Solution of All Case Study, (PU) BBA, Mathematics, First Semester

 

Dear Readers:

It may be very helpful to those students who are preparing for their board exam of Pokhara University. Here are the solutions of case studies of Business Mathematics-I asked in PU board exam. I am also trying to write and publish the solution of case studies of Business Mathematics –II, Business Statistics and DAM very soon in my blog. I need your suggestion for the further improvements.

 

 

Business Mathematics I

Case study – 1 

 (2015 Fall,                         2016 Spring                          2019 Fall)

Southwest hospital has an operating room used only for eye operation. The annual cost of rent, heat and electricity for the operating room and its equipment is Rs. 360000 and the annual salaries of the people who staff this room total Rs. 540000.

Each surgery performed requires the use of Rs. 760 worth of medical supplies and drugs. To promote the goodwill, every patient receives a bouquet of flowers the day after surgery. In addition, one quarter of the patients require dark glasses which the hospital provides free of charge. It cost the hospital Rs. 30 for each bouquet of flowers and Rs. 40 for each pair of glasses.

The hospital receive a payment of Rs. 2000 for each operation performed.

i.             Identify the revenue per case and the annual fixed and variable cost for running the operation room.

ii.           How many eye operations must the hospital perform each year in order to break even?

iii.          Southwest Hospital currently averages 70 eye operations per month. One of the nurse has just learned about a machine that would reduce by Rs. 100 per patient the amount of the medical supplies needed. It can be leased for Rs. 100000 annually. Keeping in the mind the financial cost and benefits, advice the hospital whether it should lease this machine.

iv.          An advertising agency has proposed to the hospital’s president that she spent Rs. 20000 per month on television and radio advertisement persuade people that Southwest hospital is the best place to have any eye surgery performed. Advertising account executive estimates that such publicity increases the business by 25 operations per month. If they are correct and this increase is not big enough to affect the fixed costs. What impact would this advertising have in the hospital’s profit?   

Solution:

From the given case situation, we have

Fixed cost = 360000 + 540000 = 900000

Let the number of operation performed in a year be x, then

Variable cost = 760x + 30x + ¼ x 40x = 800x

a)      Since, total Cost (C) = Fixed cost + Variable cost

 C(x) = 900000 + 800x

Revenue Function (R) = Quantity x Price

 R(x) = x. 2000 = 2000x

b)      For break-even,

C(X) = R(x)

i.e.  900000 + 800x = 2000x

or,  900000 = 2000x – 800x

or,  900000 = 1200x

or,  x = 750

 The hospital should perform 750 operations per year to cover up their cost.

c)       If the number of eye operation performed monthly is 70, then total number of operation performed annually  = 70 x 12 = 840

Also, if the machine reduces the cost of operation by Rs. 100 per operation annually,

Total amount reduced by using the machine annually = 840 x Rs. 100 = Rs. 84000

But the cost of leasing the machine for one year is Rs. 100000

Hence, Cost to lease the machine is more than the cost reduced by using the machine.

 The hospital should not lease the machine.

(Remember: In 2019 fall, cost reduced by machine is given Rs. 125/ operation. In that case Cost to lease the machine is less than the cost reduced by using the machine.  The hospital should lease the machine.)

d)      If the operation per month is increased by 25,

 then total increased operations in a year = 25x12 = 300

 increased revenue in a year = 300 x Rs. 2000 = Rs. 600000

Variable cost increased in a year = 300 x 800 = Rs. 240000

Total advertising cost in a year = Rs. 20000 x 12 = Rs. 240000

Hence, total cost is increased in a year by Rs. 24000 + Rs. 240000 =Rs. 480000

And the revenue is increased in a year by Rs. 600000

 The hospital will be beneficial by advertising their performance.

(Remember: there is given 40 operations is increased per month in the case study of 2015 Fall)

 

(Note: be sure that the amount is given in Rs. Or $ or in other currency)  

Case study- 2

(2014 Fall,       2015 Fall,         2016 Fall,        2017 Spring,          2018 Spring)

 

A politician is trying to win election to the city council and as his campaign manager you need to decide how to promote the candidates

There are three ways you can do

·         You can send glossy, full-color pamphlets to registers voters of the city

·         You can run a commercial during the television news on a local cable network and/or

·         You can buy a full page ad in the newspaper.

