2 Marks Questions

Question 12 Marks

Two tailors, A and B, earn ₹ 300 and ₹ 400 per day respectively. A can stitch 6 shirts and 4 pairs of trousers while B can stitch 10 shirts and 4 pairs of trousers per day. To find how many days should each of them work and if it is desired to produce at least 60 shirts and 32 pairs of trousers at a minimum labour cost, formulate this as an LPP.

Answer

Let tailor A work for x days and tailor B work for y days.
In one day, A can stitch 6 shirts and 4 pairs of trousers whereas B can stitch 10 shirts and 4 pairs of trousers.
Thus, in x days A can stitch 6x shirts and 4x pairs of trousers. Similarly, in y days B can stitch 10y shirts and 4y pairs of trousers.
It is given that the minimum requirement of the shirts and pairs of trousers are respectively 60 and 32 respectively.
Thus,
$6x + 10y \geq 60\\4x + 4y \geq 32$
Further it is given that A and B earn ₹ 300 and ₹ 400 per day respectively. Thus, in x days and y days, A and B earn Rs 300x and ₹ 400y respectively.
Let Z denotes the total cost
$\therefore \text{Z = ₹ } (300x + 400y)$
Number of days cannot be negative.
Therefore, $x, y \geq 0$
Hence, the required LPP is as follows:
Minimize $\text{Z = } 300x + 400y$
subject to
$6x + 10y \geq 60\\ 4x + 4y \geq 32\\ x\geq 0, y \geq 0$

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Question 22 Marks

Two tailors, A and B, earn ₹ 300 and ₹ 400 per day respectively. A can stitch 6 shirts and 4 pairs of trousers while B can stitch 10 shirts and 4 pairs of trousers per day. To find how many days should each of them work and if it is desired to produce at least 60 shirts and 32 pairs of trousers at a minimum labour cost, formulate this as an LPP.

Answer

Let tailor A work for x days and tailor B work for y days.
In one day, A can stitch 6 shirts and 4 pairs of trousers whereas B can stitch 10 shirts and 4 pairs of trousers.
Thus, in x days A can stitch 6x shirts and 4x pairs of trousers. Similarly, in y days B can stitch 10y shirts and 4y pairs of trousers.
It is given that the minimum requirement of the shirts and pairs of trousers are respectively 60 and 32 respectively.
Thus,
$6x + 10y \geq 60\\4x + 4y \geq 32$
Further it is given that A and B earn ₹ 300 and ₹ 400 per day respectively. Thus, in x days and y days, A and B earn Rs 300x and ₹ 400y respectively.
Let Z denotes the total cost
$\therefore \text{Z = ₹ } (300x + 400y)$
Number of days cannot be negative.
Therefore, $x, y \geq 0$
Hence, the required LPP is as follows:
Minimize $\text{Z = } 300x + 400y$
subject to
$6x + 10y \geq 60\\ 4x + 4y \geq 32\\ x\geq 0, y \geq 0$

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Question 32 Marks

Two tailors, A and B, earn ₹ 300 and ₹ 400 per day respectively. A can stitch 6 shirts and 4 pairs of trousers while B can stitch 10 shirts and 4 pairs of trousers per day. To find how many days should each of them work and if it is desired to produce at least 60 shirts and 32 pairs of trousers at a minimum labour cost, formulate this as an LPP.

Answer

Let tailor A work for x days and tailor B work for y days.
In one day, A can stitch 6 shirts and 4 pairs of trousers whereas B can stitch 10 shirts and 4 pairs of trousers.
Thus, in x days A can stitch 6x shirts and 4x pairs of trousers. Similarly, in y days B can stitch 10y shirts and 4y pairs of trousers.
It is given that the minimum requirement of the shirts and pairs of trousers are respectively 60 and 32 respectively.
Thus,
$6x + 10y \geq 60\\4x + 4y \geq 32$
Further it is given that A and B earn ₹ 300 and ₹ 400 per day respectively. Thus, in x days and y days, A and B earn Rs 300x and ₹ 400y respectively.
Let Z denotes the total cost
$\therefore \text{Z = ₹ } (300x + 400y)$
Number of days cannot be negative.
Therefore, $x, y \geq 0$
Hence, the required LPP is as follows:
Minimize $\text{Z = } 300x + 400y$
subject to
$6x + 10y \geq 60\\ 4x + 4y \geq 32\\ x\geq 0, y \geq 0$

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Question 42 Marks

A company produces two types of goods A and B, that require gold and silver. Each unit of type A requires 3 g of silver and 1 g of gold while that of type B requires 1 g of silver and 2 g of gold. The company can procure a maximum of 9 g of silver and 8 g of gold. If each unit of type A brings a profit of ₹ 40 and that of type B ₹ 50, formulate LPP to maximize profit.

