Q14 of 17 Page 12

A retired person wants to invest an amount of ₹ 50,000. His broker recommends investing in two types of bonds ‘A’ and ‘B’ yielding 10% and 9% return respectively on the invested amount. He decides to invest at least ₹ 20,000 in bond ‘A’ and at least ₹ 10,000 in bond ‘B’. He also wants to invest at least as much in bond ‘A’ as in bond ‘B’. Solve this linear programming problem graphically to maximize his returns. [CBSE 2016]


Let the person invest Rs x in bond A and Rs. y in bond B.


Now, the interest on bond A = (x × 1 × 10)/100 = 10x/100


and the interest on bond B = (y × 1 × 9)/100 = 9y/100


Total annual income from interest = 10x/100 + 9y/100


= 0.1x + 0.09y


Now, given he decides to invest at least 20000 in bond A and at least 10000 in bond B


So, x ≥ 20000 and y ≥ 10000


Again, total investment is x + y, and it should not exceed 50000


So, x + y ≤ 50000


Now, the LPP problem is,


Max z = 0.1x + 0.09y


subject to constraints


x + y ≤ 50000


x ≥ 20000, y ≥ 10000


x ≥ y


Now,


(x, y) z = 0.1x + 0.09y


(20000, 10000) 2950


(40000, 10000) 4900


(25000, 25000) 4750


So, when A invest Rs 40000 and B invest Rs 10000, his return is maximum.


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There are two types of fertilizers ‘A’ and ‘B’. ‘A’ consists of 12% nitrogen and 5% phosphoric acid whereas ‘B’ consists of 4% nitrogen and 5% phosphoric acid. After testing the soil conditions, the farmer finds that he needs at least 12 kg of nitrogen and 12 kg of phosphoric acid for his crops. If ‘A’ costs ₹ 10 per kg and ‘B’ cost ₹ 8 per kg, then graphically determine how much of each type of fertilizer should be used, so that nutrient requirements are met at a minimum cost. [CBSE 2016]

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A factory manufactures two types of screws A and B, each type requiring the use of two machines, an automatic and a hand – operated. It takes 4 minutes on the automatic and 6 minutes on the hand operated machines to manufacture a packet of screws ‘B’. Each machine is availble for at most 4 hours on any day. The manufacturer can sell a packet of screws ‘A’ at a profit of 70 paise and screws ‘B’ at a profit of Rs. 1. Assuming that he can sell all the screws he manufactrures, how many packets of each type should the factory owner produce in a day in order to maximize his profit ? Formulate the above LPP and solve it graphically and find the maximum profit.[CBSE 2018]