Year 12 Maths A Textbook - Chapter 2

2syllabussyllabusrrefefererenceenceStrandFinancial mathematics

Core topicManaging money 2

In thisIn this chachapterpter2A Inflation and appreciation

2B Modelling depreciation

2C Straight line depreciation

2D Declining balance or diminishing value method of depreciation

2E Depreciation tables

2F Future and present value of an annuity

Appreciation and depreciation

Maths A Yr 12 - Ch. 02 Page 53 Friday, September 13, 2002 9:01 AM

54

M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d

Introduction

With the passing years, it seems that the costof goods and services continues to rise. This

appreciation

in price is known as

inflation

.In Australia, these price movements aremeasured by the

Consumer Price Index(CPI)

. When goods lose value, this is called

depreciation

. While some assets such ashouses and land increase in value, others,such as new cars and computers, continue todecrease. Throughout this chapter we shalllook at methods of calculating appreciationand depreciation; how these may be repre-sented graphically; and how they can help usplan for the future.

1

Calculate the following, giving your answer in decimal form.

a

1

+

b

1

+

c

1

+

d

1

−

e

1

−

f

1

−

2

Calculate each of the following.

a

10% of $40

b

12.5% of $40

c

3.95% of $100

d

4.125% of $10

e

5.5% of $20

f

3.625% of $120.50

3

Calculate the following, rounding to the nearest cent.

a

Increase $40 by 10%.

b

Increase $2.50 by 20%.

c

Increase $7.45 by 2.5%.

d

Decrease $20 by 10%.

e

Decrease $145 by 6.25%.

f

Decrease $4000 by 1.5%.

4 a

Express $20 as a percentage of $50.

b

Express $350 as a percentage of $400.

c

Express $6.45 as a percentage of $10.

d

Express 45c as a percentage of $10.

e

Express $1.80 as a percentage of $3.50.

f

Express 12.5c as a percentage of $8.

5

Calculate the following, expressing your answer in decimal form correct to 2 decimalplaces.

a b c

d e f

10100--------- 1

100--------- 3.5

200---------

3.75100---------- 4.5

100--------- 3

1200------------

13

100---------+

21 3

100---------–

21

4.2600---------+

5

1 4.2600---------–

51

3.8751200-------------+

101 3.875

1200-------------–

10

Maths A Yr 12 - Ch. 02 Page 54 Wednesday, September 11, 2002 3:52 PM

C h a p t e r 2 A p p r e c i a t i o n a n d d e p r e c i a t i o n

55

Consumer Price Index

The Consumer Price Index (CPI) measures price movements in Australia. Let us investigate this further to gain an understanding of how this index is calculated.

A ‘basket’ of goods and services, representing a high proportion of household expenditure is selected. The prices of these goods are recorded each quarter. The basket on which the CPI is based is divided into eight groups, which are further divided into subgroups. The groups are: food, clothing, tobacco/alcohol, housing, health/personal care, household equipment, transportation and recreation/education. Weights are attached to each of these groups to reflect the importance of each in relation to the total household expenditure. The following table shows the weights of the eight groups.

The weights indicate that a ‘typical’ Australian household spends 19% of its income on food purchases, 7% on clothing and so on. The CPI is regarded as an indication of the cost of living as it records changes in the level of retail prices from one period to another.

Let us consider a simplified example showing how this CPI is calculated and how we are able to compare prices between one period and another.Take three items with prices as follows:• A pair of jeans costing $75• A hamburger costing $3.90• A CD costing $25.Let us say that during the next period of time, the jeans sell for $76, the hamburger for $4.20 and the CD for $29. This can be summarised in a table.

The appropriate weight of each item is multiplied by the price of the item to determine what proportion the household would spend on each item in each period. The total weighted expenditure is then calculated.

inve

stigationinvestigatio

n

CPI group Weight (% of total)

Food 19

Clothing 7

Tobacco/alcohol 8.2

Housing 14.1

Health/personal care 5.6

Household equipment 18.3

Transportation 17

Recreation/education 10.8

Period 1 Period 2

Item Weight (W) Price (P) W × P Price (P) W × P

Jeans 7 $75 525 $76 532

Hamburger 19 $ 3.90 74.1 $ 4.20 79.8

CD 10.8 $25 270 $29 313.2

Total 869.1 925


56

M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d

The first period is regarded as the base and is allocated an index number of 100. The second period is compared with the first period and is expressed as a percentage of it. So, period 2 is 925

÷

869.1

×

100%; that is, 106.4% compared with period 1 as 100%. This means that the average price of these items has risen by 6.4% (the inflation factor) from period 1 to period 2 and the CPI for period 2 is 106.4.

Task 1

1

Take 8 items with which you are quite familiar, one item from each of the CPI groups (similar to the ones chosen in the previous example — jeans, hamburger, CD). Slot them into their correct categories and note the weighting for each.

2

Draw up a table similar to the previous one, entering the item names and corresponding weightings.

3

Let the period 2 price be the typical prices of the items at this period of time. Enter these values into the table.

4

Let the period 1 price be the prices of these items one year ago (use your knowledge of these items to determine an educated realistic estimate). Enter these values into the table in the period 1 price column.

5

Complete the table as demonstrated above.

6

Determine the CPI for period 2 compared with period 1 as the base period.

7

What is your inflation factor?

Task 2

1

Conduct a search of the World Wide Web using the words ‘inflation CPI Australia’.

2

Research the history and use of the CPI in Australia. What items are in the subgroups? You should discover that separate CPI indices are calculated for each Australian State. These are then combined to produce an average CPI for the whole of Australia, like that shown in the graph below and the table at right.

3

Write a report of your findings. Remember to reference the source of your material.

Year CPI

1988–89 7.3%

1989–90 8.0%

1990–91 5.3%

1991–92 1.9%

1992–93 1.0%

1993–94 1.8%

1994–95 3.2%

1995–96 4.2%

1996–97 1.3%

1997–98 0.0%

1998–99 1.3%

Sep-

93D

ec-9

3

Jun-

94M

ar-9

4

Sep-

94D

ec-9

4M

ar-9

5Ju

n-95

Sep-

95D

ec-9

5M

ar-9

6Ju

n-96

Sep-

96D

ec-9

6M

ar-9

7Ju

n-97

Sep-

97D

ec-9

7M

ar-9

8Ju

n-98

Sep-

98D

ec-9

8M

ar-9

9Ju

n-99

Sep-

99D

ec-9

9M

ar-0

0Ju

n-00

Sep-

00D

ec-0

0M

ar-0

1Ju

n-01

–2% –1%

0%1% 2%3%4%5%6%7%CPI — Inflation — Australia GST

IntroducedJuly 2000


C h a p t e r 2 A p p r e c i a t i o n a n d d e p r e c i a t i o n

57

Inflation and appreciationOne of the measures of how an economy is performing is the rate of inflation. Inflationis the rise in prices within an economy and is generally measured as a percentage. InAustralia this percentage is called the Consumer Price Index (CPI). By looking at theinflation rate we can estimate what the cost of various goods and services will be atsome time in the future.

To estimate the future price of an item one year ahead, we increase the price of anitem by the rate of inflation. The financial functions of graphics calculators can assistthese calculations.

When calculating the future cost of an item several years ahead, the method of calcu-lation is the same as for compound interest. This is because we are adding a percentageof the cost to the cost each year.

Remember the compound interest formula is A = P and so in these

examples P is the original price, R is the inflation rate expressed as a % p.a., n is thenumber of rests per year and T is the time in years. Generally the inflation is expressedas a yearly rate, so the value of n is 1.

The cost of a new car is $35 000. If the inflation rate is 5% p.a., estimate the price of the car after one year.

THINK WRITE

Increase $35 000 by 5%. Future price = 105% of $35 000Future price = 105 ÷ 100 × $35 000Future price = $36 750

1WORKEDExample

1R

100 n×------------------+

n T×

The cost of a television set is $800. If the average inflation rate is 4% p.a., estimate the cost of the television after 5 years.

THINK WRITE

Write the values of P, R, T and n. P = $800, R = 4, T = 5, n = 1

Write down the compound interest formula.

A = P

A = 800

Substitute the values of P, R, T and n. A = $800 × (1.04)5

Calculate. A = $973.32

1

2 1 R100 n×------------------+

n T×

1 4100 1×------------------+

1 5×

34

2WORKEDExample


58 M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d

A similar calculation can be made to anticipate the future value of collectable items, suchas stamp collections and memorabilia from special occasions. This type of item increasesin value over time if it becomes rare, and rises at a much greater rate than inflation. Theamount by which an item grows in value over time is known as appreciation.

Modelling appreciation with the aid of a graphics calculator

You have purchased a rare coin that the coin dealer told you should appreciate by 15% each year. You paid $850 for the coin and hope that its value will treble within the next 10 years. The coin dealer is not sure whether this is the case, so you offer to produce a graph for him displaying the value of the coin over the next 10 years.

Using a graphics calculator greatly simplifies the calculations and will produce a graph that can be used to determine the value of the coin at any period of time. The screens displayed and instructions supplied are those of the TI-83 and Casio CFX-9850GB PLUS.1 Enter the appropriate formula into Y1, with the variable being time (X).

