1
Question
A finance company has offices located in every division, every district and every taluka in a certain state in India. Assume that there are five divisions, thirty districts and 200 talukas in the state. Each office has one head clerk, one cashier, one clerk, one typist and one peon. A district office has in addition, on clerk and one peon. The basic monthly salaries are as follows:
Office Superitendent Rs. 500, Head Clerk Rs. 200, Cashier Rs.175, clerks and typist Rs. 150 and peon Rs. 100. Using Matrix notation and find
(i) The total number of posts of each kind in all the offices taken together
(ii) the total basic monthly salary bill of each kind of office
(iii) the total basic monthly bill of all the offices taken together
A finance company has offices located in every division, every district and every taluka in a certain state in India. Assume that there are five divisions, thirty districts and 200 talukas in the state. Each office has one head clerk, one cashier, one clerk, one typist and one peon. A district office has in addition, on clerk and one peon. The basic monthly salaries are as follows:
Office Superitendent Rs. 500, Head Clerk Rs. 200, Cashier Rs.175, clerks and typist Rs. 150 and peon Rs. 100. Using Matrix notation and find
(i) The total number of posts of each kind in all the offices taken together
(ii) the total basic monthly salary bill of each kind of office
(iii) the total basic monthly bill of all the offices taken together
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Solution
The correct option is A (i)Required number of posts in all the offices taken together are 5 office superintendents, 235 head clerks, 235 cashiers, 275 clerks, 5 typists and 270 peons.(ii)Total basic monthly salary bill of each divisional, district and taluka offices are Rs. 1675, Rs. 875 and Rs. 625 respectively(iii)Total basic monthly salary bill of all the offices taken together is Rs. 159,625
Let us use the symbols Dir,Div,Tal for division, district, taluka respectively and O,H,C,Cl,T and P for office superintendent, head clerk, cashier, clerk, typist and peon respectively.Then the number of offices can be arranged as elements of a row matrix A and the composition of staff in various offices can be arranged in a 3×6 matrix B.
∴A=[DivDisTal530200]
And B=⎡⎢
⎢
⎢⎣OHCClTP1112+111+10111+101+1011101⎤⎥
⎥
⎥⎦=⎡⎢
⎢
⎢⎣OHCClTP111312011202011101⎤⎥
⎥
⎥⎦
The basic monthly salaries of various types of employees of these offices correspond to the elements of the column matrix C
∴C=OHCClTP⎡⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢⎣500200175150150100⎤⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥⎦
(i) Total number of Posts = AB
=[DivDisTal530200]×⎡⎢
⎢
⎢⎣OHCClTP111312011202011101⎤⎥
⎥
⎥⎦
=[OHCClTP52352352755270]
i.e. Required number of posts in all the offices taken together are 5 office superintendents, 235 head clerks, 235 cashiers, 275 clerks, 5 typists and 270 peons.
(ii) The total basic monthly salary bill of each kind of office = BC
=⎡⎢
⎢
⎢⎣OHCClTP111312011202011101⎤⎥
⎥
⎥⎦×⎡⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢⎣500200175150150100⎤⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥⎦OHCClTP
=⎡⎢⎣500+200+175+3×150+1×150+2×1000+1×200+1×175+2×150+0+2×1000+1×200+1×175+1×150+0+1×100⎤⎥⎦=⎡⎢⎣1675875625⎤⎥⎦
i.e. Total basic monthly salary bill of each divisional, district and taluka offices are Rs. 1675, Rs. 875 and Rs. 625 respectively.
(iii) The total basic monthly salary bill of all the offices taken together =ABC
=A(BC)=[530200]×⎡⎢⎣1675875625⎤⎥⎦
=[5×1675+30×875+200×625]=[159625]
i.e. Total basic monthly salary bill of all the offices taken together is Rs. 159,625.
Let us use the symbols Dir,Div,Tal for division, district, taluka respectively and O,H,C,Cl,T and P for office superintendent, head clerk, cashier, clerk, typist and peon respectively.
Then the number of offices can be arranged as elements of a row matrix A and the composition of staff in various offices can be arranged in a 3×6 matrix B.
∴A=[DivDisTal530200]
And B=⎡⎢ ⎢ ⎢⎣OHCClTP1112+111+10111+101+1011101⎤⎥ ⎥ ⎥⎦=⎡⎢ ⎢ ⎢⎣OHCClTP111312011202011101⎤⎥ ⎥ ⎥⎦
The basic monthly salaries of various types of employees of these offices correspond to the elements of the column matrix C
∴C=OHCClTP⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣500200175150150100⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦
(i) Total number of Posts = AB
=[DivDisTal530200]×⎡⎢ ⎢ ⎢⎣OHCClTP111312011202011101⎤⎥ ⎥ ⎥⎦
=[OHCClTP52352352755270]
i.e. Required number of posts in all the offices taken together are 5 office superintendents, 235 head clerks, 235 cashiers, 275 clerks, 5 typists and 270 peons.
(ii) The total basic monthly salary bill of each kind of office = BC
=⎡⎢ ⎢ ⎢⎣OHCClTP111312011202011101⎤⎥ ⎥ ⎥⎦×⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣500200175150150100⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦OHCClTP
=⎡⎢⎣500+200+175+3×150+1×150+2×1000+1×200+1×175+2×150+0+2×1000+1×200+1×175+1×150+0+1×100⎤⎥⎦=⎡⎢⎣1675875625⎤⎥⎦
i.e. Total basic monthly salary bill of each divisional, district and taluka offices are Rs. 1675, Rs. 875 and Rs. 625 respectively.
(iii) The total basic monthly salary bill of all the offices taken together =ABC
=A(BC)=[530200]×⎡⎢⎣1675875625⎤⎥⎦
=[5×1675+30×875+200×625]=[159625]
i.e. Total basic monthly salary bill of all the offices taken together is Rs. 159,625.
∴A=[DivDisTal530200]
And B=⎡⎢ ⎢ ⎢⎣OHCClTP1112+111+10111+101+1011101⎤⎥ ⎥ ⎥⎦=⎡⎢ ⎢ ⎢⎣OHCClTP111312011202011101⎤⎥ ⎥ ⎥⎦
The basic monthly salaries of various types of employees of these offices correspond to the elements of the column matrix C
∴C=OHCClTP⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣500200175150150100⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦
(i) Total number of Posts = AB
=[DivDisTal530200]×⎡⎢ ⎢ ⎢⎣OHCClTP111312011202011101⎤⎥ ⎥ ⎥⎦
=[OHCClTP52352352755270]
i.e. Required number of posts in all the offices taken together are 5 office superintendents, 235 head clerks, 235 cashiers, 275 clerks, 5 typists and 270 peons.
(ii) The total basic monthly salary bill of each kind of office = BC
=⎡⎢ ⎢ ⎢⎣OHCClTP111312011202011101⎤⎥ ⎥ ⎥⎦×⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣500200175150150100⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦OHCClTP
=⎡⎢⎣500+200+175+3×150+1×150+2×1000+1×200+1×175+2×150+0+2×1000+1×200+1×175+1×150+0+1×100⎤⎥⎦=⎡⎢⎣1675875625⎤⎥⎦
i.e. Total basic monthly salary bill of each divisional, district and taluka offices are Rs. 1675, Rs. 875 and Rs. 625 respectively.
(iii) The total basic monthly salary bill of all the offices taken together =ABC
=A(BC)=[530200]×⎡⎢⎣1675875625⎤⎥⎦
=[5×1675+30×875+200×625]=[159625]
i.e. Total basic monthly salary bill of all the offices taken together is Rs. 159,625.
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