Two hundred fifty thousand voters live in the city and 36% of the voters read the newspaper fifty thousand voters watch the local cable network news and 30% of them also read newspaper.

You also know that the television commercial would cost %40000, the newspaper ad $27000 and the pamphlets mailed to voters 90 cents each, including printing and bulk-rate postage.

Suppose that the success of the candidate depends upon your campaign reaching at least 125000 voters and that because your budget is limited, you must achieve your goal at a minimum cost. Based on above information, answer the following questions (Use Venn diagram, if required)

i.             How many voters in the city read newspaper but do not watch the local cable television news?

ii.           How many voters read the newspaper or watch the local television news or both?

iii.          Complete the following chart by indicating the number of voters reached by each promotional options, the total cost and the cost per voter reached

	Maximum number of voter reached	Total cost	Cost per voter reached
Pamphlets Television Newspaper

What would be your plan and the cost of plan?

Listing the given information:

Total number of voters in the city = 250000

Numbers of voters who read newspaper = 36% of 250000 = 90000

Numbers of voter watch local cable television news = 50000

30% of whom also read newspaper.

 The number of voters who watch local cable television news and read newspaper = 30% of 50000 = 15000

 

Cost of television commercial = $ 40000

Cost of newspaper ad = $ 27000

Cost of each pamphlet mailed = 90 cent = $ 0.9

i.                     Let N denote the set of voters who read newspapers and T be the set or voters who watch local cable television news.

 

 

           Number of voters who read newspaper but do not watch the local cable television news

n_o(N) = n(N) - n(N and T)

              = 90000 – 15000

=   75000

ii.                                           Number of voters who read the newspaper or watch the local television news or both n(N and T) = n(N) + n(T) - n(N and T)         

                          = 90000 + 50000 – 15000

                          = 125000

iii.                   

	Maximum number of voter reached	Total cost	Cost per voter reached
Pamphlets Television Newspaper	250000 50000 90000	$225000 $40000 $27000	$0.9 $0.8 $0.3

iv.                 Since, the candidate’s promotion should be reached to at least 125 voters, there is limited budget and three ways of promoting. Following are the alternative ways with cost

Alternative ways	Newspaper ad (90000)	Television ad (50000)	Sending pamphlets (all voters)	Total cost
First   Second   Third   Fourth	$27000   -              -              $27000	$40000   -              $40000   -	-              125000 x $0.9   75000 x $0.9 = $ 67500   35000x$0.9 = $31500	$67000   $125500   $107500   $58500

 

  From the table above we see that fourth option is cheaper than others.

Therefore, advertising in full page of the newspaper which costs $27000 and reaches to 90000 voters and sending pamphlets cost $0.9 per pamphlet to remaining 125000 – 90000 = 35000 voters minimize the total cost of promoting to the sum $ 58500.

(Be sure that the cost, number of voters, percentage of voters is same as in this question or different. Once in 2015, it is given 30% read newspaper instead of 36% here)

Case Study – 3

 2017 Fall

A newly stablished telecommunication company wants to distribute the 10 digits cell-phone numbers on different district of Nepal: Kathmandu, Kaski, Tahanu, Rupandehi, and Syangja. Because of reservation by other telecommunication companies, it was not permitted to apply 980, 981, 982, 983, 984, 895 and 986 for first three digits in its cell number.

i.             How many ways can the company select its initial three digits brand number starting with 98?

ii.           If the company makes its brand no. 987 then how many cell phone no can be formed?

iii.          If 9871 for Kathmandu, 9872 for Kaski, 9873 for Tahanu, 9874 for Rupandehi and 9875 for Syangja then how many number can be formed for each of these district?

iv.          If the company keeps those cell phone numbers which have same digits in the last 6 positions (for eg. 9871222222) then how many such type of SIMs are in its district office?

v.            Those numbers having 0 in the fifth position is postpaid and having 0 and 0 in 5^th and 6^th position as well as are 3G SIMs. How many postpaid and 3G SIMs are there?