TI: Press then enter the formula.

Casio: Enter TABLE from MENU then enter the formula.

Jenny purchases a rare stamp for $250. It is anticipated that the value of the stamp will rise by 20% per year. Calculate the value of the stamp after 10 years, correct to the nearest $10.

THINK WRITE

Write the values of P, R, T and n.

P = $250R = 20T = 10n = 1

Write down the compound interest formula.

A = P

A = 250

Substitute the values of P, R, T and n.

A = $250 × (1.2)10

Calculate and round off to the nearest $10.

A = $1550

1

2 1 R100 n×------------------+

n T×

1 20100 1×------------------+

1 10×

3

4

3WORKEDExample

inve


n

Y=


C h a p t e r 2 A p p r e c i a t i o n a n d d e p r e c i a t i o n 59

2 Set up the table to start at a minimum of 0 years, increasing in increments of 1 year.

TI: Press [TBLSET] then enter the data.

Casio: Press (RANG) then enter the data.

3 Look at the table produced.

TI: Press [TABLE]. Press (TABL).

4 Scrolling down the table, notice that the coin has trebled in value by Year 8 ($850 × 3 = $2550).

TI: Press the key to navigate. Casio: Press the key to navigate.

5 Set the window to cope with the range of x- and y-values.

TI: Press then enter the data.

Casio: Press [V-Window] then enter the data.

6 Graph the function.

TI: Press . Casio: Press , then select the GRAPH section. Press (DRAW).

2nd F5

2nd F6

� �

WINDOW SHIFT

GRAPH MENUF6



Inflation and appreciation

1 The cost of a yacht is $20 000. If the inflation rate is 4% p.a., estimate the cost of theyacht after one year.

2 For each of the following, estimate the cost of the item after one year, with the giveninflation rate.a A CD player costing $600 with an inflation rate of 3% p.a.b A toaster costing $45 with inflation at 7% p.a.c A loaf of bread costing $1.80 with inflation at 6% p.a.d An airline ticket costing $560 with inflation at 3.5% p.a.e A washing machine costing $925 with inflation at 0.8% p.a.

7 Use the TRACE option to locate the point on the curve where the value of the coin trebles.

TI: Press then move the pointer along the curve.

Casio: Press (TRACE) then move the pointer along the curve.

8 Investigate further to determine the time when the coin will be 4 times its original value.

9 Prepare a report to the coin dealer explaining how the value of the coin changes over time. Provide a table and a graph to support your figures.

10 The coin dealer asks you if he could use your graph for values other than $850 which was the initial value of the coin. Explain how the graph could be used to determine the time when any sum of money doubles, trebles, and so on, as long as the appreciation rate remains at 15% p.a.

TRACE SHIFT

remember1. Inflation is the measure of the rate at which prices increase.2. The inflation rate is given as a percentage and is called the Consumer Price

Index.3. To estimate the cost of an item after one year, we increase the price by the

percentage inflation rate.4. To estimate the cost of an item after several years, we use the compound

interest formula, using the inflation rate as the value of R.5. Rare items such as collectibles and memorabilia increase in value as time goes

on at a rate that is usually greater than inflation.

remember

2A2.1 WORKED

Example

1


C h a p t e r 2 A p p r e c i a t i o n a n d d e p r e c i a t i o n 613 An electric guitar is priced at $850 at the beginning of 2001.

a If the inflation rate is 3.3% p.a., estimate the cost of the guitar at the beginning of2002.

b The government predicts inflation to fall to 2.7% p.a. in 2002. Estimate the cost ofthe guitar at the beginning of 2003.

4 When the Wilson family go shopping, the weekly basket of groceries costs $112.50.The inflation rate is predicted to be 4.8% p.a. for the next year. How much should theWilson’s budget per week be for groceries for the next year?

5 The cost of a skateboard is $550. If the average inflation rateis predicted to be 3% p.a. estimate the cost of a newskateboard in 4 years’ time.

6 The cost of a litre of milk is $1.70. If the inflation rate is anaverage 4% p.a., estimate the cost of a litre of milk after 10years.

7 A daily newspaper costs $1.00. With an average inflationrate of 3.4% p.a., estimate the cost of a newspaper after 5years (to the nearest 5c).

8 If a basket of groceries costs $98.50 in 2001, what wouldthe estimated cost of the groceries be in 2008 if the averageinflation rate for that period is 3.2% p.a.?

9

A bottle of soft drink costs $2.50. If the inflation rate is pre-dicted to average 2% p.a. for the next five years, the cost ofthe soft drink in five years will be:

10 Veronica bought a shirt signed by the Australian cricket team after it won the 1999World Cup for $200. If the value of the shirt increases by 20% per annum for the next5 years, calculate the value of the shirt (to the nearest $10).

11 Ken purchased a rare bottle of wine for $350. If the value of the wine is predicted toincrease at 10% per annum, estimate the value of the wine in 20 years (to the nearest$10).

12 The 1968 Australian 2c piece is very rare. If a coin collector purchased one in 1999for $400 and the value of the coin increases by 15% p.a., calculate its value in 2012(to the nearest $10).

Modelling depreciationAn asset is an item that has value to its owner. Many assets such as cars and computerslose value over time. This is called depreciation.

Consider the case of a new motor vehicle. The value of the car depreciates themoment that you drive the car away from the showroom. This is because the motorvehicle is no longer new and if it were sold it would have to be sold as a used car. Thecar then continues to lose value steadily each year.

A $2.60 B $2.70 C $2.75 D $2.76

WORKEDExample

2

mmultiple choiceultiple choice

WORKEDExample

3



There are two types of depreciation, the straight line method and the decliningbalance or diminishing value method. Depreciation is a significant factor in the oper-ation of a business. The Australian Taxation Office (ATO) allows depreciation on itemsnecessary for the operation of the business as a legitimate tax deduction.

Straight line depreciationThe straight line method is where the asset depreciates by a constant amount each year.When this type of depreciation is graphed, a straight line occurs and the asset willreduce to a value of 0.

In such a case, a linear function can be derived that will allow us to calculate thevalue of the item at any time. The function can be found using the gradient–interceptmethod. The initial value of the item (Vo) will be the vertical intercept and the gradientwill be the negative of the amount that the item depreciates, D, each year. The equationof this linear function will be:

V = Vo − Dt

where V is the present value of the item and t is the age of the asset, in years.Note that gradients for depreciation will always be negative.

Depreciation of motor vehiclesChoose a make of car and find out the price for a new vehicle of this make and model. Then go through the classified advertisements in the newspaper and complete the table below or look in the RACQ publication The Road Ahead.

Draw a graph that shows the price of this car as it ages.

inve


n

Age of car (years) Price

New (0)

1

2

3

4

5

6

7

8

9

10



In the previous worked example, how long does it take for the computer todepreciate to a value of $0? The computer is said to be ‘written off’ when itreaches this value.

Declining balance or diminishing value depreciationThe other method of depreciation used is the declining balance or diminishing valuemethod of depreciation. Here, the value of the item depreciates each year by apercentage of its current value. Under such depreciation, the value of the item neveractually becomes zero.

This type of depreciation is an example of exponential decay. A graph depicting thevalue over time is non-linear, showing a downward-falling curve which never actuallyreaches a zero value.

The table below shows the declining value of a computer. Graph the value against timeand write an equation for this function.

Age (years) Value ($)

New (0) 4000

1 3500

2 3000

3 2500

4 2000

5 1500

THINK WRITE

Draw a set of axes with age on the horizontal axis and value on the vertical.

Plot each point given by the table.

Join all points to graph the function.

Write the initial value as Vo and use the gradient to state D.

Vo = 4000, D = 500

Write the equation using V = Vo − Dt. V = Vo − DtV = 4000 − 500t

1

010 2 3 4 5 6 7 8 9

Val

ue (

$)

Age (years)

50010001500 200025003000350040004500

2

3

4

5

4WORKEDExample



A graphics calculator can be used to plot the values in the tables in worked examples 4and 5. This can check the shapes of the curves. The two variables (age and value) canbe entered as lists. Because no equations are entered, we can not extrapolate valuesfrom the graph beyond the limits of the values in the lists. Later in this chapter we willexplore a graphics calculator technique which will enable us to extrapolate valuesfrom graphs.

The table at right shows the value of a car that is purchased new for $40 000.Plot the points on a set of axes and graph the depreciation of the car. Use the graph to estimate the value of the car after 10 years.

THINK WRITE

Draw a set of axes with age on the horizontal axis and value on the vertical.Plot the points from the table.Join the points with a smooth curve.

Estimate the value after 10 years from the graph you have drawn.

From the graph, the approximate value of the car after 10 years is $4000.

Age of car (years) Value ($)

New (0) 40 000

1 32 000

2 25 600

3 20 480

4 16 384

5 13 107

1

5 0000

10 00015 000

20 00025 00030 00035 00040 000

10 2 3 4 5 6 7 8 9 10

Val

ue (

$)

Age (years)

23

4

5WORKEDExample

With the aid of a graphics calculator, produce graphs showing the relationship between the age and value of the following:a the computer in worked example 4 b the car from worked example 5.

