Question

A typist charges Rs. $145$ for typing $10$ English and $3$ Hindi pages, while charges for typing $3$ English and $10$ Hindi pages are Rs. $180.$ Using matrices, find the charges of typing one English and one Hindi page separately. However typist charged only Rs. $2$ per page from a poor student Shyam for $5$ Hindi pages. How much less was charged from this poor boy? Which values are reflected in this problem?

Answer 1

Let the charge of $1$ english page be $x$ and the cost of $1$ hindi page be $y.$
We have $10x+3y=145$ and $3x+10y=180$ , If we represent this in matrices , we get
$\left(\begin{array}{cc}10& 3\\ 3& 10\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}145\\ 180\end{array}\right)$ , which implies $\left(\begin{array}{c}x\\ y\end{array}\right)={\left(\begin{array}{cc}10& 3\\ 3& 10\end{array}\right)}^{-1}\left(\begin{array}{c}145\\ 180\end{array}\right)=\frac{1}{91}\left(\begin{array}{cc}10& -3\\ -3& 10\end{array}\right)\left(\begin{array}{c}145\\ 180\end{array}\right)=\frac{1}{91}\left(\begin{array}{c}1450-540\\ -435+1800\end{array}\right)=\frac{1}{91}\left(\begin{array}{c}910\\ 1365\end{array}\right)=\left(\begin{array}{c}10\\ 15\end{array}\right)$
Therefore $x=10$ and $y=15$ , $x+y=25$
For Shyam instead of $15$ Rs , the typist charged $2$ rs . So he charged Rs$13$ less for poor boy per page , for $5$ pages the typist charged $13\times 5=65$ Rs less for poor boy.