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 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: We write the given information assuming the number of prices per page of each subject as different variables in the form of linear equations. Now we convert the linear equations into matrix form. Calculate the inverse of the matrix formed by the number of pages of each language in each case and use it to calculate the price of each page.
* Adjoint of a matrix \[M = \left[ {\begin{array}{*{20}{c}}
  a&b \\
  c&d
\end{array}} \right]\] is given by \[adjM = \left[ {\begin{array}{*{20}{c}}
  d&{ - b} \\
  { - c}&a
\end{array}} \right]\]
* Inverse of a matrix A is given by \[{A^{ - 1}} = \dfrac{1}{{\left| A \right|}}adjA\]

Complete step-by-step answer:
Let us assume the charge for typing one English page be Rs.x
Charge for typing one Hindi page be Rs.y
We can form an equation using the charge for one page of language and their respective number of pages.
We are given that typist charges Rs.145 for typing 10 English and 3 Hindi pages
 \[ \Rightarrow 10x + 3y = 145\] … (1)
Also, we are given that typist charges Rs.180 for 3 English and 10 Hindi pages
 \[ \Rightarrow 3x + 10y = 180\] … (2)
We can write the given system of linear equations in two variables using matrices. Let the system of linear equations be denoted by \[AX = B\] , where A has a number of pages of language, X has price per page for each language and B has total price paid by the child.
So we write the matrix form of the linear equation as \[AX = B\] i.e.
 \[\left[ {\begin{array}{*{20}{c}}
  {10}&3 \\
  3&{10}
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
  x \\
  y
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  {145} \\
  {180}
\end{array}} \right]\]
Here \[A = \left[ {\begin{array}{*{20}{c}}
  {10}&3 \\
  3&{10}
\end{array}} \right];X = \left[ {\begin{array}{*{20}{c}}
  x \\
  y
\end{array}} \right];B = \left[ {\begin{array}{*{20}{c}}
  {145} \\
  {180}
\end{array}} \right]\]
We know that \[AX = B\]
Take inverse of A on both sides of the equation
 \[ \Rightarrow {A^{ - 1}}AX = {A^{ - 1}}B\]
We know that \[{A^{ - 1}}A = I\]
 \[ \Rightarrow IX = {A^{ - 1}}B\]
 \[ \Rightarrow X = {A^{ - 1}}B\] … (3)
Now we calculate the inverse of A
We know that inverse of a matrix A is given by \[{A^{ - 1}} = \dfrac{1}{{\left| A \right|}}adjA\]
We calculate the adjoint, \[adjA = \left[ {\begin{array}{*{20}{c}}
  {10}&{ - 3} \\
  { - 3}&{10}
\end{array}} \right]\]
And determinant is \[\left| A \right| = 100 - 9\]
i.e.
 \[ \Rightarrow \left| A \right| = 91\]
Since we have \[{A^{ - 1}} = \dfrac{1}{{\left| A \right|}}adjA\]
 \[ \Rightarrow {A^{ - 1}} = \dfrac{1}{{91}}\left[ {\begin{array}{*{20}{c}}
  {10}&{ - 3} \\
  { - 3}&{10}
\end{array}} \right]\] … (4)
Substitute the value of inverse of A from equation (4) in equation (3)
 \[ \Rightarrow X = \dfrac{1}{{91}}\left[ {\begin{array}{*{20}{c}}
  {10}&{ - 3} \\
  { - 3}&{10}
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
  {145} \\
  {180}
\end{array}} \right]\]
We multiply the matrices on right hand side of situation
 \[ \Rightarrow X = \dfrac{1}{{91}}\left[ {\begin{array}{*{20}{c}}
  {1450 - 540} \\
  { - 435 + 1800}
\end{array}} \right]\]
 \[ \Rightarrow X = \dfrac{1}{{91}}\left[ {\begin{array}{*{20}{c}}
  {910} \\
  {1365}
\end{array}} \right]\]
Multiply term outside the matrix to terms inside the matrix
 \[ \Rightarrow X = \left[ {\begin{array}{*{20}{c}}
  {\dfrac{{910}}{{91}}} \\
  {\dfrac{{1365}}{{91}}}
\end{array}} \right]\]
 \[ \Rightarrow X = \left[ {\begin{array}{*{20}{c}}
  {10} \\
  {15}
\end{array}} \right]\]
 \[\therefore \] The value of x is 10 and y is 15.
 \[\therefore \] Price for typing 1 English page is Rs.10 and 1 Hindi page is Rs.15
Now we are given that typist charges only Rs.2 per page for 5 Hindi pages from Shyam
Total amount paid by shyam for 5 Hindi pages i.e. Rs.10
We know initially the typist charges Rs.15 per Hindi page, so Shyam would’ve paid Rs.75
The amount he was charged less \[ = 75 - 10\] i.e. Rs.65
This shows the humanity of the typist as he helped the poor boy by charging him less money.

Note:
Many students make mistakes while writing the adjoint of the matrix A as they get confused where to assume the negative sign and where not to. Keep in mind we move alternatively with positive and negative signs.