THINK WRITE/DISPLAY

a Enter the age of the computer as List1 and the value as List2.

a TI: Press , select 1:edit.

Casio: Enter MENU, select STAT.

1 STAT

6WORKEDExample



Continued over page

THINK WRITE/DISPLAY

Set the window for an age in the range 0 to 10 and a value of $4500 down to 0.

TI: Press . Casio: Press [V-WINDOW].

Set one of the stat plots ON to graph the data in L1 and L2.

TI: Press [STAT PLOT].

Casio: Press (GRPH) and (SET).

Graph the points. Use the TRACE function to check the values of the points on the line. Notice that the graph is a straight line and matches the shape of that shown in worked example 4.

TI: Press then .

Casio: Press (GRPH) and [TRACE].

b Enter the age of the car as List1 and the value as List2.

b TI: Press , and select 1:edit.

Casio: From the main MENU, select STAT and enter the information.

Set the window for an age in the range 0 to 10 and a value of $45 000 down to 0.


Set one of the stat plots ON to graph the data in L1 and L2.

TI: Press [STAT PLOT].

Casio: Press (GRPH) and (SET).

2 WINDOW SHIFT

3 2nd F1F6

4 GRAPHTRACE

F1SHIFT

1 STAT

2 WINDOW SHIFT

3 2nd F1F6



Modelling depreciation

The following questions may be solved with the aid of a graphics calculator.

1 The table below shows the depreciating value of a tractor.

a Draw a graph of the value of the tractor against the age of the tractor.b Write a function for the value of the tractor.


New (0) 100 000

1 90 000

2 80 000

3 70 000

4 60 000

5 50 000

THINK WRITE/DISPLAY

Graph the points. Use the TRACE function to check the values of the points on the line. Notice that the graph is a curve with the same shape as shown in worked example 5.

TI: Press then .

Casio: Press (GPH1) and [TRACE].

4 GRAPHTRACE

F1SHIFT

remember1. Depreciation is the loss in the value of an item over time.2. Depreciation can be of two types:

(a) straight line depreciation — the item loses a constant amount of value each year(b) declining balance or diminishing value depreciation — the value of an item

depreciates by a percentage of its value each year.3. Straight line depreciation can be graphed using a linear function in which the

new value of the item is the vertical intercept and the gradient is the negative of the annual loss in value.

4. A declining balance or diminishing value depreciation is an example of exponential decay and is graphed with a smooth curve.

5. A graphics calculator can be used to graph depreciation. Values can be determined at any point along the straight line or curve.

remember

2B

WORKEDExample

4, 6


C h a p t e r 2 A p p r e c i a t i o n a n d d e p r e c i a t i o n 672 The table at right shows the depreciating value of

a tow truck.Draw a graph of value against age and, hence,

write the value as a linear function of age.

3 The function V = 50 000 − 6000A shows the value, V, of a car when it is A years old.a Draw a graph of this function.b Use the graph to calculate the value of the car after 5 years.c After how many years would the car be ‘written off’ (that is, the value of the car

becomes $0)?

4 A computer is bought new for $6400 and depreciates at the rate of $2000 per year.a Write a function for the value, V, of the computer against its age, A.b Draw the graph of this function.c After how many years does the computer become written off?

5 The table at right shows the declining value of anew motorscooter.a Plot the points shown by the table and draw

a graph of the value of the motorscooter against age.

b Use your graph to estimate the value of the motorscooter after 8 years.

6 The table at right shows the declining value of a semi-trailer.a Plot the points as given in the table and then

draw a curve of best fit to graph the depreciation of the semi-trailer.

b Use your graph to estimate the value of the semi-trailer after 10 years.

c After what number of years will the value of the semi-trailer fall below $50 000?

7 a A gymnasium values its equipment at $200 000. Each year the value of the equip-ment depreciates by 20% of the value of the previous year. Calculate the value ofthe equipment after:

i 1 year ii 2 years iii 3 years iv 4 years.b Plot these points on a set of axes and draw a graph of the value of the equipment

against its age.


New (0) 50 000

1 42 000

2 34 000

3 26 000

4 18 000

5 10 000

WORKEDExample

5, 6Age (years) Value ($)

New (0) 20 000

1 15 000

2 11 250

3 8 500

4 6 250

5 4 750


New (0) 600 000

1 420 000

2 295 000

3 205 000

4 145 000

5 100 000



8Which of the tables below shows a straight line depreciation?

9 A car is bought new for $30 000.a The straight line method of depreciation sees the car lose $4000 in value each year.

Complete the table below.

b Draw a graph of this depreciation.

A Age (years) Value ($) B Age (years) Value ($)

New (0) 4000 New (0) 4000

1 3600 1 3600

2 3240 2 3200

3 2916 3 2800

4 2624 4 2400

5 2362 5 2000

C Age (years) Value ($) D Age (years) Value ($)

New (0) 4000 New (0) 4000

1 3600 1 3000

2 3300 2 2500

3 3100 3 1500

4 3000 4 1000

5 2950 5 500


New (0) 30 000

1

2

3

4

5



C h a p t e r 2 A p p r e c i a t i o n a n d d e p r e c i a t i o n 69c The declining balance or diminishing value method of depreciation sees the value

of the car fall by 20% of the previous year’s value. Complete the table below.

d On the same set of axes draw a graph of this depreciation.e After how many years is the car worth more under declining balance or diminishing

value than under straight line depreciation?

Straight line depreciationWe have already seen that the straight line method of depreciation is where the value ofan item depreciates by a constant amount each year.

The depreciated value of an item is called the salvage value, S. The salvage value ofan asset can be calculated using the formula:

S = Vo − Dnwhere Vo is the purchase price of the asset, D is the amount of depreciation per periodand n is the total number of periods.

By solving an equation we are able to calculate when the value of an asset falls belowa particular value.


New (0) 30 000

1

2

3

4

5

A laundry buys dry-cleaning equipment for $30 000. The equipment depreciates at a rate of $2500 per year. Calculate the salvage value of the equipment after6 years.

THINK WRITE

Write the formula. S = Vo − DnSubstitute the values of Vo, D and n. = $30 000 − $2500 × 6Calculate the value of S. = $15 000

123

7WORKEDExample



A graphics calculator can provide the answer to worked example 8 without the need tosolve an equation.

A plumber purchases equipment for a total of $60 000. The value of the equipment is depreciated by $7500 per year. When the value of the equipment falls below $10 000 it should be replaced. Calculate the number of years after which the equipment should be replaced.

THINK WRITE

Write the formula. S = Vo − Dn

Substitute for S, Vo and D. 10 000 = 60 000 − 7500n

Solve the equation to find the value of n.

7500n = 50 000n = 6

Give a written answer, taking the value of n up to the next whole number.

The equipment must be replaced after 7 years.

1

2

323---

4

8WORKEDExample

Use a graphics calculator for the problem in worked example 8.

THINK WRITE/DISPLAY

Enter the formula into Y1 with the variable X being the number of years.

TI: Press to enter data. Casio: Select the TABLE function from the MENU.

Set up the table for a minimum of 0 (years) and increments of 1.

TI: Press [TBLSET]. Casio: Press (RANG) to set the range.

Display the table produced by the formula.

TI: Press [TABLE]. Casio: Press (TABL).

1 Y=

2 2nd F5

3 2nd F6

9WORKEDExample



Straight line depreciation

1 A car that is purchased for $45 000 depreciates by $5000 each year. Calculate thesalvage value of the car after 5 years.

2 Calculate the salvage value:a after 5 years of a computer that is purchased

for $5000 and depreciates by $800 per yearb after 7 years of a motorscooter that is purchased

for $25 000 and depreciates by $2100 per yearc after 6 years of a semi-trailer that is purchased

for $750 000 and depreciates by $80 000 peryear

d after 2 years of a mobile phone that is purchasedfor $225 and depreciates by $40 per year

e after 4 years of a farmer’s plough that ispurchased for $80 000 and depreciates by$12 000 per year.

3 A bus company buys 15 buses for $475 000 each.a Calculate the total cost of the fleet of buses.b If each bus depreciates by $25 000 each year, calculate the salvage value of the

fleet of buses after 9 years.

THINK WRITE/DISPLAY

Scroll down the table to locate the first time the value falls below $10 000. Note that this occurs after 7 years. This is consistent with the answer using algebra in worked example 8.

TI: Press to navigate.

Casio: Press to navigate.

4 � �

remember1. Straight line depreciation occurs when the value of an asset depreciates by a

constant amount each year.2. The formula to calculate the salvage value, S, of an asset is S = Vo − Dn where

Vo is the purchase price of the asset, D is the amount of depreciation per period and n is the total number of periods.

3. To calculate a value of Vo, D or n we substitute all known values and solve the equation that is formed.

remember

2CWORKEDExample

7



4 The price of a new car is $25 000. The value of the car depreciates by $300 eachmonth. Calculate salvage value of the car after 4 years.

5 An aeroplane is bought by an airline for $60 million. If the aeroplane depreciates by$4 million each year, calculate when the value of the aeroplane falls below $30 million.

