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, the 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

Hint: To solve this problem, we should know the basics of solving algebraic equations. We first express the word problem in the form of two variables (say x and y). We then use the technique of matrices to solve this problem. We express the equations in the form of AX = B and them solve for X by taking inverse of A and then performing product of the two matrices ( ${{A}^{-1}}$ and B) to get the values of x and y.

Complete step-by-step answer:
We firstly start by writing the conditions in terms of x and y (x denotes charge for typing English page and y denotes charge for typing Hindi page). We know that a typist charges Rs. 145 for typing 10 English and 3 Hindi pages. Thus, we have,
10x + 3y = 145 -- (1)
Further, from the second condition, for 3 English and 10 Hindi pages, the charge is Rs 180. We have,
3x + 10y = 180 -- (2)
In this case, we first express the equations in the form of AX = B. where,
A = $\left[ \begin{aligned} & {{\text{a}}_{11}}\text{ }{{\text{a}}_{12}} \\\ & {{\text{a}}_{21}}\text{ }{{\text{a}}_{22}} \\\ \end{aligned} \right]$ = $\left[ \begin{aligned} & 10\text{ 3} \\\ & \text{3 10} \\\ \end{aligned} \right]$ --(3)
Also,
X = $\left[ \begin{aligned} & x \\\ & y \\\ \end{aligned} \right]$ -- (4)
B = $\left[ \begin{aligned} & 145 \\\ & 180 \\\ \end{aligned} \right]$ -- (5)
Now, X = ${{A}^{-1}}$ B -- (A)
Now, finding ${{A}^{-1}}$, we have,
${{A}^{-1}}$ = $\dfrac{adj(A)}{|A|}$ -- (B)
(where, adj(A) is adjoint of matrix A and |A| is the determinant of A)

For 2 $\times$ 2 matrix ,we have,
adj(A) = $\left[ \begin{aligned} & {{\text{a}}_{\text{22}}}\text{ -}{{\text{a}}_{\text{12}}} \\\ & \text{-}{{\text{a}}_{\text{21}}}\text{ }{{\text{a}}_{\text{11}}} \\\ \end{aligned} \right]$
For, A = $\left[ \begin{aligned} & {{\text{a}}_{11}}\text{ }{{\text{a}}_{12}}\text{ } \\\ & {{\text{a}}_{21}}\text{ }{{\text{a}}_{22}} \\\ \end{aligned} \right]$, |A| = ${{a}_{11}}{{a}_{22}}-{{a}_{12}}{{a}_{21}}$ -- (C)
Now, adjoint of A (by substituting the values from (1)) is,
$\left[ \begin{aligned} & \text{10 -3} \\\ & \text{-3 10} \\\ \end{aligned} \right]$
Also, |A| = 100 – 9 = 91 (from substituting the respective values in formula of (C))
X = ${{A}^{-1}}$ B
X = $\dfrac{adj(A)}{|A|}$B
X = $\dfrac {1}{91} \cdot \left[ \begin{aligned} & \text{10 -3} \\\ & \text{-3 10} \\\ \end{aligned} \right] \cdot$ $\left[ \begin{aligned} & 145 \\\ & 180 \\\ \end{aligned} \right]$
X = $\dfrac{1}{91}\left[ \begin{aligned} & 10\times 145-3\times 180 \\\ & -3\times 145+10\times 180 \\\ \end{aligned} \right]$
X = $\dfrac{1}{91}\left[ \begin{aligned} & 910 \\\ & 1365 \\\ \end{aligned} \right]$
X = $\left[ \begin{aligned} & 10 \\\ & 15 \\\ \end{aligned} \right]$ = $\left[ \begin{aligned} & x \\\ & y \\\ \end{aligned} \right]$
Thus, x = 10 and y = 15.
Thus, charges of typing one English page are Rs 10 and one Hindi page is Rs 15.
Since, the typist charged Rs. 2 per page from a poor student Shyam for 5 Hindi pages. Thus, total charge was $2\times 5$ = Rs 10 (for 5 pages). Normally, the charge was Rs 15 per Hindi page separately, for 5 pages, it would be $15\times 5$ = Rs 75. Thus, Rs 65 less was charged for the poor boy. The values of humanity and kindness are reflected.

Note: An alternative way to solve the problem after converting the word problem into an algebraic equation in terms of x and y, we can solve these equations manually. That is, we can substitute the value of x (in terms of y) from one of the equations into the second equation to get the value of y and then find the value of x from the first equation.