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?

#### Solution

Let charges for typing one English page be Rs. x.

Let charges for typing one Hindi page be Rs.y.

Thus from the given statements, we have,

10x+3y=145

3x+10y=180

Thus the above system can be written as,

`[(10,3),(3,10)][(x),(y)]=[(145),(180)]`

⇒ AX = B, where, `A=[(10,3),(3,10)],x=[(x),(y)] " and " B = [(145),(180)]`

Multiply A^{-1} on both the sides, we have,

A^{-1 }x^{ }AX = A^{-1}B

⇒ IX = A^{-1}B

⇒ X = A^{-1}B

Thus, we need to find the inverse of the matrix A.

We know that, if `P=[(a,b),(c,d)] " then " P^(-1) = 1/(ad-bc)[(d,-b),(-c,a)]`

Thus, `A^(-1)=1/(10xx10-3xx3)[(10,-3),(-3,10)]`

`= 1/(100-9)[(10,-3),(-3,10)]`

`=1/91[(10,-3),(-3,10)]`

Therefore, `X=1/91[(10,-3),(-3,10)][(145),(180)]`

`=1/91[(10xx145-3xx180),(-3xx145+10xx180)]`

`=1/91[(910),(1365)]`

`=[(10),(15)]`

`=>[(x),(y)][(10),(15)]`

⇒ x = 10 and y=15

Amount taken from Shyam = 2 × 5 = Rs.10

Actual rate = 15 × 5 =75

Difference amount = Rs.75 – Rs.10 = Rs.65

Rs. 65 less was charged from the poor boy Shyam.

Humanity and sympathy are reflected in this problem.