6 Calculate the length of time for each of the following items to depreciate to the value given.a A computer purchased for $5600 to depreciate to less than $1000 at $900 per yearb An electric guitar purchased for $1200 to depreciate to less than $500 at $150 per yearc An entertainment unit purchased for $6000 to become worthless at $750 per yeard Office equipment purchased for $12 000 to depreciate to less than $2500 at $1500

per year

7 A motor vehicle depreciates from $40 000 to $15 000 in 10 years. Assuming that it isdepreciating in a straight line, calculate the annual amount of depreciation.

8 Calculate the annual amount of depreciation in an asset that depreciates:a from $20 000 to $4000 in 4 yearsb from $175 000 to $50 000 in 10 yearsc from $430 000 to $299 500 in 9 years.

9 A computer purchased for $3600 is written off in 4 years. Calculate the annualamount of depreciation.

10 A car that is 5 years old has an insured value of $12 500. If the car is depreciating ata rate of $2500 per year, calculate its purchase price.

11 Calculate the purchase price of each of the following assets given that:a after 5 years the value is $50 000 and is depreciating at $12 000 per yearb after 15 years the value is $4000 and is depreciating at $1500 per yearc after 25 years the value is $200 and is depreciating at $50 per year.

12 An asset that depreciates at $6500 per year is written off after 12 years. Calculate thepurchase price of that asset.

Declining balance or diminishing value method of depreciation

The declining balance or diminishing value method of depreciation occurs when thevalue of an asset depreciates by a given percentage each period.

Consider the case of a car purchased new for $30 000, which depreciates at the rateof 20% p.a. Each year the salvage value of the car is 80% of its value at the end of theprevious year.

After 1 year: S = 80% of $30 000= $24 000

After 2 years: S = 80% of $24 000= $19 200

After 3 years: S = 80% of $19 200= $15 360

WORKEDExample

8, 9

2.1



The salvage value under a declining balance can be calculated using the formula:

S = Vo

where S is the salvage value, Vo is the purchase price, R is the annual percentagedepreciation and T is the number of years. Note the similarity to the compound interestformula with a minus sign in place of a plus sign (because the value is decreasing). Thevalue of n is 1 since depreciation is usually calculated yearly.

To calculate the amount by which the asset has depreciated, we subtract the salvagevalue from the purchase price.

A small truck that was purchased for $45 000 depreciates at a rate of 25% p.a. By calculating the value at the end of each year, find the salvage value of the truck after 4 years.

THINK WRITE

The salvage value at the end of each year will be 75% of its value at the end of the previous year.Find the value after 1 year by calculating 75% of $45 000.

After 1 year: S = 75% of $45 000= $33 750

Find the value after 2 years by calculating 75% of $33 750.

After 2 years: S = 75% of $33 750= $25 312.50

Find the value after 3 years by calculating 75% of $25 312.50.

After 3 years: S = 75% of $25 312.50= $18 984.38

Find the value after 4 years by calculating 75% of $18 984.38.

After 4 years: S = 75% of $18 984.38= $14 238.28

1

2

3

4

5

10WORKEDExample

1 R100---------–

T

The purchase price of a yacht is $15 000. The value of the yacht depreciates by 10% p.a. Calculate (correct to the nearest $1) the salvage value of the yacht after 8 years.

THINK WRITE

Write the formula. S = Vo

Substitute values for Vo, R and T. = 15 000

= $15 000 × 0.98

Calculate the salvage value. = $6457.00

1 1 R100---------–

T

2 1 10100---------–

8

3

11WORKEDExample



Declining balance or diminishing value method of depreciation

1 The purchase price of a forklift is $50 000. The value of the forklift depreciates by20% p.a. By calculating the value of the forklift at the end of each year, find thesalvage value of the forklift after 4 years.

2 A trailer is purchased for $5000. The value of the trailer depreciates by 15% eachyear. By calculating the value of the trailer at the end of each year, calculate:a the salvage value of the trailer after 5 years (to the nearest $10)b the amount by which the trailer depreciates:

i in the first year ii in the fifth year.

The purchase price of a computer for a music studio is $40 000. The computer depreciates by 12% p.a. Calculate the amount by which the computer depreciates in 10 years.

THINK WRITE

Write the formula. S = Vo

Substitute the value of Vo, R and T. = 40 000

= $40 000 × 0.8810

Calculate the value of S. = $11 140.04Calculate the amount of depreciation by subtracting the salvage value from the purchase price.

Depreciation = $40 000 − $11 140.04= $28 859.96

1 1 R100---------–

T

2 1 12100---------–

10

34

12WORKEDExample

remember1. The declining balance or diminishing value method of depreciation occurs

when the value of an asset depreciates by a fixed percentage each year.2. The salvage value of an asset can be calculated by subtracting the percentage

depreciation each year.

3. The salvage value can be calculated using the formula S = Vo .

4. To calculate the amount of depreciation, the salvage value should be subtracted from the purchase price.

1 R100---------–

T

remember

2D

2.2 WORKEDExample

10


C h a p t e r 2 A p p r e c i a t i o n a n d d e p r e c i a t i o n 753 A company purchases a mainframe computer for $3 000 000. The value of the

computer depreciates by 15% p.a. By calculating the value at the end of each year findthe number of years that it takes for the salvage value of the mainframe to fall below$1 000 000.

4 Use the declining balance depreciation formula to calculate the salvage value after 7years of a power generator purchased for $800 000 that depreciates at a rate of10% p.a. (Give your answer correct to the nearest $1000.)

5 Calculate the salvage value of an asset (correct to the nearest $100) with a purchaseprice of:a $10 000 that depreciates at 10% p.a. for 5 yearsb $250 000 that depreciates at 15% p.a. for 8 yearsc $5000 that depreciates at 25% p.a. for 5 yearsd $2.2 million that depreciates at 30% p.a. for 10 yearse $50 000 that depreciates at 40% p.a. for 5 years.

6 A plumber has tools and equipment valued at $18 000. If the value of the equipmentdepreciates by 30% each year, calculate the value of the equipment after 3 years.

7 A yacht is valued at $950 000. The value of the yacht depreciates by 22% p.a.Calculate the amount that the yacht will depreciate in value over the first 5 years(correct to the nearest $1000).

8 A new car is purchased for $35 000. The owner plans to keep the car for 5 years thentrade the car in on another new car. The estimate is that the value of the car willdepreciate by 16% p.a. Calculate:a the amount the owner can expect as a trade in for the car in 5 years (correct to the

nearest $100)b the amount by which the car will depreciate in 5 years.

9A shop owner purchases fittings for her store that cost a total of $120 000. Three yearslater, the shop owner is asked to value the fittings for insurance. If the shop ownerallows for depreciation of 15% on the fittings, which of the following calculationswill give the correct estimate of their value?A 120 000 × 0.853 B 120 000 × 0.153 C 120 000 × 0.55 D 120 000 × 0.45

10A computer purchased for $3000 will depreciate by 25% p.a. The salvage value of thecomputer after 4 years will be closest to:A $0 B $10 C $950 D $2000

11 An electrician purchases tools of trade for a total of $8000. Each year the electricianis entitled to a tax deduction for the depreciation of this equipment. If the rate ofdepreciation allowed is 33% p.a., calculate:a the value of the equipment at the end of one year (correct to the nearest $1)b the tax deduction allowed in the first yearc the value of the equipment at the end of two years (correct to the nearest $1)d the tax deduction allowed in the second year.

12 An accountant purchased a computer for $6000. The value of the computer depreci-ates by 33% p.a. When the value of the computer falls below $1000, it is written offand a new one is purchased. How many years will it take for the computer to bewritten off?

WORKEDExample

11

WORKEDExample

12





1 The price of a new CD player is $1250. The CD player will depreciate under straightline depreciation at a rate of $200 per year. Calculate the value of the CD player after3 years.

2 An asset that was valued at $39 000 when new depreciates to $22 550 in 7 years.Calculate the annual amount of depreciation under straight line depreciation.

3 A computer that is purchased new for $9000 depreciates at a rate of $1350 per year.Calculate the length of time before the computer is written off.

4 A car dealer values a used car at $7000. If the car is 8 years old and the rate ofdepreciation is $1750 per year, calculate the value of the car when new.

5 Write the formula for depreciation under the declining balance method.

6 A truck is valued new at $50 000 and depreciates at a rate of 32% p.a. Calculate thevalue of the truck after 5 years (correct to the nearest $50).

7 An asset that has a purchase price of $400 000 depreciates at a rate of 45% p.a.Calculate the asset’s value after 6 years (correct to the nearest $1000).

8 For the asset in question 7, calculate the amount by which it has depreciated in6 years.

9 Office equipment valued at $250 000 depreciates at a rate of $15% p.a. Calculate theamount by which it depreciates in the first year.

10 Calculate the length of time it will take for the salvage value of the office equipmentin question 9 to fall below $20 000.

Rates of depreciationIn the previous investigation you chose a make and model of car and researched the salvage value of this car after each year.

1 Calculate the percentage depreciation for each year.

2 Calculate if this percentage rate is approximately the same each year.

3 Using the average annual depreciation, calculate a table of salvage values for the first 10 years of the car’s life.

4 Draw a graph showing the depreciating value of the car.

5 Some people claim that it is best to trade your car in on a new model every three years. With reference to your figures and graph, express your views on this statement.

inve


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1



Depreciation tablesThe following computer application will prepare a table that will show the depreciatedvalue of an asset with a purchase price of $1 over various periods of time and variousrates of depreciation.

