A manufacturer produces two types of products - A and B, which are subjected to two types of operations, viz. grinding and polishing. Each unit of product A takes 2 hours of grinding and 3 hours of polishing whereas product B takes 3 hours of grinding and 2 hours of polishing. The manufacturer has 10 grinders and 15 polishers. Each grinder operates for 12 hours/day and each polisher 10 hours/day. The profit margin per unit of A and B are Rs. 5/ - and Rs. 7/ - respectively. If the manufacturer utilises all his resources for producing these two types of items, what is the maximum profit that the manufacturer can earn in a day?

Let the number of units of A and B produced be $$x$$ and $$y$$ respectively.

For product A, time taken for grinding = $$2x$$ and polishing = $$3x$$

For product B, time taken for grinding = $$3y$$ and polishing = $$2y$$

Total number of hours of grinding done per day = $$10 \times 12 = 120$$ hrs

Total number of hours of polishing done per day = $$15 \times 10 = 150$$ hrs

=> $$2x + 3y = 120$$ ---------Eqn(I)

and $$3x+ 2y = 150$$ ----------Eqn(II)

Applying the operation : 3* Eqn(I) - 2* Eqn(II), we get :

=> $$(6x - 6x) + (9y - 4y) = 360 - 300$$

=> $$y = \frac{60}{5} = 12$$

=> $$x = 42$$

$$\therefore$$ Profit made by the manufacturer = $$(42 \times 5) + (12 \times 7)$$

= $$210 + 84 = Rs. 294$$