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प्रश्न
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?
उत्तर
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-1B
⇒ IX = A-1B
⇒ X = A-1B
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.
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