Introductory Mathematics
& Statistics
Chapter 2
Percentages
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Learning Objectives
 Understand and use percentages
 Apply percentages to common commercial situations
 Calculate commission (including brokerage)
 Calculate discounts (including chain, trade and cash
discounts)
 Calculate tax (including GST, personal tax, company
tax, FBT and land tax)
 Calculate profit and loss
 Calculate stamp duty
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2-2
2.1 Conversion to and from percentages
• Conversion of a fraction to a percentage
– Multiply by 100 and use % sign
5
Express as a percentage
 E.g.
8
5
  100  62.5%
8
• Conversion of a decimal to a percentage
– Multiply by 100 by moving the decimal point 2 places to the
right and then add a % sign
 E.g.
Express 0.269 as a percentage
0.269  100  26.9%
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Conversion to and from percentages
(cont…)
• Conversion of a percentage to a fraction
– Divide by 100 and remove % sign, then simplify
 E.g. Express 72% as a fraction

•
72 18

100 25
Conversion of a percentage to a decimal
– Divide the percentage by 100 by moving the decimal point 2
places to the left, and then remove the % sign
 E.g.
Express 45.3% as a decimal
45.3  100  0.453
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2.2 Commission
•
An agent is paid a commission when he or she sells goods and
services
•
Commission can be paid by looking at a either
– fixed amount (irrespective of sales)
C  F  S  R
– straight commission with no fixed amount
CSR
•
Where:
 S = sale amount
 R = rate of commission per sale
 F= fixed amount paid (irrespective of sales)
 C= commission earned
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2.2 Commission (cont…)
• Brokerage
– Brokerage is commission paid to a stockbroker who acts on
a client’s behalf
– Brokerage rates vary according to the type of transaction,
but general rules that apply are:
 When selling, brokerage is subtracted from the proceeds
of the sale
 When buying, brokerage is added to the amount that
you must pay for the stock or shares
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2.3 Discounts
•
Discount is an inducement for the customer to make a
purchase
E.g.
 two for the price of one
 an advertised special with an expiry date
 a fixed amount off the price
 a reduction in the unit price if large quantities are
purchased
•
On some occasions there may be more than one discount
on an item. These multiple discounts are called chain
discounts
•
A reduction in price is referred to as a discount and is often
expressed in the form of a percentage
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2.3 Discounts (cont…)
– To calculate the amount of discount,
– Where:
 L = list price
 D = amount of discount
 R = rate of discount
 DP = discount price
– The rate of discount is:
D
R
L
– The amount of discount
D  R L
– The discount price (or net price) is:
DP  L  D
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2.3 Discounts (cont…)
• Trade discount
– A special discount when goods and services are purchased
from one business by another business.
E.g.
 a builder purchasing timber from a timber yard
 a service station obtaining tyres from a manufacturer
 an electrician purchasing cables and switches from an
electrical supplier
– A typical trade discount would range between 10% and
25%, depending on the trade and the item
– The method for calculating trade discounts is the same as in
the previous examples
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2.3 Discounts (cont…)
• Cash discount
– A form of discount that is given if the purchaser pays in cash
or by cheque
– The percentage of the discount may depend on how quickly
the bill is paid.
