Stage 2 Mathematics Pathways
Assessment Type 3: Investigation 20-credit
Mathematics and Small Business focus: Business Feasibility Study
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Topic: Mathematics and Small Business:
Costing Calculations; Business Structure and Taxation
Introduction
Ann and Bob own and operate a building and hardware business in a joint partnership; Bob
is the builder and Ann runs the shop and does cashier duties. They also have a son, Dave,
who works part-time in the business. Dave considers that there is a market for gift packs in
the form of basic handyman starter kits consisting of hammer, pincers and assorted nails. He
decides to investigate the feasibility of setting up this venture as a sideline within the
Hardware shop.
Mathematical Investigations
After some initial market research, Dave decides to make 3 types of Gift Packs that range in
price from $50 to $100. These are shown below with the associated variable and fixed costs:
Type of Gift Pack
Basic
Regular
Deluxe
Sale Price
$50
$70
$100
Variable Costs
$32
$53
$78
Fixed Costs
$1000
$1000
$1000
INTRODUCTION: Read through the whole paper and then write a paragraph explaining
the problems and the maths needed to find solutions to this investigation.
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1. List some of the fixed and variable costs, which are likely to contribute to the costs
shown in the table above.
Fixed costs:
Variable Costs:
Page 1 of 9
Ref: A197152
Last Updated: 2/6/2016 10:57 AM
Stage 2 Mathematics Pathways task for use from 2012
External Investigation – 20-credit MSB (revised July 2012)
© SACE Board of South Australia 2011
2. Calculate the number of packs required to break-even in each of the three scenarios
listed above and calculate the revenue from these sales.
Basic
Regular
Deluxe
Breakeven number
Revenue
3. Explain why these solutions do not give Dave a useful indication of the number of packs
that he needs to sell in his business plan.
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Ref: A197152
Last Updated: 2/6/2016 10:57 AM
Stage 2 Mathematics Pathways task for use from 2012
External Investigation – 20-credit MSB (revised July 2012)
© SACE Board of South Australia 2011
4. Dave has now realised that he needs to focus on one specific type of gift pack. Using the
‘regular’ pack above and using the figures for variable and fixed costs given, graphically
display Dave’s breakeven scenario, using appropriate technology.
Paste printout or graph paper here.
5. Discuss how realistic the breakeven number is for Dave’s business.
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Ref: A197152
Last Updated: 2/6/2016 10:57 AM
Stage 2 Mathematics Pathways task for use from 2012
External Investigation – 20-credit MSB (revised July 2012)
© SACE Board of South Australia 2011
6. a) Assuming that Dave makes a profit in his first year of operating this business, predict
the number of gift packs he might sell and the profit made in this year.
b) Give some indication of the placement of the business premises and the effect that
this might have on sales.
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7. In addition to Dave’s salary of $35,000 from working for ‘Bob’s Building and
Hardware’, he retains the profits from the sale of the gift packs.
Calculate his taxable income, tax payable and his net income for 2011/2012, including
the Medicare levy and assuming he has private health cover.
Page 4 of 9
Ref: A197152
Last Updated: 2/6/2016 10:57 AM
Stage 2 Mathematics Pathways task for use from 2012
External Investigation – 20-credit MSB (revised July 2012)
© SACE Board of South Australia 2011
8. Using the ‘Regular’ pack chosen previously, and initially assuming 300 sales in a year,
use appropriate technology to investigate the effect of varying the sales price.
Complete the table below for sales figures of 300 ‘Regular’ packs.
Selling
Price
Revenue
Total
costs
Profit
Breakev
en no.
Dave’s
income
before
tax
Tax
payable
Dave’s
net
income
$70
$80
$65
Now investigate sales per year of 400 ‘Regular’ packs. Choose two further appropriate
selling prices to investigate mathematically other than $70.
Selling
price
Revenue
Total
costs
Profit
Breakev
en no.
