Question 14 Marks
What is the maximum profit earned by the firm?
Answer
View full question & answer→5440
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Case study (4 Marks)
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Question 24 Marks
Read the following text carefully and answer the questions that follow:
A small firm manufacturers gold rings and chains. The total number of rings and chains manufactured per day is almost 24. It takes 1 hour to make ring and 30 minutes to make a chain. The maximum number of hours available per day is 16. If the profit on a ring is ₹ 300 and that on a chain is ₹ 190. Firm is concerned about earning maximum profit on the number of rings (x) and chains (y) that have to be manufactured per day.
i. what is the expression for objective function?
ii. For maximum profit, firm has to make,what should be the number of rings and chains?
iii. What are the Corner points of the feasible region?
A small firm manufacturers gold rings and chains. The total number of rings and chains manufactured per day is almost 24. It takes 1 hour to make ring and 30 minutes to make a chain. The maximum number of hours available per day is 16. If the profit on a ring is ₹ 300 and that on a chain is ₹ 190. Firm is concerned about earning maximum profit on the number of rings (x) and chains (y) that have to be manufactured per day.
i. what is the expression for objective function?
ii. For maximum profit, firm has to make,what should be the number of rings and chains?
iii. What are the Corner points of the feasible region?
Answer
View full question & answer→i. 300x + 190y
ii. 8, 16
iii. both (0, 24) and (8, 16)
ii. 8, 16
iii. both (0, 24) and (8, 16)
Question 34 Marks
Aeroplane is an important invention for three reasons. It shortens travel time, is more comfortable and facilitates the transport of heavy cargo.
An aeroplane can carry a maximum of 200 passengers.
A profit of ₹400 is made on each executive class ticket and a profit of ₹300 is made on each economy class ticket. The airline reserves at least 20 seats for executive class. However at least 4 times as many passenger prefer to travel by economy class than by executive class.
Based on above information answer the following questions.
i. If x tickets of executive class and y tickets of economy class be sold, then write the constraint.
ii. Write the correct pair of constraints.
iii. If profit earned by airlines is represented by Z, then write the constraint.
iv. Airlines are interested to maximise the profit. For this what are the value of x and y i.e.r number of executive class ticket and economy class ticket to be sold?
v. What is the maximum profit earned by airlines?
An aeroplane can carry a maximum of 200 passengers.
A profit of ₹400 is made on each executive class ticket and a profit of ₹300 is made on each economy class ticket. The airline reserves at least 20 seats for executive class. However at least 4 times as many passenger prefer to travel by economy class than by executive class.
Based on above information answer the following questions.
i. If x tickets of executive class and y tickets of economy class be sold, then write the constraint.
ii. Write the correct pair of constraints.
iii. If profit earned by airlines is represented by Z, then write the constraint.
iv. Airlines are interested to maximise the profit. For this what are the value of x and y i.e.r number of executive class ticket and economy class ticket to be sold?
v. What is the maximum profit earned by airlines?
Answer
View full question & answer→i. Since, Aeroplane can carry a maximum of 200 passengers
$\therefore x + y \leq 200$
ii. Since, Airline reserves at least 20 seats for executive class
$\Rightarrow x \leq 200$
Also atleast four times as many passengers prefer to travel by economy class than by executive class.
$\begin{array}{l}\Rightarrow y=4 x \\ \Rightarrow x+4 x \leq 200[\because x+y \leq 200]\end{array}$
$\begin{array}{l}\Rightarrow 5 x \leq 200 \Rightarrow x \leq 40 \\ \Rightarrow x \geq 20 \text { and } x \leq 40\end{array}$
$\begin{array}{l}\text { iii. Profit on executive class }=400 x \\ \text { Profit on executive class }=300 y \\ \therefore \text { Total profit } Z =400 x +300 y \end{array}$
iv. We have
Z = 400x + 300y which is to be maximise under constraints
$\begin{array}{l}x+y \leq 200 \\ x \leq 40 \\ x \geq 20, y \geq 0\end{array}$
Here, ABCD in bounded feasible region with comer points A(20, 0), B(40, 0), C(40,160), D (20,180).
Now we evaluate Z at each corner points.
For maximum profit x = 40, y = 160
v. We have
Z = 400x + 300y
= 400 * 40 + 300 * 160
= 16000 + 48000
$=₹ 64000[\because$ For maximum profit $x=40, y=160]$
$\therefore x + y \leq 200$
ii. Since, Airline reserves at least 20 seats for executive class
$\Rightarrow x \leq 200$
Also atleast four times as many passengers prefer to travel by economy class than by executive class.
$\begin{array}{l}\Rightarrow y=4 x \\ \Rightarrow x+4 x \leq 200[\because x+y \leq 200]\end{array}$
$\begin{array}{l}\Rightarrow 5 x \leq 200 \Rightarrow x \leq 40 \\ \Rightarrow x \geq 20 \text { and } x \leq 40\end{array}$
$\begin{array}{l}\text { iii. Profit on executive class }=400 x \\ \text { Profit on executive class }=300 y \\ \therefore \text { Total profit } Z =400 x +300 y \end{array}$
iv. We have
Z = 400x + 300y which is to be maximise under constraints
$\begin{array}{l}x+y \leq 200 \\ x \leq 40 \\ x \geq 20, y \geq 0\end{array}$
Here, ABCD in bounded feasible region with comer points A(20, 0), B(40, 0), C(40,160), D (20,180).