Depreciation table1 Open a new spreadsheet and type in the headings shown.2 In cell B3 enter the formula =(1-B$2)^$A3. 3 Highlight the range of cells B3 to K12. Select the Edit function, and then use

(a) the Fill and (b) the Right and Fill and Down functions to copy the formula throughout the table.

4 The table that you now have should have the values shown in the table.

5 Use the spreadsheet’s graphing facility to draw a depreciation graph for each of the depreciation rates shown in the table.

6 Print out a copy of the graph for one of these rates.The table produced shows the depreciated value of $1 and can be used to make

calculations about depreciation.

inve


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Time (years)

Rate of depreciation (per annum)

5% 10% 15% 20% 25% 30% 35% 40% 45% 50%

1 0.9500 0.9000 0.8500 0.8000 0.7500 0.7000 0.6500 0.6000 0.5500 0.5000

2 0.9025 0.8100 0.7225 0.6400 0.5625 0.4900 0.4225 0.3600 0.3025 0.2500

3 0.8574 0.7290 0.6141 0.5120 0.4219 0.3430 0.2746 0.2160 0.1664 0.1250

4 0.8145 0.6561 0.5220 0.4096 0.3164 0.2401 0.1785 0.1296 0.0915 0.0625

5 0.7738 0.5905 0.4437 0.3277 0.2373 0.1681 0.1160 0.0778 0.0503 0.0313

6 0.7351 0.5314 0.3771 0.2621 0.1780 0.1176 0.0754 0.0467 0.0277 0.0156

7 0.6983 0.4783 0.3206 0.2097 0.1335 0.0824 0.0490 0.0280 0.0152 0.0078

8 0.6634 0.4305 0.2725 0.1678 0.1001 0.0576 0.0319 0.0168 0.0084 0.0039

9 0.6302 0.3874 0.2316 0.1342 0.0751 0.0404 0.0207 0.0101 0.0046 0.0020

10 0.5987 0.3487 0.1969 0.1074 0.0563 0.0282 0.0135 0.0060 0.0025 0.0010



The computer application on page 77 will produce a general table for a declining balanceor diminishing value depreciation. We should be able to use the formula to create a tableand graph showing the salvage value of an asset under both straight line and diminishingvalue depreciation.

An item is purchased for $500 and depreciates at a rate of 15% p.a. Use the depreciation table on page 77 to calculate the value of the item after 4 years.

THINK WRITE

Look up the table to find the depreciated value of $1 at 15% p.a. for 4 years.Multiply the depreciated value of $1 by $500.

Depreciated value = 0.5220 × $500= $261

1

2

13WORKEDExample

A car is purchased new for $20 000. The depreciation can be calculated under straight-line depreciation at $2500 per year and under diminishing value at 20% p.a.a Complete the table at right.

(Give all values to the nearest $1.)b Draw a graph of both the straight

line and diminishing value depreciation and use the graph to show the point at which the straight-line value of the car falls below the diminishing value.

THINK WRITE

a Copy the table.Complete the straight line column by subtracting $2500 from the previous year’s value.Complete the diminishing value by multiplying the previous year’s value by 0.8.

Age of car (years)

Straight line value ($)

Diminishing value ($)

New (0) 20 000 20 000

1

2

3

4

5

6

7

8

12

3

a Age of car (years)



New (0) 20 000 20 000

1 17 500 16 000

2 15 000 12 800

3 12 500 10 240

4 10 000 8 192

5 7 500 6 554

6 5 000 5 243

7 2 500 4 194

8 0 3 355

14WORKEDExample



THINK WRITE

b Plot the points generated by the table.

b

Join the points for the straight line depreciation with a straight line.

Join the points for the diminishing value depreciation with a smooth curve.

The graph shows the straight line going below the curve after 6 years.

The straight line depreciation becomes less than the diminishing value depreciation after 6 years.

1

0

5 000

10 000

15 000

20 000

25 000

1(New)0 2 3 4 5 6 7 8

Val

ue (

$)

Age (years)

Straight line valueDiminishing value2

3

4

Use a graphics calculator to solve worked example 14.

Continued over page

THINK WRITE/DISPLAY

Enter a formula for the straight line depreciation in Y1 and a formula for diminishing value depreciation in Y2.

TI: Press . Casio: From the main MENU, select TABLE.

Set up the table to start at 0 and progress in increments of 1.

TI: Press and [TBLSET]. Casio: Press (RANG).

Display the table and write down the results (rounding to the nearest $).

TI: Press [TABLE]. Casio: Press (TABL).

1 Y=

2 2nd F5

3 2nd F6

15WORKEDExample



Depreciation is an allowable tax deduction for people in many occupations. A taxdeduction for depreciation is allowed when equipment used in earning an incomedepreciates in value and will eventually need replacing. Depending on the equipmentand the occupation, either straight line or diminishing value depreciation may beused.

Under diminishing value depreciation, when the salvage value falls below a certainpoint the equipment may be written off. This means that the entire remaining balancecan be claimed as a tax deduction and as such is considered worthless. From this pointon, no further tax deductions can be claimed for this equipment. A visit to the AustralianTaxation Office’s website provides more detailed information on this subject.

Comparing straight line depreciation and diminishing value depreciation

using a spreadsheetLet us look at a solution to worked example 14 using a spreadsheet. The value of the new car is $20 000. Depreciation using the straight line method is $2500 per year. Depreciation using the diminishing value method is 20% p.a. Let us draw up a spreadsheet similar to the one following.

THINK WRITE/DISPLAY

Set the window to display the age of the car (x-value) in the range 0 to 10 and the car’s value (y-value) from $20 000 down to 0.


Graph the functions. Use the options to determine the co-ordinates of the point of intersection. Beyond this point the value using straight line depreciation becomes less than the value using diminishing value depreciation. Notice that this gives a more accurate result than the manual method.

TI: Press , then [CALC] and select 5:intersect.At the question ‘First curve’, press (to confirm that the cursor is on one of the curves), press again (toconfirm the second curve) and press to confirm that you now want to see the ‘Guess’.

Casio: Press , select GRAPH, press (DRAW) then (G-Solv) and (ISCT) to give the first intersection point (0, 20 000), Then press for the second intersection point.

4 WINDOW SHIFT

5 GRAPH 2nd

ENTER

ENTER

ENTER

MENUF6

SHIFTF5

�

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1 Start by entering the main heading in cell A1 and the side headings in cells A3, A4 and A5.

2 Enter the new value, 20000, in cell B3 and format it as currency with no decimal places.

3 Enter the value 2500 in cell B4 and format as currency with no decimal places.

4 Enter the numeric value of 20 in cell B5.5 Head up the three columns in row 7.6 In cell position A8, start the years by entering 0.7 In cell A9, enter the formula =A8+1. Copy this formula down to row 16.8 In cell B8, enter the formula =$B$3-$B$4*A8 (the formula for straight line

depreciation). Copy this formula down to cell B16.9 The diminishing value formula to be entered into cell C8 is

=$B$3*(1-$B$5/100)^A8. Copy this formula down to cell C16.10 Check that all your figures agree with those in the spreadsheet above. Notice

that they are the same as those in the table of worked example 14.11 Use the graphing facility of the spreadsheet to produce graphs similar to the

ones shown.12 From the table and the graphs it is evident that a critical point occurs around

the 6-year mark. This is consistent with the graph shown in the worked example.

13 Try changing values in cells B3, B4 and B5. Notice how the spreadsheet adjusts.

14 Save your spreadsheet.



Depreciation tables

1 Use the table of depreciated values of $1 to calculate:a the value of a computer purchased for $5000 after 5 years, given that it depreciates

at 20% p.a.b the value of a car after 8 years with an initial value of $35 000, given that it

depreciates at 15% p.a.c the value of a boat with an initial value of $100 000 after 10 years, given that it

depreciates at 10% p.a.

A builder has tools of trade that are purchased new for $14 000. He is allowed atax deduction of 33% p.a. for depreciation of this equipment. When the salvagevalue of the equipment falls below $3000, the electrician is allowed to write theequipment off on the next year’s return. Complete the depreciation table below.(Use whole dollars only.)

Years Salvage value ($) Tax deduction ($)

1

2

3

4

5

THINK WRITE

Calculate the salvage value by multiplying the previous year’s value by 0.67.Calculate the tax deduction by multiplying the previous year’s value by 0.33.When the salvage value is less than $3000, claim the entire amount as a tax deduction.

1

YearSalvage value

($)Tax deduction

($)

1 9380 4620

2 6285 3095

3 4211 2074

4 2821 1390

5 0 2821

2

3

16WORKEDExample

remember1. Graphs can be drawn to compare the salvage value of an asset under different

rates of depreciation, or to compare diminishing value and straight line depreciation.

