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 Rs. $x$ be the cost per page for typing English and
Rs. $y$ be the cost per page for typing Hindi.

Now, using statement of the question, we have

$10x+3y=145$

$3x+10y=180$

The given equations can be written as:

$AX=B$.

where $A=\left[\begin{array}{cc}10& 3\\ 3& 10\end{array}\right],X=\left[\begin{array}{c}x\\ y\end{array}\right],B=\left[\begin{array}{c}145\\ 180\end{array}\right]$

Now, $|A|=100-9=91\ne 0$

$Adj.A=\left[\begin{array}{cc}10& -3\\ -3& 10\end{array}\right]$

$A^{-1}=\frac{1}{|A|}adj.A=\frac{1}{91}\left[\begin{array}{cc}10& -3\\ -3& 10\end{array}\right]$

$\therefore X=A^{-1}B=\frac{1}{91}\left[\begin{array}{cc}10& -3\\ -3& 10\end{array}\right]$ $\left[\begin{array}{c}145\\ 180\end{array}\right]$

$=\frac{1}{91}\left[\begin{array}{c}1450-540\\ -435+1800\end{array}\right]$ $=\frac{1}{91}\left[\begin{array}{c}910\\ 1365\end{array}\right]$

i.e., $\left[\begin{array}{c}x\\ y\end{array}\right]$ = $\left[\begin{array}{c}10\\ 15\end{array}\right]$, thus, $x=10;y=15$

So, charges for typing one English page is Rs. 10 and charges for typing one Hindi page is Rs. 15.

Total pages typist typed for Shyam = 5

Actual charge for Hindi pages = Rs. 15 $\times$ 5 = Rs.75

But, amount charged by typist = Rs. 2 $\times$ 5 = Rs. 10

Hence, discount given to Shyam = Rs. (75-10) = Rs. 65

Typist showing humanity and helping nature.