Answer

Let number of goods A = x units, number of goods B = y units
LPP is: Maximize profit, P = 40x + 50y
subject to following:
$\text{3x + y} \leq 9\\ \text{x + 2y} \geq 8\\ \text{x} \geq 0, \text{y} \geq 0$

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Question 52 Marks

A company produces two types of goods A and B, that require gold and silver. Each unit of type A requires 3 g of silver and 1 g of gold while that of type B requires 1 g of silver and 2 g of gold. The company can procure a maximum of 9 g of silver and 8 g of gold. If each unit of type A brings a profit of ₹ 40 and that of type B ₹ 50, formulate LPP to maximize profit.

Answer

Let number of goods A = x units, number of goods B = y units
LPP is: Maximize profit, P = 40x + 50y
subject to following:
$\text{3x + y} \leq 9\\ \text{x + 2y} \geq 8\\ \text{x} \geq 0, \text{y} \geq 0$

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Question 62 Marks

A company produces two types of goods A and B, that require gold and silver. Each unit of type A requires 3 g of silver and 1 g of gold while that of type B requires 1 g of silver and 2 g of gold. The company can procure a maximum of 9 g of silver and 8 g of gold. If each unit of type A brings a profit of ₹ 40 and that of type B ₹ 50, formulate LPP to maximize profit.

Answer

Let number of goods A = x units, number of goods B = y units
LPP is: Maximize profit, P = 40x + 50y
subject to following:
$\text{3x + y} \leq 9\\ \text{x + 2y} \geq 8\\ \text{x} \geq 0, \text{y} \geq 0$

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Question 72 Marks

A small firm manufactures necklaces and bracelets. The total number of necklaces and bracelets that it can handle per day is at most 24. It takes one hour to make a bracelet and half an hour to make a necklace. The maximum number of hours available per day is 16. If the profit on a necklace is ₹ 100 and that on a bracelet is ₹ 300. Formulate on L.P.P. for finding how many of each should be produced daily to maximize the profit? It is being given that at least one of each must be produced.

Answer

Let x necklaces and y bracelets are manufactured
$\therefore \text{L.P.P. is}$
Maximize profit, P = 100x + 300y
subject to constraints
$\text{x + y} \leq 24$
$\frac{1}{2} \text{x + y} \leq 16 \text{ or x + 2y} \leq 32$
$\text{x, y,} \geq 1$

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Question 82 Marks

A small firm manufactures necklaces and bracelets. The total number of necklaces and bracelets that it can handle per day is at most 24. It takes one hour to make a bracelet and half an hour to make a necklace. The maximum number of hours available per day is 16. If the profit on a necklace is ₹ 100 and that on a bracelet is ₹ 300. Formulate on L.P.P. for finding how many of each should be produced daily to maximize the profit? It is being given that at least one of each must be produced.

Answer

Let x necklaces and y bracelets are manufactured
$\therefore \text{L.P.P. is}$
Maximize profit, P = 100x + 300y
subject to constraints
$\text{x + y} \leq 24$
$\frac{1}{2} \text{x + y} \leq 16 \text{ or x + 2y} \leq 32$
$\text{x, y,} \geq 1$

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Question 92 Marks

A small firm manufactures necklaces and bracelets. The total number of necklaces and bracelets that it can handle per day is at most 24. It takes one hour to make a bracelet and half an hour to make a necklace. The maximum number of hours available per day is 16. If the profit on a necklace is ₹ 100 and that on a bracelet is ₹ 300. Formulate on L.P.P. for finding how many of each should be produced daily to maximize the profit? It is being given that at least one of each must be produced.

Answer

Let x necklaces and y bracelets are manufactured
$\therefore \text{L.P.P. is}$
Maximize profit, P = 100x + 300y
subject to constraints
$\text{x + y} \leq 24$
$\frac{1}{2} \text{x + y} \leq 16 \text{ or x + 2y} \leq 32$
$\text{x, y,} \geq 1$

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