# Introduction to Quantitative Methods

Introduction to Quantitative Methods

Introduction to Quantitative Methods

Assignment 2 Set 4

- A father dies and leaves £950,000 in his will. First, funeral and legal expenses of £35,000 have to be paid. Then inheritance tax of 40% is charged on all amounts above £650,000. The remaining money is to be divided between his three sons, Martin aged 50, Norman aged 54 and Oliver aged 59 in the ratio of their ages, how much will they each .

** **

**Taxable Estate = Total Estate – Less Funeral and Legal Expenses**

**Total estate = £950,000**

**Less funeral and legal expenses = £35,000**

**Taxable estate: £950,000 – £35,000 = £915,000**

** **

**Inheritance tax=**

**Nil-rate band: £650,000**

**Taxable amount: £915,000 – £650,000 = £265,000**

**Inheritance tax at 40%: £265,000 x 0.4 = £106,000**

**Inheritance Tax = £****106,000**

** **

**The remaining amount after tax:**

**Taxable estate: £915,000**

**Less inheritance tax: £106,000**

**Remaining amount: £915,000 – £106,000 = £809,000**

** **

**Division Ratios**

**Martin (50): 50 parts**

**Norman (54): 54 parts**

**Oliver (59): 59 parts**

**Total parts: 50 + 54 + 59 = 163 parts**

** **

**Total for Each son’s share:**

**Martin: £809,000 x (50/163) = £248,159.50**

**Norman: £809,000 x (54/163) = £268,012.30**

**Oliver: £809,000 x (59/163) = £292,828.20**

- A clothing manufacturer makes 50,000 shirts at a total cost of £75,000. When the manufacturer sold them to the wholesaler he made 30% profit. The wholesaler bought boxes to put the shirts in which cost £30 per hundred. He then sold the shirts in the boxes to clothing shops making 40% profit.

- How much did the wholesaler pay?

**Manufacturer’s cost per shirt: £75,000 / 50,000 = £1.50 per shirt**

** – Manufacturer’s selling price to wholesaler: £1.50 + (£1.50 x 0.3) = £1.95 per shirt**

**The wholesaler paid £1.95 per shirt.**

**Cost for packaging; 50,000/100= 500 x £30 =£1500**

**Total wholesaler paid = 1.95 ****× 50000 = 97,500**

**=£97,500 – ****£1500**

**=****£****96,000**

- How much did the clothing shops pay for each shirt?

**– Wholesaler’s cost per shirt: £1.95**

** – Wholesaler’s cost for boxes: £30 per 100 = £0.30 per shirt**

** – Wholesaler’s selling price to clothing shops: £1.95 + (£1.95 + £0.30) x 0.4 = £2.85 per shirt**

** – The clothing shops paid £2.85 per shirt.**

- The clothing shop sold the shirts at £6.99. What percentage profit did the clothing shop make?

** – Clothing shop’s cost per shirt: £2.85**

** – Clothing shop’s selling price per shirt: £6.99**

** – Clothing shop’s profit per shirt: £6.99 – £2.85 = £4.14**

** – Clothing shop’s profit margin: (£4.14 / £2.85) x 100% = 145%**

- A clothing shop brought 200 shirts. It was able to sell 150 shirts at the full price of £6.99, but sold the rest to a discount shop at 20% more that it paid. How much actual profit did it make on selling the shirts?

** – Shirts sold at full price (150): 150 x £6.99 = £1,048.50**

** – Shirts sold to discount shop (50): 50 x (£2.85 x 1.2) = £171**

** – Total revenue: £1,048.50 + £171 = £1,219.50**

** – Total cost: 200 x £2.85 = £570.00**

** – Actual profit: £1,219.50 – £570.00 = £649.50**

** **

** **

- An account with an interest rate of 3.4% has been left untouched for 5 years. If the account now has £7560 in it, how much was originally put into the account?

**The compound interest =**

** **

**A = P (1 + r/2) ^{2t}**

** **

**Where:**

**A = £7,560**

**P = the original principal amount**

**r = 3.4% or 0.034**

**n = 4 years**

**t = 5 years**

** **

**P = A / (1 + r/n)^(nt)**

**P = £7,560 / (1 + 0.034/4)^(4*5)**

**P = £7,560 / 1.00850^20**

**P = £7,560 / 1.184453596**

**P = £6,382.68**

- A company has decided to set up a fund for its employees with an annual investment on the 1
^{st}January of £5000. If the interest rate is 4% per year how much will be in the fund at the end of four years?

** **

**Year 1:**

**– Principal: £5000**

**– Interest: £5000 x 0.04 = £200**

**– Total: £5000 + £200 = £5200**

** **

**Year 2:**

**– Principal: £5200**

**– Interest: £5200 x 0.04 = £208**

**– Total: £5200 + £208 = £5408**

** **

**Year 3:**

**– Principal: £5408**

**– Interest: £5408 x 0.04 = £216.32**

**– Total: £5408 + £216.32 = £5624.32**

** **

**Year 4:**

**– Principal: £5624.32**

**– Interest: £5624.32 x 0.04 = £224.96**

**– Total: £5624.32 + £224.96 = £5849.28**

- Tickets for a concert cost either £15.00 or £12.00. The number of the cheaper tickets sold was four times the number of dearer tickets sold and the total receipts were £5292. How many of each type of ticket were sold?