2. The amount by which an asset depreciates can, in many cases, be claimed as a tax deduction.

remember

2EWORKEDExample

13


C h a p t e r 2 A p p r e c i a t i o n a n d d e p r e c i a t i o n 832 A taxi owner purchases a new taxi for $40 000. The taxi depreciates under straight-

line depreciation at $5000 per year and under diminishing value depreciation at20% p.a.a Copy and complete the table below. Give all values to the nearest $100.

b Draw a graph of the salvage value of the taxi under both methods of depreciation. c State when the value under straight line depreciation becomes less than under

diminishing value depreciation.

3 A company has office equipment that is valued at $100 000. The value of the equip-ment can be depreciated at $10 000 each year or by 15% p.a.a Draw a table that will show the salvage value of the office equipment for the first

ten years. (Give all values correct to the nearest $50.)b Draw a graph of the depreciating value of the equipment under both methods of

depreciation.

4 A computer purchased new for $4400 can be depreciated at either 20% p.a. or 35% p.a.Draw a table and a graph that compare the salvage value of the computer at each rate ofdepreciation.

5 A teacher purchases a laptop computer for $6500. A tax deduction for depreciation of the computer is allowed at the rate of 33% p.a. When the value of the computer falls below $1000, the computer can be written off. Copy and complete the table at right. (Give all values correct to the nearest $1.)

Age of car (years) Straight line value ($) Diminishing value ($)

New (0) 40 000 40 000

1

2

3

4

5

6

7

8

WORKEDExample

14, 15

2.4

2.3

2.5

WORKEDExample

16 YearSalvage value

($)Tax deduction

($)

1

2

3

4

5

6



6 A plumber purchases a work van for $45 000. The van can be depreciated at a rate of 25% p.a. for tax purposes and the van can be written off at the end of 8 years. Copy and complete the depreciation schedule at right. (Give all answers correct to the nearest $1.)

7 A truck is purchased for $250 000. The truck can be depreciated at the rate of $25 000each year or over 10 years at 20% p.a.a Copy and complete the table

at right. (Give all values correct to the nearest $1.)

b Draw a graph of the depreciating value of the truck under both methods of depreciation.

c Complete a depreciation schedule for each method of calculation.

8 Leilay is a cinematographer and on 1 March purchases a camera for work purposes.The cost of the camera is $40 000 and for tax purposes the camera depreciates at therate of 25% p.a.a Calculate the amount that the camera will depreciate in the first year.b The financial year ends on 30 June. For what fraction of the financial year did

Leilay own the camera?c Leilay is allowed a tax deduction for depreciation of her camera. Calculate the

amount of tax deduction that Leilay is allowed for the financial year ending on30 June.

YearSalvage value

($)Tax deduction

($)

1

2

3

4

5

6

7

8

Age of truck (years)



New (0) 250 000 250 000

1

2

3

4

5

6

7

8

9

10


C h a p t e r 2 A p p r e c i a t i o n a n d d e p r e c i a t i o n 859 Calculate the amount of depreciation

on each of the following assets:a a tractor with an initial value of

$80 000 that depreciates at 15%p.a. for 3 months

b a bicycle with an initial value of$600 that depreciates at 25% p.a.for 6 months

c office furniture with an initialvalue of $8000 that depreciates at30% p.a. for 8 months

d a set of encyclopedias with aninitial value of $2500 that depreci-ates at 40% p.a. for 9 months.

Future and present value of an annuityAn annuity is a form of investment involving regular periodic contributions to anaccount. On such an investment, interest compounds at the end of each period and thenext contribution to the account is then made.

Superannuation is a common example of an annuity. Here, people invest in a fund ona regular basis, the interest on the investment compounds, while the principal is addedto for each period. The annuity is usually set aside for a person’s entire working lifeand is used to fund retirement. It may also be used to fund a long-term goal, such as atrip in 10 years time.

To understand the growth of an annuity, we first need to revise compound interest.

The compound interest formula is:

A = P

where A is the amount that an investment will become, P is the principal, R is the interest rate per annum, n is the number of interest periods per year and T is the time in years.

1 R100 n×------------------+

n T×

Calculate the value of a $5000 investment made at 8% p.a. for 4 years, with interest compounded annually.

THINK WRITE

Write the formula. A = P

Substitute values for P, R, n and T. = 5000

= $5000 × (1.08)4

Calculate the value of A. = $6802.44

1 1 R100 n×------------------+

n T×

2 1 8100 1×------------------+

1 4×

3

17WORKEDExample



An annuity takes the form of a sum of compound interest investments. Consider thecase of a person who invests $1000 at 10% p.a. at the end of each year for five years.

To calculate this, we would need to calculate the value of the first $1000 which isinvested for four years, the second $1000 which is invested for three years, the third$1000 which is invested for two years, the fourth $1000 which is invested for one yearand the last $1000 which is added to the investment.

To find out more about the future value of an annuity, click here when using the MathsQuest Maths A Year 12 CD-ROM.

Annuity calculationsThe amount to which an annuity grows is called the future value of an annuity and

can be calculated using the formula:

A = M

Calculate the value of an annuity in which $1000 is invested each year at 10% p.a. for 5 years.

THINK WRITE

The value of n is 1 in all cases. n = 1

Use the compound interest formula to calculate the amount to which the first $1000 will grow.

A = P

= $1000 × 1.14

= $1464.10

Use the compound interest formula to calculate the amount to which the second $1000 will grow.

A = P

= $1000 × 1.13

= $1331.00

Use the compound interest formula to calculate the amount to which the third $1000 will grow.

A = P

= $1000 × 1.12

= $1210.00

Use the compound interest formula to calculate the amount to which the fourth $1000 will grow.

A = P

= $1000 × 1.1= $1100.00

Find the total of the separate $1000 investments, remembering to add the final $1000.

Total value = $1464.10 + $1331.00 + $1210.00 + $1100.00 + $1000

= $6105.10

1

2 1R

100---------+

T

3 1R

100---------+

T

4 1R

100---------+

T

5 1R

100---------+

T

6

18WORKEDExample

extensioneextensionxtension Future value of an annuityINTE

RACTIVE

C

D- ROM

1 r+( )n 1–r

----------------------------


C h a p t e r 2 A p p r e c i a t i o n a n d d e p r e c i a t i o n 87M is the amount of each periodical investment made at the end of the period; r is theinterest rate per period expressed as a decimal and n is the number of deposits.

The present value of an annuity, P, can be calculated using the formula

P = M

This formula allows us to calculate the single sum needed to be invested to give thesame financial result as an annuity where we are given the size of each contribution.

The financial calculator section of the later model graphics calculators makes shortwork of annuity calculations. Let us use the TI-83 and Casio CFX-9850GB PLUS todemonstrate this facility.

1 r+( )n 1–

r 1 r+( )n----------------------------

Christina invests $500 into a fund every 6 months at 9% p.a. interest,compounding six-monthly for 10 years. Calculate the future value of the annuityafter 10 years.

THINK WRITE/DISPLAY

Enter the financial section of the calculator.Enter the following data as shown at right.

n = 10 × 2 (Interest is calculated twice a year for 10 years.)

I% = 9 (Interest rate is 9% p.a.)PV = 0 (No deposit is made

initially – only regular 6-monthly payments.)

PMT = 500 (Regular $500 payments are made.)

FV = 0 (This value will be calculated.)

P/Y = 2 (Regular payments are made twice a year.)

C/Y = 2 (Interest is calculated twice a year.)

TI: Press Casio: From MENU, [FINANCE]and select select TVM, then 1:TVMSolver. F2:Compound Interest.

Solve for FV. Take the cursor to the FV line and press [SOLVE] on the TI and press (FV) for the Casio.

Write the answer. The future value of the annuity after 10 years is $15 685.71.

1

2 2nd

3ALPHA

F5

4

19WORKEDExample



Annuity calculatorStart up the Maths Quest Maths A Year 12 CD-ROM which accompanies this book and download the spreadsheet ‘Annuity Calculator’. The spreadsheet will show you the growth of an annuity where $1000 is invested at the end of each year for 20 years, at the rate of 8% p.a. interest, compounding annually.

1 The spreadsheet shows that after 20 years the value of this investment is $45 761.96. Below is the growth of the annuity after each deposit is made. This will allow you to see the growth for up to 30 deposits. You can use Edit, then Fill and Down functions on the spreadsheet to see further.

2 Browse the spreadsheet, taking note of the formulas in the various cells.

Vicky has the goal of saving $10 000 in the next five years. The best interest rate that she can obtain is 8% p.a., with interest compounded annually. Calculate the amount of each annual contribution that Vicky must make.

THINK WRITE/DISPLAY

Enter the following data using the methods described for the previous worked example.

n = 5 × 1 (Interest is calculated once a year for 5 years.)

I% = 8 (Interest rate is 8% p.a.)PV = 0 (No deposit is made

initially.)PMT = 0 (This is the unknown

value which will be calculated.)

FV = 10 000 (The future value is $10 000.)

P/Y = 1 (Vicky makes one payment per year.)

C/Y = 1 (Interest is calculated yearly.)

TI Casio

Solve for PMT. Take the cursor to the PMT line and press [SOLVE] for the TI. For the Casio, press (PMT).

Write the answer. A payment of $1704.56 is required as the annual contribution.

1

2ALPHA

F4

3

20WORKEDExample

Annuity calculator

inve


n



Annuity values using tablesTo compare an annuity with a single sum investment, we need to use the presentvalue of the annuity. The present value of an annuity is the single sum of moneywhich, invested on the same terms as the annuity, will produce the same financialresult.