 

Solution:-

 We know that:

The permutation of n different digits taking r at a time = n^r , if the repetition of the digits is allowed.

i.         For first 3 digits starting with 98,

The digits 0, 1, 2, 3, 4, 5 and 6 are already used after 98 by other companies,

7, 8 and 9 are left.

So, in only 3 ways the company can select its initial three digits brand number.

ii.       If the company makes its brand number 987, then initial 3 digits are fixed and remaining 7 digits can be put from 10 digits in 10⁷ ways ( since repetition is allowed for mobile number)

 Required no. of cell phone numbers = 10⁷ = 1000000

 

iii.      In each district, initial 4 digits are fixed. Remaining 6 digits can be put from 10 digits in 10⁶ ways.

 Required no. of cell phone numbers in each district = 10⁶ = 1000000

 

iv.     First 4 digits are fixed in each district. If remaining 6 digits is the recurrence of same digit then it may be any one of 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9 but 1 cannot be put for Kathmandu because it causes same 7  last digits, similarly 1 digit be excluded from each district to make same last 6 digits.

 Required no. of cell phone numbers with last six recurrence of same digit = 10-1= 9

 

v.       In each district, initial 4 digits are fixed,

 If 0 is in the fifth position then remaining 5 digits can be put from 10 digits in 10⁵ ways.

 Required number of Postpaid SIMs in each district = 10⁵ = 100000

And hence total number of Postpaid SIMs with the company has = 5 x 100000 = 500000

(Since, the company provides service only in 5 districts)

 

If 0 is in 5^th and 6^th position, then remaining 4 digits be put from 10 digits in 10⁴ ways.

 Required no. of 3G SIMs in each district = 10⁴ = 10000

                     And hence, total number of 3G SIMs the company has = 5 x 10000 = 50000

 

(Note: while solving the problems related to arrangements of digits, be careful about whether the repetition is allowed or not allowed) 

Case study – 4    

                       2014 Spring

 

The XYZ Bakery Ltd. Produces three basic pastry mixes A, B,and C. In the past, the mix of ingredients has shown in the following matrix:

Type	Flour	Fat	Sugar
A	5	1	1
B	6.5	2.5	0.5
C	4.5	3	2

 

Due to change of the customer’s tastes, it has been decided to change the mixes using the following amendment matrix.

Type	Flour	Fat	Sugar
A	0	+1	0
B	-0.5	+0.5	+0.5
C	+0.5	0	0

 

Using matrix algebra you are required to calculate:

a.    The matrix for the new mix

b.    The production requirement to meet an order for 50 units of type A, 30 units of B, and 20 units of type C of new matrix.

c.    The amount of each type (by Cramer’s rule) that must be made to totally use up 3700 lbs of flour, 1700 lbs of fat and 800 lbs of sugar that are present.

Solution:-

 

c. Let x, y and z denotes the number of pastry mixes A, B and C respectively.

Then according to question

 5x + 6y + 5z = 3700

2x + 3y + 3z= 1700

x + y + 2z = 800

 Case study – 6

2015 Spring

Aakash Construction, one of the leading construction company in Pokhara is going to construct three types of apartments in Chiple Dhunga, Pokhara. Currently the managers and engineers of the company are analyzing the cost and the selling strategies of the apartments. For the apartment of type 1 all the raw materials except the sand and concrete will be imported from India and also the labors will be imported from India. For type 2 and 3 local Nepali raw materials will be used and Nepali labors will be used. The general perception in Nepali buyers is that imported things have better quality than the local things. The following table summarizes the requirements per unit of each type of apartments.