**– Let x be the number of dearer tickets sold at £15 each**

**– Let 4x be the number of cheaper tickets sold at £12 each**

**15x + 12(4x) = 5292**

**15x + 48x = 5292**

**63x = 5292**

**x = 5292 / 63**

**x = 84**

**4x = 4 × 84 = 336**

** **

**84 dearer tickets at £15 each **

** **

**336 cheaper tickets at £12 each **

- The cost of 5 kg of parsnips and 8 kg of swedes is £9.84. The cost of 9 kg of parsnips and 10 kg of swedes is £14.72. Find the cost of parsnips and swedes per kg.

**Let x be =1kg of parsnips **

**Let y be = 1kg of Swedes**

** **

**Formulate Equation**

** 5x+8y=9.84**

** **

** ****9x+10y=14.72**

**Step 5Solve the system of equations.**

**5x+8y=9.84**

** 9x+10y=14.72**

**Multiply both sides of the first equation by 5**

**25x+40y=49.2**

**9x+10y=14.72**

**Multiply both sides of the second equation by 4**

**25x+40y=49.2**

**-36x-40y=-58.88**

**Add the equations vertically to eliminate y**

**-11x=-9.68**

** **

******Divide both sides of the equation by -11.**

**x=0.88**

**Substitute x=0.88 into the equation to find the value of y.**

**5x+8y=9.84**

**(5****×**** 0.88+8y=9.84**

**4.4+8y=9.84**

**8y=9.84-4.4**

**8y=5.44**

**Divide both sides of the equation by 8.**

**y=0.68**

** **

**Parsnips cost= £0.88 per kg **

** swedes cost = £0.68 per kg.**

- It is estimated that the fixed costs of production are £2500 and the variable costs are £65 per unit. The price is given by: p = 170 – x where x is the number of units.

- Find an expression for revenue.

– The price is given by the equation: p = 170 – x

– Revenue = Price × Quantity

– Revenue = (170 – x) × x

– Revenue = 170x – x^2

- Find an expression for profit.

– Profit = Revenue – Fixed Costs – Variable Costs

– Profit = (170x – x^2) – 2500 – 65x

– Profit = 105x – x^2 – 2500

- Find the break-even points.

– At the break-even point, Profit = 0

– 105x – x^2 – 2500 = 0

– x^2 – 105x + 2500 = 0

– Using the quadratic formula:

x = (105 ± √(105^2 – 4 × 1 × 2500)) / (2 × 1)

x = (105 ± √(11025 – 10000)) / 2

x = (105 ± √1025) / 2

x = (105 ± 32.03) / 2

– Therefore, the break-even points are:

x1 = (105 – 32.03) / 2 = 36.49 units

x2 = (105 + 32.03) / 2 = 68.52 units

- Find the prices at the break-even points.

– At the first break-even point (x1 = 36.49 units):

Price = 170 – x1 = 170 – 36.49 = £133.51

– At the second break-even point (x2 = 68.52 units):

Price = 170 – x2 = 170 – 68.52 = £101.48

- A company wants to determine the size of orders that will give it the lowest annual inventory cost. It has to take into consideration that for each order it places it incurs order costs but if it orders larger quantities it has higher storage costs.

You are determining the stock policy for Part P and have the following data:

Cost of placing an order £17.00

Storage cost per unit for one year £2.70

Annual demand 1000 units

Annual Inventory Cost = Order Cost + Storage Cost

where the annual costs for an order quantity of ** n** units will be:

Order Cost = (Number of Orders) x (Cost per Order)

= __1000__ x 17

*n*

Storage Cost = (Average Stock Levels) x (Storage Costs per unit)

= **n** x 2.70

2

- Calculate the order cost, storage cost and inventory cost for order

quantities 50, 100, 150, 200 and 250.

**Order Quantity = 50**

**Order Cost = (1000/50) x 17 = £340**

**Storage Cost = (50/2) x 2.70 = £67.50**

**Total Inventory Cost = £340 + £67.50 = £407.50**

** **

**Order Quantity = 100**

**Order Cost = (1000/100) x 17 = £170**

**Storage Cost = (100/2) x 2.70 = £135**

**Total Inventory Cost = £170 + £135 = £305**

** **

**Order Quantity = 150**

**Order Cost = (1000/150) x 17 = £113.59**

**Storage Cost = (150/2) x 2.70 = £202.50**

**Total Inventory Cost = £113.59 + £202.50 = £316.09**

** **

**Order Quantity = 200**

**Order Cost = (1000/200) x 17 = £85**

**Storage Cost = (200/2) x 2.70 = £270**

**Total Inventory Cost = £85 + £270 = £355**

** **

**Order Quantity = 250**

**Order Cost = (1000/250) x 17 = £68**

**Storage Cost = (250/2) x 2.70 = £337.50**

**Total Inventory Cost = £68 + £337.50 = £405.50**

- Plot the graph of the inventory costs.

** **

- Use the graph to find the order quantity which gives the minimum inventory cost and find this minimum cost.

**The minimum total inventory cost is £305, which occurs at an order quantity of 100 units.**