 E.g. A supplier of electrical goods informs retailers that the
following discounts are available for early payment of
purchases:
• Within 7 days – 10.0 %
• Within 14 days – 7.5 %
• Within 28 days – 2.5 %
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2.4 Goods and Services Tax (GST)
• An Australian broad-based tax of 10% on most supplies of
goods and services
• The GST replaced a number of other taxes, including
wholesale sales tax, that were applied at varying rates to a
range of products
• The amount of GST payable can be calculated easily by
working out 10% of the cost of the goods or service
• To work out how much a customer has paid in GST, divide
the final price by 11
GST inclusive price  11  GST
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2.5 Personal income tax
• Individuals who receive income are liable for
personal income
• Definitions
– Gross income is the total amount received or accrued
– Assessable income is gross income less exempt income
– Allowable deductions are costs of producing income and
certain concessional deductions
– Taxable income is assessable income less allowable
deductions
– Tax payable is tax according to the table on taxable income
less rebates
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2.5 Personal income tax (cont…)
•
The tax-free threshold for most resident individuals is $6000
•
Taxpayers are subject to a Medicare levy, normally calculated at
the rate of 1.5% of your taxable income
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2.6 Company tax
•
A company is a distinct legal entity with its own income tax
liability that is separate from personal income tax
•
The income tax of companies is calculated on taxable income,
which is the income earned by the company less any allowable
deductions
•
The amount of tax to be paid is reduced by any PAYG (pay as
you go) instalments paid during the year
•
The general rate of tax payable by companies on 2009–10
income is 30%
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2.7 Fringe Benefits Tax (FBT)
•
FBT is an Australian government tax paid on certain benefits
employers provide to their employees in place of, or in addition
to, salary
•
FBT is separate from income tax
•
FBT is payable by the employer and is based on the taxable
value of the various fringe benefits provided
•
The rate of FBT is currently aligned with the top marginal
income tax rate and was set at 46.5% in 2009–10
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2.8 Land tax
•
Land tax is a state tax levied on the owners of land in various
states of Australia
•
A principal place of residence (your home) or land used for
primary production (a farm) is exempt from land tax
•
You may be liable for land tax if you own or part-own:
– vacant land, including vacant rural land
– a holiday home
– investment properties
– company title units
– residential, commercial or industrial units
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2.8 Land tax (cont…)
•
Land tax is calculated on the combined value of all the taxable
land you own
•
The land tax threshold varies from state to state
•
For example, for 2009 it was
– $368 000 in New South Wales
– $250 000 in Victoria
– $110 000 in South Australia
– $600 000 in Queensland
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2.9 Profit and loss
•
The actual profit is the difference between the price the item is sold
for (the selling price) and the cost of that item (the cost price)
•
An item may be sold for an amount that is less than the cost price.
The profit is therefore negative and is referred to as a loss
•
The actual cost of an item is often difficult to calculate, since it
involves not only the cost of obtaining the item from the supplier but
other costs as well.
– These include the general costs of running a business, e.g.
 wages
 insurance
 taxes
 stationery
 equipment
 electricity
 rent
 other overhead expenses
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2.9 Profit and loss (cont…)
•
The following definitions will be useful for profits:
SP = selling price (not including any applicable GST)
CP = cost price
P = actual profit (when the value of SP exceeds the value of
CP)
Ps = profit rate (or mark-up rate) expressed as a fraction of
selling price
Pc = profit rate (or mark-up rate) expressed as a fraction of cost
price
(The profit rates expressed as percentages are 100 × Ps and
100 × Pc, respectively)
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2.9 Profit and loss (cont…)
•
The following definitions will be useful for loss:
L = actual loss (when the value of CP exceeds the value of SP)
Ls = loss rate (or mark-down rate) expressed as a fraction of
selling price
Lc = loss rate (or mark-down rate) expressed as a fraction of
cost price
(The loss rates expressed as percentages are 100 × Ls and
100 × Lc , respectively)
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2.9 Profit and loss (cont…)
• The following relationships
hold:
P  SP  CP
L  CP  SP
P
Ps 
SP
L
Ls 
SP
P
Pc 
CP
L
Lc 
CP
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2.10 Stamp duty
•
•
Stamp duty is a tax on transactions that is levied by the
individual states in Australia
Stamp duty is really a tax on the document of transfer (the title
transfer), and not on the property itself
• Stamp duty on vehicles
– Stamp duty is based on the market value of the vehicle or
the price that was paid, whichever is greater
• Stamp duty on real estate
– Stamp duty is payable on the purchase price of property
– The amount payable depends on the price of the property
and which state it was bought in
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Summary
•
The application of percentages in modern business practice is
widespread and this chapter has presented some of the more
common examples
•
The introduction of the GST into Australia was part of significant
tax reform including substantial personal income tax cuts and
the removal of a number of indirect taxes
•
In using the taxation tables and related information, it is
important to be aware that rates charged may vary from year to
year
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