix}
   6650 \\
   7000 \\
\end{matrix}\ \ \ \ \ \ \ \ \ \ \begin{matrix}
   7055 \\
   7500 \\
\end{matrix}\ \ \ \ \ \ \ \ \ \ \ \begin{matrix}
   8905 \\
   10200 \\
\end{matrix}\ \right]\begin{matrix}
   Suresh \\
   Ganesh \\
\end{matrix} \\
\end{align}$
Then
(i) find the increase in sales in rupees from July to August 09
(ii) If both the book shops got 10% profit in the month of August 09, find the profit for each book seller in each subject in that month.

Answer 1

Hint: i) We start solving this question by first finding the total sales in July by adding all the sales of both shops in all the subjects. Then similarly we find the total sales in August by adding all the sales in all subjects in both shops. Then we subtract the total sales in July from the total sales in August to find the increase in sales.
ii) We solve this question by finding the relation between S.P and C.P using the formula for profit percentage $\dfrac{S.P-C.P}{C.P}\times 100$ and then use the formula $\text{Profit}=S.P-C.P$ to find the profit in terms of S.P. Then we substitute the sales in August in the relation to find the profits in August.

Complete step by step answer:
We are given that the sales in July are given by
$A=\left[ \begin{matrix}
   5600 & 6750 & 8500 \\
   6650 & 7055 & 8905 \\
\end{matrix} \right]$
We are also given that sales in August are given by
$B=\left[ \begin{matrix}
   6650 & 7055 & 8905 \\
   7000 & 7500 & 10200 \\
\end{matrix} \right]$
i) We need to find the increase in sales from July to August.
First, let us find the total sales in July
To find that value we need to add all the sales in July. So, adding them we get
$\begin{align}
  & \Rightarrow Total\ Sales=5600+6750+8500+6650+7055+8905 \\
 & \Rightarrow Total\ Sales=43460 \\
\end{align}$
Now let us find the total sales in August.
So, let us add all the sales in August. Then we get,
$\begin{align}
  & \Rightarrow Total\ Sales=6650+7055+8905+7000+7500+10200 \\
 & \Rightarrow Total\ Sales=47310 \\
\end{align}$
As we need to find the increase in the sales from July to August, let us subtract the total sales in July from the total sales in August.
$\begin{align}
  & \Rightarrow Sales\ from\ August-Sales\ from\ July \\
 & \Rightarrow 47310-43460 \\
 & \Rightarrow 3850 \\
\end{align}$
So, the increase in the sales from July to August is Rs.3445.
Hence the answer is Rs.3850.

ii) We are given that the book shops had 10% profit in the month August 09.
Let us consider the formula for percentage of profit.
$\dfrac{S.P-C.P}{C.P}\times 100$
So, using that formula we have,
$\begin{align}
  & \Rightarrow \dfrac{S.P-C.P}{C.P}\times 100=10 \\
 & \Rightarrow \dfrac{S.P-C.P}{C.P}=\dfrac{1}{10} \\
 & \Rightarrow 10S.P-10C.P=C.P \\
 & \Rightarrow 11C.P=10S.P \\
 & \Rightarrow C.P=\dfrac{10}{11}S.P \\
\end{align}$
Now, let us consider the formula for Profit,
$\text{Profit}=S.P-C.P$
Using that formula, we get,
$\begin{align}
  & \Rightarrow \text{Profit}=S.P-\dfrac{10}{11}S.P \\
 & \Rightarrow \text{Profit}=\dfrac{1}{11}S.P \\
\end{align}$
As the sales is the selling price in this question, we can give the profit in August as,
$\begin{align}
  & \Rightarrow \text{Profit}=\dfrac{1}{11}B \\
 & \Rightarrow \text{Profit}=\dfrac{1}{11}\left[ \begin{matrix}
   6650 & 7055 & 8905 \\
   7000 & 7500 & 10200 \\
\end{matrix} \right] \\
 & \\
 & \Rightarrow \text{Profit}=\left[ \begin{matrix}
   \dfrac{6650}{11} & \dfrac{7055}{11} & \dfrac{8905}{11} \\
   \dfrac{7000}{11} & \dfrac{7500}{11} & \dfrac{10200}{11} \\
\end{matrix} \right] \\
\end{align}$
Hence answer is $\left[ \begin{matrix}
   \dfrac{6650}{11} & \dfrac{7055}{11} & \dfrac{8905}{11} \\
   \dfrac{7000}{11} & \dfrac{7500}{11} & \dfrac{10200}{11} \\
\end{matrix} \right]$.

Note:
i) One can also solve this question by first subtracting the matrix of sales in July from the matrix of sales in August and then add the values in the obtained matrix. Then we get,
$\begin{align}
  & \Rightarrow B-A=\left[ \begin{matrix}
   6650 & 7055 & 8905 \\
   7000 & 7500 & 10200 \\
\end{matrix} \right]-\left[ \begin{matrix}
   5600 & 6750 & 8500 \\
   6650 & 7055 & 8905 \\
\end{matrix} \right] \\
 & \Rightarrow B-A=\left[ \begin{matrix}
   1050 & 305 & 405 \\
   350 & 445 & 1295 \\
\end{matrix} \right] \\
\end{align}$
Adding them we get,
$\begin{align}
  & \Rightarrow Increase\ in\ Sales=1050+305+405+350+445+1295 \\
 & \Rightarrow Increase\ in\ Sales=3850 \\
\end{align}$
Hence the answer is Rs.3850.
ii) The common mistake one makes while solving this problem is one night make a mistake by taking the formula for percentage of profit as $\dfrac{C.P-S.P}{S.P}\times 100$. But it is the formula for loss percentage not for the profit percentage.