Question

A manufacturer produces three products x, y, z which he sells in two markets. Annual sales are indicated below :

Market

Products

I

10,000

2,000

18,000

II

6,000

20,000

8,000

(a) If unit sale prices of x, y and z are Rs. 2.50, Rs. 1.50 and Rs. 1.00, respectively, find the total revenue in each market with the help of matrix algebra.
(b) If the unit costs of the above three commodities are Rs. 2.00, Rs. 1.00 and 50 paise respectively. Find the gross profit.

Solution

left parenthesis straight a right parenthesis space Let space straight A space be space the space product space matrix comma space straight B space be space the space price space matrix space and space straight C space be space the space cost space matrix straight A equals space open square brackets table row 10000 2000 18000 row 6000 20000 8000 end table close square brackets space comma space straight B equals space open square brackets table row cell 2.50 end cell row cell 1.50 end cell row cell 1.00 end cell end table close square brackets straight C equals space open square brackets table row cell 2.00 end cell row cell 1.50 end cell row cell 0.50 end cell end table close square brackets therefore space space AB equals space open square brackets table row 10000 2000 18000 row 6000 20000 8000 end table close square brackets space open square brackets table row 10000 2000 18000 row 6000 20000 8000 end table close square brackets space space space space space space space space space space space equals space space open square brackets table row cell 10 space 000 space cross times space 2.50 plus 2 space 000 space cross times space 1.50 space plus 18000 space cross times space 1.00 end cell row cell 6000 cross times space 2.50 plus space 20 space 000 space cross times space 1.50 plus 8000 space cross times 1.00 end cell end table close square brackets space space space space space space space space space space space equals space open square brackets table row cell 25000 plus 3000 plus 18000 end cell row cell 15000 plus 30000 plus 8000 end cell end table close square brackets space equals space open square brackets table row 46000 row 53000 end table close square brackets therefore space revenue space in space market space straight I space equals space Rs. space 46 space 000 space space and space revenue space in space market space II space equals space Rs. space 53000. space space Total space revenue space equals space Rs. space left parenthesis 46000 space plus space 53000 right parenthesis space equals space Rs. space 99000 left parenthesis straight b right parenthesis therefore space space AC equals space open square brackets table row 10000 2000 18000 row 6000 20000 8000 end table close square brackets space open square brackets table row 10000 2000 18000 row 6000 20000 8000 end table close square brackets space space space space space space space space space space space space space space space space space equals space open square brackets table row cell 10000 cross times 2.00 plus 2 space 000 space cross times 1.00 space plus 18000 space cross times space 0.50 end cell row cell 6000 cross times space 2.00 plus 20 space 000 space plus 1.00 plus 8000 space cross times 0.50 end cell end table close square brackets space space space space space space space space space space space space space space space space space equals space open square brackets table row cell 20000 plus 2000 plus 9000 end cell row cell 12000 plus 20000 plus 4000 end cell end table close square brackets space equals space open square brackets table row 31000 row 36000 end table close square brackets therefore space revenue space in space market space straight I space equals space Rs. space 31000 space space and space revenue space in space market space II space equals space Rs. space 36000. space space Total space revenue space equals space Rs. space left parenthesis 31000 space plus space 36000 right parenthesis space equals space Rs. space 67000 therefore space gross space profit space equals space Revenue space – space cost space price therefore space gross space profit space equals space Revenue space – space cost space price space space equals space Rs. space left parenthesis 99000 space – space 67000 right parenthesis space equals space Rs. space 32000.

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