Question

To raise money for an orphanage, students of three schools A, B, and C organized an exhibition in their locality, where they sold old paper bags, scrapbooks, and pastel sheets made by them using recycled paper at the rate of Rupees 20, 15 and 10 per unit respectively. School A sold 25 paper bags, 10 scrapbooks, and 30 pastel sheets. School B sold 20 paper bags, 15 scrapbooks, and 30 pastel sheets while School C sold 25 paper bags, 18 scrapbooks, and 35 pastel sheets. Using matrices, find the total amount raised by each school.

Answer 1

Hint: We represent the number of units sold by each school in a 3 $\times $ 3 matrix and the cost of individual units in a 3 $\times $ 1 matrix. The resulting product of these matrices will give us the total amount raised by each school.

Complete step-by-step answer:
Let us denote the number of paper bags, scrapbooks, and pastel sheets made by school A by ${{x}_{A}},{{y}_{A}},{{z}_{A}}$ respectively. We know that ${{x}_{A}}=25,{{y}_{A}}=10,{{z}_{A}}=30$ .
Let us denote the number of paper bags, scrapbooks, and pastel sheets made by school B by ${{x}_{B}},{{y}_{B}},{{z}_{B}}$ respectively. We know that ${{x}_{B}}=20,{{y}_{B}}=15,{{z}_{B}}=30$ .
Let us denote the number of paper bags, scrapbooks, and pastel sheets made by school C by ${{x}_{C,}}{{y}_{C}},{{z}_{C}}$ respectively. We know that${{x}_{C}}=25,{{y}_{C}}=18,{{z}_{C}}=35$ .
We can represent this by a 3 $\times $ 3 matrix M such that
M = $\left( \begin{matrix}
   {{x}_{A}} & {{y}_{A}} & {{z}_{A}} \\
   {{x}_{B}} & {{y}_{B}} & {{z}_{B}} \\
   {{x}_{C}} & {{y}_{C}} & {{z}_{C}} \\
\end{matrix} \right)=\left( \begin{matrix}
   25 & 10 & 30 \\
   20 & 15 & 30 \\
   25 & 18 & 35 \\
\end{matrix} \right)$
Now we will write the cost of paper bags, scrapbooks and pastel sheets, i.e. Rupees 20, 15, 10 in 3 $\times $ 1 matrix N such that
N = $\left( \begin{matrix}
   20 \\
   15 \\
   10 \\
\end{matrix} \right)$
We know that to find the total amount raised by each school, we must multiply the number of items sold by their price. This can be done using matrices.
Thus, let the 3 $\times $ 1 matrix P denote the total amount raised. We know that P is the product of matrices M and N.
For any two matrices A and B given by
 $\begin{align}
  & A=\left( \begin{matrix}
   {{a}_{11}} & {{a}_{12}} & {{a}_{13}} \\
   {{a}_{21}} & {{a}_{22}} & {{a}_{23}} \\
   {{a}_{31}} & {{a}_{32}} & {{a}_{33}} \\
\end{matrix} \right)\text{ and B=}\left( \begin{matrix}
   p \\
   q \\
   r \\
\end{matrix} \right)\text{ } \\
 & \\
 & \text{A}\times \text{B = }\left( \begin{matrix}
   {{a}_{11}}\times p+{{a}_{12}}\times q+{{a}_{13}}\times r \\
   {{a}_{21}}\times p+{{a}_{22}}\times q+{{a}_{23}}\times r \\
   {{a}_{31}}\times p+{{a}_{32}}\times q+{{a}_{33}}\times r \\
\end{matrix} \right) \\
\end{align}$

 Thus, M $\times $ N = P
$\begin{align}
  & \Rightarrow \left( \begin{matrix}
   25 & 10 & 30 \\
   20 & 15 & 30 \\
   25 & 18 & 35 \\
\end{matrix} \right)\times \left( \begin{matrix}
   20 \\
   15 \\
   10 \\
\end{matrix} \right)=P \\
 & \\
 & \Rightarrow \left( \begin{matrix}
   25\times 20+10\times 15+30\times 10 \\
   20\times 20+15\times 15+30\times 10 \\
   25\times 20+18\times 15+35\times 10 \\
\end{matrix} \right)=P \\
 & \\
 & \Rightarrow \left( \begin{matrix}
   500+150+300 \\
   400+225+300 \\
   500+270+350 \\
\end{matrix} \right)=P \\
 & \\
 & \Rightarrow \left( \begin{matrix}
   950 \\
   925 \\
   1120 \\
\end{matrix} \right)=P \\
\end{align}$
Hence, the amount raised by school A is Rupees 950, school B is Rupees 925 and school C is Rupees 1120.

Note: There is a chance of making a mistake while writing matrix M as students may interchange the rows and columns which can lead to the wrong answer. There is also the possibility of making a calculation mistake during matrix multiplication.