Problems associated with annuities can be simplified by creating a table that willshow either the future value or present value of an annuity of $1 invested per interestperiod.

3 Click on the Tab ‘Chart 1’. This is a line graph that shows the growth of the annuity for up to 30 deposits.

4 Change the size of the deposit to $500 and the compounding periods to 2. This will show how much benefit can be achieved by reducing the compounding period.

Future value of $1Consider $1 is invested into an annuity each interest period. The table we are going to construct on a spreadsheet shows the future value of that $1.

1 Open a new spreadsheet.

2 Type in the headings shown on the following page.

3 In cell B4 enter the formula =((1+B$3)^$A4-1)/B$3. (This is the future value formula.) Format the cell, correct to 4 decimal places.

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n



4 Highlight the range of cells B3 to K13. Choose Edit and then the Fill and Down functions followed by the Fill and Right functions to copy the formula to all other cells in this range.

5 Save the spreadsheet as ‘Future value of $1’.

This completes the table. The table shows the future value of an annuity of $1 invested for up to 10 interest periods at up to 10% per interest period. You can extend the spreadsheet further for other interest rates and longer investment periods.

The above table is the set of future values of $1 invested into an annuity. This is the table you should have obtained.

A table such as this can be used to find the value of an annuity by multiplying the amount of the annuity by the future value of $1.

Future values of $1

Period

Interest rate (per period)

1% 2% 3% 4% 5% 6% 7% 8% 9% 10%

1 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

2 2.0100 2.0200 2.0300 2.0400 2.0500 2.0600 2.0700 2.0800 2.0900 2.1000

3 3.0301 3.0604 3.0909 3.1216 3.1525 3.1836 3.2149 3.2464 3.2781 3.3100

4 4.0604 4.1216 4.1836 4.2465 4.3101 4.3746 4.4399 4.5061 4.5731 4.6410

5 5.1010 5.2040 5.3091 5.4163 5.5256 5.6371 5.7507 5.8666 5.9847 6.1051

6 6.1520 6.3081 6.4684 6.6330 6.8019 6.9753 7.1533 7.3359 7.5233 7.7156

7 7.2135 7.4343 7.6625 7.8983 8.1420 8.3938 8.6540 8.9228 9.2004 9.4872

8 8.2857 8.5380 8.8923 9.2142 9.5491 9.8975 10.2598 10.6366 11.0285 11.4359

9 9.3685 9.7546 10.1591 10.5828 11.0266 11.4913 11.9780 12.4876 13.0210 13.5795

10 10.4622 10.9497 11.4639 12.0061 12.5779 13.1808 13.8164 14.4866 15.1929 15.9374



Just as we have a table for the future value of an annuity, we can create a table for thepresent value of an annuity.

Present value tableThe table we are about to make on a spreadsheet shows the present value of an annuity of $1 invested per interest period.1 Open a new spreadsheet.

2 Enter the headings shown in the spreadsheet.3 In cell B4 type the formula =((1+B$3)^$A4-1)/(B$3*(1+B$3)^$A4). 4 Drag from cell B4 to K13 and then use the Edit and then the Fill and Down and

the Fill and Right functions to copy this formula to the remaining cells in your table.

5 Save your spreadsheet as ‘Present value of $1’.The table created shows the present value of an annuity of $1 per interest period

for up to 10% per interest period and for up to 10 interest periods.

Use the table to find the future value of an annuity where $1500 is deposited at the end of each year into an account that pays 7% p.a. interest, compounded annually for 9 years.

THINK WRITE

Look up the future value of $1 at 7% p.a. for 9 years.

Future value = $1500 × 11.9780

Multiply this value by 1500. = $17 967

1

2

21WORKEDExamplein

vestigation

investigation



The result that you should have obtained is shown below.

This table can be used in the same way as the future value table.

Present values of $1

Period

Interest rate (per period)

1% 2% 3% 4% 5% 6% 7% 8% 9% 10%

1 0.9901 0.9804 0.9709 0.9615 0.9524 0.9434 0.9346 0.9259 0.9174 0.9091

2 1.9704 1.9416 1.9135 1.8861 1.8594 1.8334 1.8080 1.7833 1.7591 1.7355

3 2.9410 2.8839 2.8286 2.7751 2.7232 2.6730 2.6243 2.5771 2.5313 2.4869

4 3.9020 3.8077 3.7171 3.6299 3.5460 3.4651 3.3872 3.3121 3.2397 3.1699

5 4.8534 4.7135 4.5797 4.4518 4.3295 4.2124 4.1002 3.9927 3.8897 3.7908

6 5.7955 5.6014 5.4172 5.2421 5.0757 4.9173 4.7665 4.6229 4.4859 4.3553

7 6.7282 6.4720 6.2303 6.0021 5.7864 5.5824 5.3893 5.2064 5.0330 4.8684

8 7.6517 7.3255 7.0197 6.7327 6.4632 6.2098 5.9713 5.7466 5.5348 5.3349

9 8.5660 8.1622 7.7861 7.4353 7.1078 6.8017 6.5152 6.2469 5.9952 5.7590

10 9.4713 8.9826 8.5302 8.1109 7.7217 7.3601 7.0236 6.7101 6.4177 6.1446

Liam invests $750 per year into an annuity at 6% per annum for 8 years, with interest compounded annually. Use the table to calculate the present value of Liam’s annuity.

THINK WRITE

Use the table to find the present value of a $1 annuity at 6% for 8 interest periods.Multiply this value by 750. Present value = $750 × 6.2098

= $4657.35

1

2

22WORKEDExample

remember1. The compound interest formula is A = P , where A is the

amount that the investment will grow to, R is the interest rate per year, n is the number of periods per year and T is the time of the investment in years.

2. An annuity is a form of investment where periodical equal contributions are made to an account, with interest compounding at the end of each period.

3. The value of an annuity is calculated by adding the value of each amount contributed as a separate compound interest investment.

4. A table of future values shows the future value of an annuity where $1 is invested per interest period.

5. A table of present values shows the present value of an annuity where $1 is invested per interest period.

6. A table of present or future values can be used to compare investments and determine which will give the greater financial return.

1 R100 n×------------------+

n T×

remember



Future and present value of an annuity

1 Calculate the value after 5 years of an investment of $4000 at 12% p.a., with interestcompounded annually.

2 Calculate the value to which each of the following compound interest investmentswill grow.a $5000 at 6% p.a. for 5 years, with interest calculated annuallyb $12 000 at 12% p.a. for 3 years, with interest calculated annuallyc $4500 at 8% p.a. for 4 years, with interest compounded six-monthlyd $3000 at 9.6% p.a. for 3 years, with interest compounded six-monthlye $15 000 at 8.4% p.a. for 2 years, with interest compounded quarterlyf $2950 at 6% p.a. for 3 years, with interest compounded monthly.

3 At the end of each year for four years Rodney invests $1000 into an investment fundthat pays 7.5% p.a. interest, compounded annually. By calculating each investment of$1000 separately, use the compound interest formula to calculate the future value ofRodney’s investment after four years.

4 At the end of every six months Jason invests $800 into a retirement fund which paysinterest at 6% p.a., with interest compounded six-monthly. Jason does this for25 years. Calculate the future value of Jason’s annuity after 25 years.

5 Calculate the future value of each of the following annuities on maturity.a $400 invested at the end of every six months for 12 years at 12% p.a. with interest

compounded six-monthlyb $1000 invested at the end of every quarter for 5 years at 8% p.a. with interest com-

pounded every quarterc $2500 invested at the end of each quarter at 7.2% p.a. for 4 years with interest

compounded quarterlyd $1000 invested at the end of every month for 5 years at 6% p.a. with interest com-

pounded monthly

6

Tracey invests $500 into a fund at the end of each year for 20 years. The fund pays12% p.a. interest, compounded annually. The total amount of interest that Traceyearns on this fund investment is:A $4323.15 B $4823.23 C $26 026.22 D $36 026.22

7 Thomas has the goal of saving $400 000 for his retirement in 25 years. If the bestinterest rate that Thomas can obtain is 10% p.a., with interest compounded annually,calculate the amount of each annual contribution that Thomas will need to make.

8 Calculate the amount of each annual contribution needed to obtain each of thefollowing amounts.a $25 000 in 5 years at 5% p.a., with interest compounded annuallyb $100 000 in 10 years at 7.5% p.a., with interest compounded annuallyc $500 000 in 40 years at 8% p.a., with interest compounded annually

9 Use the table of future values on page 90 to determine the future value of an annuityof $800 invested per year for 5 years at 9% p.a., with interest compounded annually.

2FWORKEDExample

17

WORKEDExample

18

WORKEDExample

19


WORKEDExample

20

WORKEDExample

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10 Use the table of future values on page 90 to determine the future value of each of thefollowing annuities.a $400 invested per year for 3 years at 10% p.a., with interest compounded annuallyb $2250 invested per year for 8 years at 8% p.a., with interest compounded annuallyc $625 invested per year for 10 years at 4% p.a., with interest compounded annuallyd $7500 invested per year for 7 years at 6% p.a., with interest compounded annually

11 Samantha invests $500 every 6 months for 5 years into an annuity at 8% p.a., withinterest compounded every 6 months.a What is the interest rate per interest period?b How many interest periods are there in Samantha’s annuity?c Use the table to calculate the future value of Samantha’s annuity.

