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## Introduction to Quantitative Methods

Task 1 of 1: Numeracy Assessment and Exercise
Question 1:
Given are the revenue and expenses of DYI Stores Ltd., a high street chain selling DYI goods.
Year Revenue Expenses Actual profits budgeted profits
2013 £ 316,449,668.60 £ 313,395,191.59 £ 3,046,000.00
2014 £ 354,423,628.83 £ 344,734,710.74 £ 9,561,000.00
2015 £ 396,954,464.29 £ 379,208,181.82 £ 18,115,000.00

a. The above table requires computation of Actual Profits.
Actual Profits for each year = Revenue of the year – Expenses of the year.
Year Revenue Expenses Actual profits budgeted profits
2013 £ 316,449,668.60 £ 313,395,191.59 = £ 316,449,668.60 – £ 313,395,191.59
= £ 3,054,477.01 £ 3,046,000.00
2014 £ 354,423,628.83 £ 344,734,710.74 = £ 354,423,628.83 – £ 344,734,710.74
= £ 9,688,918.09 £ 9,561,000.00
2015 £ 396,954,464.29 £ 379,208,181.82 = £ 396,954,464.29 – £ 37,9,208,181.82
= £ 1,77,46,282.47 £ 18,115,000.00

b. Computation of likely profit of 2016:
Annual Revenue Increase Rate = (Revenue of Year 2 – Revenue of Year 1) / Revenue of Year 1
Year Revenue Annual Revenue Increase rate
2013 £ 316,449,668.60
2014 £ 354,423,628.83 = (£ 354,423,628.83 – £ 316,449,668.60) / £ 316,449,668.60
= 12.00%
2015 £ 396,954,464.29 = (£ 396,954,464.29 – £ 354,423,628.83) / £ 354,423,628.83
= 12.00%

Similarly,
Annual Expense Increase Rate = (Expense of Year 2 – Expense of Year 1) / Expense of Year 1
Year Expenses Annual Expense Increase rate
2013 £ 313,395,191.59
2014 £ 344,734,710.74 = (£ 344,734,710.7 – £ 313,395,191.59) / £ 313,395,191.59
= 10.00%
2015 £ 379,208,181.82 = (£379,208,181.82 – £ 344,734,710.74) / £ 344,734,710.74
= 10.00%

Budgeted figures for 2016:
Budgeted Revenue of 2016 = Revenue of 2015 + 12% of Revenue of 2015
Budgeted Expense of 2016 = Expense of 2015 + 10% of Expense of 2015
Budgeted Profit = Budgeted Revenue of 2016 – Budgeted Expense of 2016
Year Revenue Expenses budgeted profits
2016 = £ 396,954,464.29 + (12% * £ 396,954,464.29)
= £ 444,589,000.00 = £ 379,208,181.82 + (10% * £ 379,208,181.82)
= £ 417,129,000.00 = £ 444,589,000.00 – £ 417,129,000.00
= £ 27,460,000.00

c. Law of Probability and likelihood of actual profits being higher than the budgeted profits:
Probability is a computed estimation of the accruing of an event. In normal business, the law of probability can be applied to compute several possible outcomes in various arenas of business. Investments, client services and competitive strategies are examples of such fields. In case of investments, the entity should predict how many products they should sell to be able to invest the appropriate quantum of resources into the production. In case of Customer behaviour, prediction is needed for companies to be able to know how they can promote their services.
In the given case, the total number of years for which the profit data is available is 3.
Of the 3 years given, Number of years where Actual Profit is greater than Budgeted Profit = 2
Number of years where Actual Profit is less than Budgeted Profit = 1
Ratio of Actual profit being higher than budgeted profit over actual profit being lower than budgeted profit = 2:1
In terms of percentage, it is 66.67% : 33.33%
Thus, the probability that the actual profit shall be greater than budgeted profit is 66.67%.