 

 

 

 

	Cement (sacks)	Bricks (units)	Iron (KG)	Sand (trucks)	Concrete (trucks)	Labor (hours)
Apartment Type – 1	650	50000	4500	42	30	8000
Apartment Type – 2	700	55500	5000	45	35	7000
Apartment Type – 3	950	65000	7000	50	25	6000

 

If it is imported from India, cement costs Rs. 1000 per sack, bricks cost Rs. 25 per unit, iron costs Rs. 180 per kg and the labor cost Rs. 100 per hour. If all domestic products is used then the cost of cement is Rs. 750 per sack, bricks costs Rs. 12 per unit, iron cost Rs. 110 per kg, sand rs. 7000 per truck, concrete Rs. 14000 per truck and the labor cost Rs. 60 per hour. From the meeting of the board of directors it is decided that they will construct 10 apartments of type 1, 12 apartments of type 2, and 8 apartments of type 3.

i.       Perform matrix multiplication to calculate the cost of each type of apartment and the total cost of the entire project.

ii.     Perform a matrix multiplication which compute the total quantities of six resources required to produce the desired number of apartments of type 1, 2 and 3.

iii.    If the total area of the apartment of type 1 is 2800 sq. feet, type 2 is 2500 sq. feet and type 3 is 3400 sq. feet and if you were the buyer of the apartment, which apartment would you buy? Give logical reason.

iv.    If you were a sales manager of the company, depending the information given above and the result which you have calculated, describe how would you promote your product?

 Solution:-   

Let us symbolize Ce for cement, I for iron, B for bricks, S for sand, Co for concrete, and L for labor.

All materials are in the given units, T1, T2 and T3 denotes the apartment of type 1, type 2 and type 3 respectively.

Since, the apartment of type 1 is made by imported materials whereas apartments of type 2 and type 3 are made by domestic materials.

i.                     Let the matrix Q denote the quantity of material needed to construct the apartments and the matrix  R denote rate per unit of material used in construction of different apartments, then                                          

Case study - 9

(2014 Fall            2015 Spring         2017 Fall          2018 Fall)

 

Twins graduated from a college together and start their careers. Twin 1 invests $2000 at the end of each of 8 years in an account that earns 10% compounded annually. After the initial 8 years no additional contributions are made, but the investment continues to earn 10% compounded annually.

Twin 2 invests no money for 8 years but then contributes $2000 at the end of each year for a period of 36 years (to age 65) to an account that pays 10% compounded annually.

i.             How much money does each twin contribute?

ii.           How much money does each twin have at the age of 65?

iii.          If you are supposed to choose one of these schemes, which scheme will you choose? Explain with reason.

Solution:-

Case study – 10

(2014 Spring,       2015 Fall,             2016 Spring,                 2018 Spring)

 

Mrs. Shrestha (Mrs. Gurung, Mrs. Sharma) is going to buy a piece of property. The owner of the property has her three options of payments plans which are as follows:

She may pay Rs. 75000 on the spot. (Mrs. Sharma may pay 70000 on the spot given in 2018)

She may pay Rs. 100000 at the end of 4 years.

She may pay Rs. 120000 at the end of 9 years.

The owner of the property also says that for the future payments he will add 4% (5%  given in 2015)  compound interest annually ( semi-annually). Mrs. Shrestha does not have much idea about mathematical calculations and further investment. Hearing these many options she seems quite confused. Being a student of Business Mathematics suggest Mrs. Shrestha

i.             Which option should she select in an ideal condition?

ii.           If she has no any money to pay right now, which future option is better for her? Why?

iii.          If any of the payment plan is ok for Mrs. Shrestha, which payment plan would the owner of the property most likely to implement? Why?

iv.          Will she make any changes in the above decisions if the rate of interest increases to 8% per annum compounded semi-annually?

Solution:-

We compute the price of the property at present for each options;

First option:

Mrs. Shrestha may pay Rs. 75000 on the spot.

That is, present worth of the property by this option is Rs. 75000

Second option:

Mrs. Shrestha may pay Rs. 100000 after 4 years

Where 4% interest compounded annually is calculated.

Let present worth of the property be P

Rate of interest (r) =4% p.a.

Time (n) = 4 years

Amount to this present worth (A) = 100000