Antony wishes to buy black shirts and white shirts. He intends to buy at most black shirts. A black shirt costs Tsh. 24,000 while a white shirt costs Tsh. 30,000 and he is planning to spend up to Tsh. 180,000 for buying shirts.

1. How many shirts of each kind should be bought so as to have maximum number of shirts?
2. Find the greatest number of shirts that should be bought

A trader has a space for 5 refrigerators. The trader plans to spend 2,400,000 shillings to buy refrigerators of two brands, Hitachi and Sony. Each Hitachi refrigerator costs 600,000 shillings whereas each Sony refrigerator costs 400,000 shillings. The unit profits for Hitachi and Sony refrigerators are 200,000 shillings and 150,000 shillings respectively. Denoting x and y as the number of Hitachi and Sony refrigerators respectively, determine the number of refrigerators for each brand that maximizes the profit. 

  A farmer needs to buy up to 25 cows for a new herd. He can buy either brown cows at 50,000/= each or black cows at 80,000/= each and he can spend a total of not more than 1,580,000/=. He must have at least 9 cows of each type. On selling the cows he will make a profit of 5,000/= on each brown cow and 6,000/= on each black cow. How many of each type he should buy to maximize profit?

Zelda wants to buy oranges and mangoes for her children. The oranges are sold at sh. 150 each and mangoes at sh. 200 each. She must buy at least two of each kind of fruit but her shopping bag cannot hold more than 10 fruits. If the owner of the shop makes a profit of sh. 40 on each orange and sh. 60 on each mango, determine how many fruits of each kind Zelda must buy for the shop owner to realise maximum profit.

A shopkeeper sells refrigerators and washing machines. Each refrigerator takes up 1.8 m ² of space and costs 500,000 ² of space and costs 300,000 shillings; whereas each washing machine takes up 1.5 mshillings. The owner of the shop has 6,000,000 shillings to spend and has 27 m 2 of space.

(a)  Write down all the inequalities which represent the given information.

(b)  If he makes a profit of 30,000 shillings on each refrigerator and 40,000 shillings on each washing machine, find how many refrigerators and washing machines he should sell for maximum profit.

A small industry makes two types of clothes namely type A and type B. Each type A takes 3 hours to produce and uses 6 metres of material and each type B takes 6 hours to produce and uses 7 metres of material. The workers can work for a total of 60 hours and there is 90 metres of material available. If the profit on a type A cloth is 4,000 shillings and on a type B is 6,000 shillings, find how many of each type should be made for maximum profit.

A businessman plans to buy at most 210 sacks of Irish and sweet potatoes. Irishpotatoes cost shs. 30,000 per sack and sweet potatoes cost shs. 5,000 per sack. He can spend up to shs. 2,500,000 for his business. The profit on a single sack of Irish potatoes is shs. 12,000 and for sweet potatoes is shs. 10,000. How many sacks of each type of potatoes the businessman will buy in order to realize the maximum profit?

(a) Jennifer makes two types of garments, Batiki and Kitenge. Batiki requires 2.5 metres of material while Kitenge requires 2 metres of material. The business uses up to 400 metres of materials daily for the production of both types of garments but produces at most 80 metres of Batiki and at least 60 metres of Kitenge daily. Taking x to represent the number of Batiki and y the number of Kitenge produced daily;

(i) write down the inequalities satisfying the given information.

(ii) find the number of each type of garments the business can produce in order to get the maximum income if the income is given by f (x, y) = 300x+ 200y .

(b) What is the importance of studying linear programming? Give 2 points.

A transport company is hired to transport 420 people it has two types, P and Q of vehicle to be used. Type P carries 35 passengers and type Q carries 14 passengers. There are at least 10 vehicles of type Q and not more than 9 vehicles of type P. Write down inequalities to represent this information.

A craftsman wishes to decide how many of each type A and B charcoal stove he has to fabricate in order to maximize profit for this month. Unit profit for type A stove is shs. 1000 and Unit profit for type B is shs. 1500. Type A stove requires 1m² of mild steel sheet per unit and type B requires 2m². He has only 12 m² of mild steel available. He can fabricate a total of 8 stoves of either type per month. How many of each type should he fabricate?

A farmer has 20 hectares for growing tomatoes and cabbages. The cost per hectare for tomatoes is sh 48,000 and for cabbages is sh 32,000. The farmer has budgeted sh 768,000. Tomatoes require one man­day per hectare and cabbages require two man­days per hectare. There are 36 man­days available. The profit on tomatoes is sh 160,000 per hectare and on cabbages is sh 192,000 per hectare. Find the number of hectares of each crop the farmer should plant to maximize the profit.

(a) Solve by graphical method the following system of simultaneous equations:

4x + y = 6

5x + 2y = 9

(b) A farm is to be planted with sorghum and maize while observing the following constraints:

If sorghum yields a profit of 800,000 shillings per hectare while maize yields 600,000 shillings per hectare, how many hectares should be planted with each crop for maximum profit?

Anna and Mary are tailors. They make x blouses and y skirts each week. Anna does all the cutting and Mary does all the sewing. To make a blouse it takes 5 hours of cutting and 4 hours of sewing. To make a skirt it takes 6 hours of cutting and 10 hours of sewing. Neither tailor works for more than 60 hours a week.

(a) For sewing show that 2x + 5y ? 30

(b) Write down another inequality in x and y for the cutting.

(c) If they make at least 8 blouses each week, write down another inequality.

(d) Using 1cm to represent 1 unit on each axis, show the information in parts (a), (b) and (c) graphically. Shade only the required region.

(e) If the profit on a blouse is shs. 3,000/= and on a skirt is shs. 10,000/=, calculate the maximum profit that Anna and Mary can make in a week.

The number of units of proteins and starch contained in each of two types of food A and B are shown in the table below:

What is the cheapest way of satisfying the minimum daily requirement?

Maximize f = 2y — x subject to the following constraints:

x ? 0

y ? 0

2x+y ? 6

x+2y ? 6

(b) Sara had 300 shillings to buy erasers and pencils. An eraser cost 20 shillings while a pencil costs 30 shillings. If the number of erasers bought is at least twice the number of pencils, formulate the inequalities that represent this information.