Dave’s
income
before
tax
Tax
payable
Dave’s
net
income
$70
Include calculations completed by hand in the tables above.
Include excel printouts showing calculations completed electronically and graphs on the
following page.
This page could include:
 Graphs representing the information for the sales of 300 and 400‘Regular’ packs
Excel printouts of tables if completed electronically rather than by hand.
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Ref: A197152
Last Updated: 2/6/2016 10:57 AM
Stage 2 Mathematics Pathways task for use from 2012
External Investigation – 20-credit MSB (revised July 2012)
© SACE Board of South Australia 2011
9. Analysis/Discussion for question 8.
Critically analyse your results, considering the appropriateness of each of the chosen sale
price scenarios.
Consider:
 Highest and lowest realistic possible prices for the gift packs and why.
 How the packs could be made and sold more cheaply, and the effect this would have on
his profit.
 How Dave could increase the number of sales made in a year and what effect this would
have on his profit.
 The effect of employing a second person to prepare and distribute the gift packs if he
were unable to cope with the demand by himself.
 How many packs would need to be sold to enter the next tax bracket and would it be
feasible or worth the effort?
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Ref: A197152
Last Updated: 2/6/2016 10:57 AM
Stage 2 Mathematics Pathways task for use from 2012
External Investigation – 20-credit MSB (revised July 2012)
© SACE Board of South Australia 2011
10. Conclusion
In concluding this section, justify your choice of the best scenario for Dave, that is, the
recommended selling price and predicted number to be sold in a year, referring to the
limitations of the original problem as well as appropriate refinements or extensions.
11. a) If Bob and Ann were to offer Dave a 30% (Bob) – 30% (Ann) – 40% (Dave)
partnership in the business for 2012-13, instead of the wage he currently receives, by
how much would Dave be better off if the business were expected to earn $120,000 net
profit for the following financial year? Compare Dave’s net income for each year,
assuming the tax table is unchanged.
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Ref: A197152
Last Updated: 2/6/2016 10:57 AM
Stage 2 Mathematics Pathways task for use from 2012
External Investigation – 20-credit MSB (revised July 2012)
© SACE Board of South Australia 2011
12. Analysis/Discussion/Conclusion for question 11.



Discuss who will benefit most from a partnership split, justifying your points with
figures.
What is your advice to Bob, Ann and Dave on what action to take for the following
year?
Should Dave continue with his gift pack venture in 2012-13, if he were to take up the
partnership offer, or not bother?
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Ref: A197152
Last Updated: 2/6/2016 10:57 AM
Stage 2 Mathematics Pathways task for use from 2012
External Investigation – 20-credit MSB (revised July 2012)
© SACE Board of South Australia 2011
Formula Sheet for Investigation Under Supervision
 Revenue = Selling Price per item X Number of items sold, i.e.
 Revenue = price X turnover
 Total costs = Fixed costs + Variable costs per item X No. of items.
 Marginal income = Selling price – Variable costs per item.
 Breakeven number occurs when Revenue = Total costs.
 Breakeven number =
 Profit = Revenue – Total Costs
Tax rates 2010-11
Taxable income
Tax on this income
0 - $6,000
Nil
$6,001 - $37,000
15c for each $1 over $6,000
$37,001 - $80,000
$4,650 plus 30c for each $1 over $37,000
$80,001 - $180,000
$17,550 plus 37c for each $1 over $80,000
$180,001 and over
$54,550 plus 45c for each $1 over $180,000
The above rates do not include the Medicare levy of 1.5% of taxable income.

Net income = Taxable Income – Tax payable(including Medicare
levy)
(Include a photocopy of Tax spreadsheet here)
Page 9 of 9
Ref: A197152
Last Updated: 2/6/2016 10:57 AM
Stage 2 Mathematics Pathways task for use from 2012
External Investigation – 20-credit MSB (revised July 2012)
© SACE Board of South Australia 2011