Question 2:
Part 1:
a. Given,
Purchase price = £ 350
Reselling price = £ 450
Garage space rent = £ 800 per month
Let the number of units sold in a month be x
Total sales = 450x
Total purchases = 350x
Gross profit = 450x – 350x = 100x
Net profit = Gross Profit – Other expenses
= 100x – 800
Therefore, equation to calculate profit = 100x -800

b. In order to attain Breakeven,
Breakeven units = Fixed Costs / Contribution per unit
Breakeven units = 800 / 100
= 8 units.

c. If John sells 23 pushchairs, his profit shall be :
Equation: 100x – 800
= (100 * 23) – 800
= 2300 – 800
= £ 1500
Part 2:
a. Given,
Purchase Price = £ 0.90
Sales price = £ 1
Store Rent = £ 2000 per month.
Let the number of units sold in a month be x
Total sales = 1x
Total purchases = 0.90x
Gross profit = 1x – 0.90x = 0.10x
Net profit = Gross Profit – Other expenses
= 0.10x – 2000
Therefore, equation to calculate profit = 0.10x – 2000

b. In order to attain Breakeven,
Breakeven units = Fixed Costs / Contribution per unit
Breakeven units = 2000 / 0.10
= 20,000 units.

c. If Hector sells 15000 products, his loss shall be :
Equation: 0.10x – 2000
= (0.10 * 15000) – 2000
= 1500 – 2000
= £ 500 loss

Question 3:
Part 1:
a. Equation for computation of Maximum Profit:
Relationship between sales and Price:
First, we need to write the equation to represent the calculation
Profit = Revenue – Cost
Revenue = Quantity sold * Price
R = Revenue
Using the High-Low method,
At price 15, quantity = 0
At price 14, quantity = 15
Quantity change per change in price = (15-0) / (15-14)
= 15
Now, at price = 14, quantity = 1
Quantity Per Day)*( = (225 – 15P) * 10
Revenue Per Day = P * (225 – 15P) * 10
= 2250P – 150P2
Cost Per Day = 10*4.5 * (225 – 15P) + 130
= 10125 – 675P + 130
= 10255 – 675P
Profit Per Day = Revenue Per Day – Cost Per Day
= 2250P – 150P2 – 10255 + 675P
= 2925P – 150P2 – 10255
This equation can be used to create a table for price range from £15 to £3

b. Table presenting price and profit:
Price Quantity per hour Revenue per day(£) Cost (£) Profit (£)
3 180 5400 8230 -2830
4 165 6600 7555 -955
5 150 7500 6880 620
6 135 8100 6205 1895
7 120 8400 5530 2870
8 105 8400 4855 3545
9 90 8100 4180 3920
10 75 7500 3505 3995
11 60 6600 2830 3770
12 45 5400 2155 3245
13 30 3900 1480 2420
14 15 2100 805 1295
15 0 0 130 -130

Note: The cost also includes the fixed cost of rent space.

c. Graph presentation:

d. The maximum possible profit is £ 3,995 per day.
The selling price to achieve this profit is £ 10 per unit.

Part 2:
a. Tabular Presentation of Price and Profit:
In order to derive a relationship between price and quantity, the quantity sold = 180 – 40P.
This has been computed using the high low method.
At price 4, quantity = 20
At price 1, quantity = 140
quantity change per change in price = (140 -20) / (4 – 1) = 40
Say for Price = 1, Quantity is 140.
Quantity = 180 – (40*price)
Quantity sold = 180 – 40P
Revenue = P * (180 – 40P)
= 180P – 40P2
Cost = (Quantity * 0.65) + 60
= (180 – 40P) * 0.65 + 60
= 117 – 26P + 60
= 177 – 26P
Profit = Revenue – Cost
= 180P – 40P2 – (177 – 26 P)
= 206P – 40P2 – 177

Price Quantity Revenue Cost Profit/(Loss)
4 20 80 73 7
3.5 40 140 86 54
3 60 180 99 81
2.5 80 200 112 88
2 100 200 125 75
1.5 120 180 138 42
1 140 140 151 -11
0.5 160 80 164 -84

b. Diagrammatic presentation of the above:

c. Optimal Selling Price = £ 2.5 as it yields the maximum profit of £ 88.