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Question
A coaching institute of English (subject) conducts classes in two batches I and II and fees for rich and poor children are different. In batch I, it has 20 poor and 5 rich children and total monthly collection is Rs 9,000, whereas in batch II, it has 5 poor and 25 rich children and total monthly collection is Rs 26,000. Using matrix method, find monthly fees paid by each child of two types. What values the coaching institute is inculcating in the society?
Solution
Let the monthly fees paid by poor and rich children be Rs x and Rs y, respectively.
For batch I:
20x + 5y = 9000 .....(1)
For batch II:
5x + 25y = 26000 .....(2)
The system of equations can be written as
\[AX = B\]
\[\begin{matrix}20 & 5 \\ 5 & 25\end{matrix}\binom{x}{y} = \binom{9000}{26000}\]
\[\text { Here }, A = \begin{matrix}20 & 5 \\ 5 & 25\end{matrix}, X = \binom{x}{y} \text { and } B = \binom{9000}{26000}\]
\[\left| A \right| = \begin{vmatrix}20 & 5 \\ 5 & 25\end{vmatrix} = 500 - 25 = 475 \neq 0\]
\[C_{11} = \left( - 1 \right)^{1 + 1} \left( 25 \right) = 25, C_{12} = \left( - 1 \right)^{1 + 2} \left( 5 \right) = - 5\]
\[ C_{21} = \left( - 1 \right)^{2 + 1} \left( 5 \right) = - 5, C_{22} = \left( - 1 \right)^{2 + 2} \left( 20 \right) = 20\]
\[\text { Adj }A = \begin{bmatrix}25 & - 5 \\ - 5 & 20\end{bmatrix}^T = \begin{bmatrix}25 & - 5 \\ - 5 & 20\end{bmatrix}\]
\[ \therefore A^{- 1} = \frac{AdjA}{\left| A \right|} = \frac{1}{475}\begin{bmatrix}25 & - 5 \\ - 5 & 20\end{bmatrix}\]
So, the given system has a unique solution given by X = A−1B.
\[\therefore X = A^{- 1} B\]
\[ \Rightarrow \binom{x}{y} = \frac{1}{475}\begin{bmatrix}25 & - 5 \\ - 5 & 20\end{bmatrix}\binom{9000}{26000}\]
\[ \Rightarrow \binom{x}{y} = \frac{1}{475}\binom{95000}{475000}\]
\[ \Rightarrow \binom{x}{y} = \binom{200}{1000}\]
\[ \Rightarrow x = 200, y = 1000\]
Hence, the monthly fees paid by each poor child is Rs 200 and the monthly fees paid by each rich child is Rs 1000.
By offering discount to the poor children, the coaching institute offers an unbiased chance for the development and enhancement of the weaker section of our society.
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