Now we evaluate Z at each corner points.
| Corner point | Z=400x+300y |
| A(20,0) | 8000 |
| B(40,0) | 16000 |
| C(40,160) | 64000 ← Maximum |
| D(20,180) | 62000 |
v. We have
Z = 400x + 300y
= 400 * 40 + 300 * 160
= 16000 + 48000
$=₹ 64000[\because$ For maximum profit $x=40, y=160]$
Question 44 Marks
What amount is received at the end of every 6 months forever, if ₹ 72000 kept in a bank earns 8% per annum compounded half yearly?
Answer
View full question & answer→2880
Question 54 Marks
Read the text carefully and answer the questions:
Understanding Perpetuity
An annuity is a stream of cash flows. A perpetuity is a type of annuity that lasts forever, into perpetuity. The stream of cash flows continues for an infinite amount of time. In finance, a person uses the perpetuity calculation in valuation methodologies to find the present value of a company's cash flows when discounted back at a certain rate.
An example of a financial instrument with perpetual cash flows was the British-issued bonds known as consols, which the Bank of England phased out in 2015. By purchasing a consol from the British government, the bondholder was entitled to receive annual interest payments forever.
Perpetuity Present Value Formula
The formula to calculate the present value of perpetuity or security with perpetual cash flows is as follows:
$PV =\frac{C}{(1+r)^1}+\frac{C}{(1+r)^2}+\frac{C}{(1+r)^3} \cdots=\frac{C}{r}$
where:
PV present value
C = cash flow
r = discount rate
(a) Find the present value of a perpetuity of ₹ 900 payable at the end of each year, if money is worth 5% per annum.
(b) Find the present value of a perpetuity of ₹ 500 payable at the end of each quarter, if money is worth 8% per annum.
(c) Find the present value of a perpetuity of ₹ 300 payable at the beginning of every 6 months, if money is worth 6% per annum.
Understanding Perpetuity
An annuity is a stream of cash flows. A perpetuity is a type of annuity that lasts forever, into perpetuity. The stream of cash flows continues for an infinite amount of time. In finance, a person uses the perpetuity calculation in valuation methodologies to find the present value of a company's cash flows when discounted back at a certain rate.
An example of a financial instrument with perpetual cash flows was the British-issued bonds known as consols, which the Bank of England phased out in 2015. By purchasing a consol from the British government, the bondholder was entitled to receive annual interest payments forever.
Perpetuity Present Value Formula
The formula to calculate the present value of perpetuity or security with perpetual cash flows is as follows:
$PV =\frac{C}{(1+r)^1}+\frac{C}{(1+r)^2}+\frac{C}{(1+r)^3} \cdots=\frac{C}{r}$
where:
PV present value
C = cash flow
r = discount rate
(a) Find the present value of a perpetuity of ₹ 900 payable at the end of each year, if money is worth 5% per annum.
(b) Find the present value of a perpetuity of ₹ 500 payable at the end of each quarter, if money is worth 8% per annum.
(c) Find the present value of a perpetuity of ₹ 300 payable at the beginning of every 6 months, if money is worth 6% per annum.
Answer
View full question & answer→Read the text carefully and answer the questions:
Understanding Perpetuity
An annuity is a stream of cash flows. A perpetuity is a type of annuity that lasts forever, into perpetuity. The stream of cash flows continues for an infinite amount of time. In finance, a person uses the perpetuity calculation in valuation methodologies to find the present value of a company's cash flows when discounted back at a certain rate.
An example of a financial instrument with perpetual cash flows was the British-issued bonds known as consols, which the Bank of England phased out in 2015. By purchasing a consol from the British government, the bondholder was entitled to receive annual
interest payments forever.
Perpetuity Present Value Formula
The formula to calculate the present value of perpetuity or security with perpetual cash flows is as follows:
$PV =\frac{C}{(1+r)^1}+\frac{C}{(1+r)^2}+\frac{C}{(1+r)^3} \cdots=\frac{C}{r}$
where:
PV present value
C = cash flow
r = discount rate
(i) ₹ 18000
(ii) ₹ 25000
(iii)₹ 10300
Understanding Perpetuity
An annuity is a stream of cash flows. A perpetuity is a type of annuity that lasts forever, into perpetuity. The stream of cash flows continues for an infinite amount of time. In finance, a person uses the perpetuity calculation in valuation methodologies to find the present value of a company's cash flows when discounted back at a certain rate.
An example of a financial instrument with perpetual cash flows was the British-issued bonds known as consols, which the Bank of England phased out in 2015. By purchasing a consol from the British government, the bondholder was entitled to receive annual
interest payments forever.
Perpetuity Present Value Formula
The formula to calculate the present value of perpetuity or security with perpetual cash flows is as follows:
$PV =\frac{C}{(1+r)^1}+\frac{C}{(1+r)^2}+\frac{C}{(1+r)^3} \cdots=\frac{C}{r}$
where:
PV present value
C = cash flow
r = discount rate
(i) ₹ 18000
(ii) ₹ 25000
(iii)₹ 10300
Question 64 Marks
What should be the number of apartments, that the complex should have in order to minimize the maintenance cost?
Answer
View full question & answer→The complex must have 4500 apartments to minimise the maintenance cost.
Question 74 Marks
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