# small company

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**Mathematics and Economics for Business International Year One 2015**

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**Maths Report April 2015**

**Instructions**

The report is worth 30% of your final mark in the module.

The report will be marked out of 100.

Marks will be given for:

- Responding to the report brief and the completion of the tasks
- Correct answers
- Showing working out clearly
- Giving appropriate units and correct d.p. as requested
- Graphs done on graph paper (or Excel) with titles, correct axis and labels given; use of sensible scales etc
- Tables typed (or Excel) with titles, columns labelled, units given etc
- Presentation – readable font size, consistent use of heading, bold, page numbering
- Layout – follows task order, has short introduction and conclusion
- Use of English – clear, concise, use of mathematical and business terms where appropriate
- The report being 1000 words (+/- 20%) in length

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**Marking scheme**

80 marks awarded for doing the tasks, showing working out and providing explanations. The breakdown of these marks is indicated in the brief.

20 marks awarded for presentation, layout, correct number of words, use of English etc

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**Now read the report brief.
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**Report Brief**

You are a consultant business analyst. You have been employed to help a small company called ‘Blogs Tools’ plan an expansion of its tool making business based on financial and other data that is available. Your task is to use your mathematical skills and the data to solve a series of tasks and suggest the best ways forward to Mr Big Cheese the company CEO (Chief Executive Officer).

In your report you should list each task and sub-task and present your responses to them giving all necessary working out and explanation. Assume the CEO has basic mathematical background.

In the short introduction to your report you should give a suitable report title, your aim, your job title, the date and include the customer’s name and company. The report should have a short conclusion.

**Task 1 Capital currency conversion (8 marks)**

The company has capital (money) that it can use to invest. This capital is located in several countries around the world as given in the table below.

Country |
Capital |

UK | (UK£) 13,124 |

Germany | (Euro €) 18,764 |

Canada | (Canadian $) 21,762 |

Japan | (Japanese Yen ¥) 154,864 |

Turkey | (Turkish Lira) 25,231 |

Switzerland | (Swiss Franc SF) 17,257 |

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- Calculate in UK£ how much the company has to invest in each country. For your calculations use the US dollar exchange rates given in the table below. Lay out clearly how you work out each exchange rate. Work to four d.p.

0.6513 | UK£ |

0.816 | Euro € |

1.1444 | Canadian $ |

121.1586 | Japanese ¥ |

2.2727 | Turkish Lira |

0.9813 | Swiss Francs |

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- Inform the CEO how much capital in total (in 2 d.p.) in UK£ has the company to invest?

**Task 2 Investment loan (10 marks)**

To invest the company needs to borrow another £110,000. There are two banks that are offering a ten year (the term) loan where interest is paid each year and the loan repaid at the end. Bank A is offering to lend the money at an interest rate of 2.1% compounded monthly. Bank B is offering to lend the money at an interest rate of 2.05% compounded weekly.

- How much interest in pounds (2 d.p.) would each bank charge on the loan over the term?

- What would the APR rate (2 d.p.) of each of the loans be?

- Explain to the CEO what “APR” and “compound interest” mean.

- Use this information to recommend to the CEO what is the best loan to take.

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**Task 3 VAT table (5 marks)**

The CEO has provided you with a list of goods and prices. There are some parts missing. Complete the table below to fill these out (2 d.p.).

Good |
Pre-VAT price £ |
VAT rate % |
Total price £ |

A | 220 | 15 | |

B | 17.5 | 286.70 | |

C | 166 | 203.35 |

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**Task 4 The demand function (14 marks)**

The demand function Q = f(P) for a good the company supplies is given by:

Q = 100 – 2P

where Q is demand and P is price.

- Convert this into the inverse demand function P = f
^{-1}(Q).

- Plot the
**inverse**function as a graph on a suitable set of axes.

- Explain to the CEO the economic significance of the negative portions of the graph.

- What is the y intercept? Explain in words to the CEO what this means in this graph.

- What is the x intercept? Explain in words to the CEO what this means in this graph.

- What is the gradient? Explain in words to the CEO what this means in this graph.

- Use the
**inverse**function to work out what is the value of P when Q is 40? Confirm this by using the graph.

- Use the
**inverse**function to work out what value Q is when P is 20. Confirm this by using the graph.

**Task 5 The supply function (14 marks)**

The supply function for a good the company supplies is given by:

Q = 2P – 30

where Q is supply and P is price.

- Convert this into the inverse supply function P = f
^{-1}(Q).

- Plot the
**inverse**function as a graph on a suitable set of axes.

- Explain to the CEO the economic significance of the negative portions of the graph.

- What is the y intercept? Explain in words to the CEO what this means in this graph.

- What is the x intercept? Explain in words to the CEO what this means in this graph.

- What is the gradient? Explain in words to the CEO what this means in this graph.

- Use the
**inverse**function to work out what is the value of P when Q is 70? Confirm this by using the graph.

- Use the
**inverse**function to work out what value Q is when P is 80. Confirm this by using the graph.

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**Task 6 Condition of market equilibrium (12 marks)**

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**Goods market equilibrium for a company’s good occurs when** **the quantity demanded by consumers equals the quantity supplied by the company**.

- Use the
**inverse**demand and supply functions you worked out for task 4 and 5 and your knowledge of**simultaneous**equations to work out the equilibrium quantity for demand and supply for your company’s good by equating the two functions of P.

- Use the
**inverse**demand function to find what price equilibrium occurs at.

- Other than using a graph how could you check your answer for b)?

- Plot on one graph with suitable scales etc the
**inverse**supply and demand functions from task 4 and 5 to confirm your answer for the**equilibrium**demand and supply quantity and the equilibrium price you worked out above.

**Task 7 Which six month period to expand (12 marks)**

The CEO supplies you with monthly revenue data (in UK£) for the last five years.

Month |
2009 |
2010 |
2011 |
2012 |
2013 |

January | 2302 | 2575 | 2677 | 2314 | 2799 |

February | 2568 | 2690 | 2395 | 2656 | 2403 |

March | 2468 | 2484 | 2429 | 1972 | 2563 |

April | 2567 | 2368 | 2654 | 2651 | 2763 |

May | 2345 | 2531 | 2786 | 2655 | 2230 |

June | 2988 | 2235 | 2454 | 2545 | 2871 |

July | 3030 | 2576 | 2599 | 2445 | 2775 |

August | 2466 | 2684 | 2377 | 2457 | 2336 |

September | 2399 | 2485 | 2425 | 1977 | 2565 |

October | 2561 | 2499 | 2672 | 2347 | 2641 |

November | 2436 | 2657 | 2764 | 2984 | 2455 |

December | 2898 | 2486 | 2655 | 2640 | 2683 |

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**For each calendar month**work out the average revenue (2 d.p.) over the five year period 2009-2013.

The CEO wants to invest and expand production for one of the two six month periods either January to June or July to December.

**For each six month period**work out the average monthly revenue (2 d.p.). What six month period does this suggest would be best to expand production?

**For each six month period**work out the range in monthly revenue (2 d.p.). What six month period does this suggest would be best to expand production?

**For each six month period**work out what is the standard deviation in monthly revenue (1 d.p.). What six month period does this suggest would be best to expand production?

**Use your answers above**to explain to the CEO which six month period would be best overall to expand production in.

**Task 8 Investment projection (5 marks)**

Overall you have projected a 14.87% year on year return in investment for the company. If the company start with an investment of £132,000 how long will it take the company to at least double its investment? ** **

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