Mathematics For Business: Essay On Powers, Roots, Interests, Probabilities, And Linear Relationships.

Question 1: Powers and Roots

 QUESTION                                                                                                        

1. Powers and Roots:

1. a) Simplify                                                                          
2. b) Simplify                                                                        
3. c) Evaluate                                                                                   

QUESTION 2                                                                                                                         

1. a) Express the power 100 ^1/2 using the root notation and evaluate.                     
2. b) Evaluate                                                                                      
3. c) Simplify 7                                                                                    
4. d) Scientific notation allows one to express large or small numbers in a simpler form. Express the UK population of 65,648,000 in a scientific notation                               

QUESTION 3

            Ann Miller invests £150,000 at an interest rate of 6% p.a. Calculate the final balance after 5 years.

           a) Using simple interest?             

          b) Using interest compounded annually?            

          c) Using interest compounded semi-annually?              

          d) Using interest compounded quarterly?              

QUESTION 4                                                                                                   

 a) Eliza invests £22,000 at a 2% interest rate annually. Compounding the interest annually, how long will it take her to receive the balance of £33,000?

 b) Using Rule 72, calculate how long will it take Eliza to double her investments?                                          

 c) Mr Ramsbottom invests £32,000 in a bank savings account and after 10 years his balance is £45,200.20. Calculate the compound interest rate he received and round your answer to the second decimal place.                                

 QUESTION 5                           

   a) Find the value of x if                                                 

    b) Solve the equation   X + 20  = 70                                                         

c) Solve the equation        = 10                                                           

 d) To plot the linear graph of y = 3x + 10 complete the following table:   

Question 2

      

x	- 8	-5	0	7	12	24
y

 Question 6                                                                                                                            

Sarah Hair Saloon is considering an investment project to purchase and run a Hair Saloon business. The initial cost is £55,000. The annual cash inflows (income) are projected to be as follows:

Year 1	Year 2	Year 3	Year 4
£15,000	£25,000	£45,000	£15,000

The discount rate for this investment is 8% p.a., compounded annually.

 a) Work out the Net Present Value (NPV) of this investment.            

 b) Should Sarah proceed with this project?

    Explain your reasoning. 

Question 7                                                                                                                       

A set of test scores, marked out of 100, is as follows:

                                                                     

66	93	75	58	68
53	65	92	94	62
63	74	93	92	95
58	94	62	78	96
62	64	87	66	57

a) Produce a tally of this data set suitable for the production of a histogram                                                                                                          

b) Draw a histogram of this data set                  

c) Comment on the distribution of these marks. 

Question 8

Probability is a measure of the likelihood and can be stated as a ratio, percentage or generally as a number between zero and one.

 a) What is the probability when the likelihood is impossible?                                                  

b) What is the probability when the likelihood is certain?

c) Express the probability of 0.06 as a %                                                                                                                                   

d) Josiah tossed a coin and thrown a die at the same time (simultaneously). Work out the probability of getting a head on the coin and a 5 on the die.

Question 9. Two Real life examples (100 words each)                                      

Give two real-life situations or problems in businesses that involve the topics studied in this module (e.g. powers and roots, simple and compound interests, linear relationships, graphs, probabilities and Net Present values (NPV)).

Question 10. Reflective Log (150 words)

This reflective log should develop as the course proceeds, and may be the last part to be completed. Reflect honestly on your experiences throughout the semester. Start your reflective log from week one by completing the skills audits and by writing personal weekly notes after each topic. Please ask for your Tutor’s support if needed.

 You may wish to consider the following points when providing your reflective comments: 

• Which topics do you feel most confident about? (e.g. powers and roots, interest rates, NPV etc.)
• Are there areas for improvement (e.g. in probability, I need do practice more or research etc.)?
• How would you evaluate your participation on the module (e.g. contribution to classes, independent study etc.)?

1. 

Part a

 

Part b

 

Part c

2. 

Part a

 

Part b

 

Part c

 

Part d  65,648,000 in scientific notation

3.

Part a

Part b

  

Part c

 

Part d

 

4.

Part a

 

Dividing both sides of the equation by 22000 and simplifying we obtain

 

 Introducing logs on both sides of the equation we obtain

 

Part b

At 2% interest, rule 72 confirms that it will take 36 years to double the investment.

Part c

Divide both sides by 32000 to obtain

 

 Introducing logs to both side of the equation we obtain

 

5. 

Part a

Part b

 

Part c

 

  d)

  

x	- 8	-5	0	7	12	24
y	-14	-5	10	31	46	82

6. 

Part a

Part b

 Sarah should proceed with the project. The project is viable since the NPV is positive.

7. 

Part a

We group the data in Excel using the data analysis function to obtain the table below

Range	Frequency
50	0
55	1
60	3
65	6
70	3
75	2
80	1
85	0
90	1
95	7
100	1

Part b

Then, drawing the histogram we obtain

Part c

The histogram above shows that most of the scores lie between 60-74 implying that the median lies there.

8. 

Part a

 The probability when the likelihood is impossible equals zero.

Part b

The probability when the likelihood is certain equals 1.

Part c

Probability of 0.06 as a percentage equals 

Part d

Table 1 below shows the possible outcomes when tossing a coin and throwing a dice simultaneously. H represents a head while T stands for the Tail. Also the numbers 1-6 represent the number that appears on top when a dice is thrown.

Table 1: Total possible outcomes

	1	2	3	4	5	6
H	H1	H2	H3	H4	H5	H6
T	T1	T2	T3	T4	T5	T6

9.

(1)Net Present Value

NPV is a capital budgeting technique that takes into account the time value of money when making calculations. Investors use the method as the basis for selecting or rejecting a project. As a result, NPV can be positive, negative or zero. Positive NPV implies that cash inflows are higher than cash outflows meaning that the project is viable and should be accepted. A zero NPV denotes an equal amount of cash inflows and cash outflows. A project may be considered to be acceptable when it has zero NPV. On the other hand, a project with negative NPV should be ignored since it brings losses. 

10. Reflective Log

Actually, before the semester I started I was clueless about how the semester could unfold, bearing in mind of what I overheard from our predecessors. They could instil fear in us that the course is tough. However, I was amazed to realize how simple and informative the course was. Learning about powers, simple and compound interest, probability and linear relationships in the first few classes of the semester was a bit challenging. Particularly, probability. However, after concerted efforts through topical tutorials, I was able to understand everything from the topic. This made me feel more confident about the topic. Truly, I was amazed to know that these topics are applicable in real life. Especially, the simple and compound interests which are crucial tools in financial institutions.

Throughout the next couple of classes, I was thrilled to see how simpler the classes became. I think this is attributed to the positive attitude and high self-esteem I developed with time. I could perform calculations more easily and interpret real-life word problems based on the understanding of the topics. However, I need to improve my speed, especially when using a calculator to compute the calculations.

(2) Linear relationships                                                                               

Linear relations use one or more variables where one depends on another. Almost every situation in life with an unknown quantity can be represented using linear relationships. For instance, calculating mileage rates and predicting profit. Besides, linear equations can be applied in calculating variable costs. For example, if a taxi charges $8 to pick a person from a hotel and another $0.12 per kilometre travelled. One can get a linear equation to find the total cost of the taxi over a given distance. That is, setting x to represent distance covered and setting y to represent the total cost. Therefore, the linear relationship will be y=0.12x+8