12 Use the table to calculate the future value of each of the following annuities.a $400 invested every 6 months for 4 years at 14% p.a., with interest compounded

six-monthlyb $600 invested every 3 months for 2 years at 12% p.a., with interest compounded

quarterlyc $100 invested every month for 5 years at 10% p.a., with interest compounded six-

monthly

13 Use the table of future values todetermine whether an annuity at 5% p.a.for 6 years or an annuity at 6% p.a. for 5years will produce the greatest financialoutcome. Explain your answer.

14

Use the table of future values to deter-mine which of the following annuitieswill have the greatest financial outcome.A 6% p.a. for 8 years, with interest

compounded annuallyB 8% p.a. for 6 years, with interest

compounded annuallyC 7% p.a. for 7 years, with interest

compounded annuallyD 10% p.a. for 5 years, with interest

compounded six-monthly

15 Use the table of present values on page 92to determine the present value of anannuity of $1250 per year for 8 yearsinvested at 9% p.a.

16 Use the table of present values todetermine the present value of each of thefollowing annuities.a $450 per year for 5 years at 7% p.a., with interest compounded annuallyb $2000 per year for 10 years at 10% p.a., with interest compounded annuallyc $850 per year for 6 years at 4% p.a., with interest compounded annuallyd $3000 per year for 8 years at 9% p.a., with interest compounded annually


WORKEDExample

22

WorkS

HEET 2.2



1 Calculate the amount of interest earned on $10 000 invested for 10 years at 10% p.a.,with interest compounding annually.

2 Calculate the future value of an annuity of $1000 invested every year for 10 years at10% p.a., with interest compounding annually.

3 Calculate the future value of an annuity where $200 is invested each month for5 years at 5% p.a., with interest compounding quarterly.

4 Calculate the amount of each annual contribution to an annuity that will have a futurevalue of $15 000 if the investment is for 8 years at 7.5% p.a., with interestcompounding annually.

5 Calculate the amount of each annual contribution to an annuity that will have a futurevalue of $500 000 in 25 years when invested at 10% p.a., with interest compoundingannually.

6 Calculate the present value of an annuity that will have a future value of $50 000 in10 years at 10% p.a., with interest compounding annually.

7 Calculate the present value of an annuity that will have a future value of $1 000 000 in40 years at 10% p.a., with interest compounding annually.

8 Calculate the present value of an annuity where annual contributions of $1000 aremade at 10% p.a., with interest compounding annually for 20 years.

9 Use the table on page 90 to find the future value of $1 invested at 16% p.a. for4 years, with interest compounding twice annually.

10 Use the answer to question 9 to calculate the future value of an annuity of $1250every six months for 4 years, with interest of 16% p.a., compounding twice annually.

A growing investmentBindi is investing $20 000 in a fixed term deposit earning 6% p.a. interest. When Bindi has $30 000 she intends to put a deposit on a house.

1 Write the compound interest function that will model the growth of Bindi’s investment.

2 Use your graphics calculator to graph this function

3 Find the length of time (correct to the nearest year) that it will take for Bindi’s investment to grow to $30 000.

4 Suppose that Bindi had been able to invest at 8% p.a. How much quicker would Bindi’s investment have grown to the $30 000 she needs?

5 Calvin has $15 000 to invest. Find the interest rate at which Calvin must invest his money, if his investment is to grow to $30 000 in less than 8 years.

2

inve


n



Inflation• The price of goods and services rises from year to year. To predict the future price

of an item we can use the compound interest formula taking the rate of inflation as R.• The same method is used to predict the future value of collectibles and of

memorabilia, which tend to rise at a rate greater than inflation.

Modelling depreciation• Depreciation can be calculated in two ways. The depreciation can be straight line

depreciation or declining balance depreciation.• Straight line depreciation occurs when the value of an asset decreases by a constant

amount each year. The graph of salvage value is a straight line, the vertical intercept is the purchase price and the gradient is the negative of the annual depreciation.

• Declining balance depreciation occurs when the salvage value of the item is a percentage of the previous year’s value. The graph of a declining balance depreciation will be an exponential decay graph.

Straight line depreciation• The salvage value of an asset under straight line depreciation can be calculated

using the formula S = Vo − Dn, where S is the salvage value, Vo is the purchase price of the asset, D is the amount of depreciation per period and n is the number of periods of depreciation.

• Values of Vo, D or n can be calculated by substitution and solving the equation.

Declining balance method of depreciation• Under declining balance depreciation the salvage value of an asset can be

calculated using the formula S = Vo , where R is the percentage

depreciation per year.• To calculate the amount by which an asset depreciates in a year we subtract the

salvage value at the end of the year from the salvage value at the beginning of the year.

Depreciation tables• Depreciation can be compared using either a table or a graph.• Tax deductions are allowed for depreciation of assets that are used as part of

earning an income.• A depreciation schedule is used to calculate tax deductions over a period of years.

Future value of an annuity• An annuity is where regular equal contributions are made to an investment. The

interest on each contribution compounds as additions are made to the annuity.• The future value of an annuity is the value that the annuity will have at the end of a

fixed period of time.

Use of tables• A table can be used to find the present or future value of an annuity.• The table shows the present or future value of $1 under an annuity.• The present or future value of $1 must be multiplied by the contribution per period

to calculate its present or future value.

summary

1 R100---------–

T



1 A walkman is currently priced at $80. If the current inflation rate is 4.3%, estimate the price of the walkman after one year.

2 It is predicted that the average inflation rate for the next five years will be 3.7%. If a skateboard currently costs $125, estimate the cost of the skateboard after five years.

3 In 1975, Cherie bought a limited edition photograph autographed by Sir Donald Bradman for $120. If the photograph appreciates in value by 15% per annum, calculate the value of the photograph in 2005 (to the nearest $100).

4 The table below shows the depreciating value of a yacht.

a Draw a graph of the value of the yacht against its age.b Write a function for the value of the yacht.

5 The table below shows the depreciating value of a racing bike.

a Draw a graph of the value of the bike against age.b Write a function for the straight line depreciation.c Use your graph to estimate the value of the bike after 9 years.


New (0) 200 000

1 180 000

2 160 000

3 140 000

4 120 000

5 100 000


New (0) 3500

1 3250

2 3000

3 2750

4 2500

5 2250

CHAPTERreview

2A

2A

2A

2B

2B



6 The function V = 15 000 − 900A shows the value, V, of a motorcycle when it isA years old.a Draw a graph of this function.b Use the graph to calculate the value of the motorcycle after 5 years.c After how many years would the motorcycle be ‘written off’ (the value of the

motorcycle become $0)?

7 The table below shows the declining value of a 4-wheel drive.

a Plot the points as given in the table and then draw a curve of best fit to graph the depreciation of the 4-wheel drive.

b Use your graph to estimate the value of the 4-wheel drive after 10 years.c After what number of years will the value of the 4-wheel drive fall below $10 000?

8 A laundry buys dry-cleaning equipment for $8000. Each year the equipment depreciates by 25% of the previous year’s value. Calculate the value of the equipment at the end of the first five years and use the results to draw a graph of the depreciation.

9 The purchase price of a car is $32 500. The car depreciates by $3250 each year. Use the formula S = Vo − Dn to calculate the salvage value of the car after 8 years.

10 Calculate the salvage value of an asset:a after 6 years, that was purchased for $4000

and depreciates by $450 each yearb after 10 years, that was purchased for

$75 000 and depreciates by $6000 each year

c after 9 years, that was purchased for $640 000 and depreciates by $45 000 each year.

11 A movie projector is purchased by a cinema for $30 000. The projector depreciates by $2500 each year. Calculate the length of time it takes for the projector to be written off.

12 A camera that was purchased new for $1500 has a salvage value of $500 four years later. Calculate the annual amount of depreciation on the camera.


New (0) 60 000

1 48 000

2 38 400

3 30 720

4 24 576

5 19 660

2B

2B

2B

2C

2C

2C

2C


C h a p t e r 2 A p p r e c i a t i o n a n d d e p r e c i a t i o n 9913 Arthur buys a car for $25 000. The depreciation on the car is $2250 each year. He decides

that he will trade the car in on a new car in the final year before the salvage value falls below $10 000. When will Arthur trade the car in?

14 The purchase price of a mobile home is $40 000. The value of the mobile home depreciates by 15% p.a. By calculating the value of the mobile home at the end of each year, find the salvage value of the mobile home after 4 years. (Give your answer correct to the nearest $1.)

15 Use the declining balance depreciation formula to calculate the salvage value after 7 years of a crop duster that was purchased for $850 000 and depreciates at 8% p.a. (Give your answer correct to the nearest $1000.)

16 Calculate the salvage value of an asset (correct to the nearest $10) with a purchase price of:a $40 000 that depreciates at 10% p.a. for 5 yearsb $1500 that depreciates at 4% p.a. for 10 yearsc $180 000 that depreciates at 12.5% p.a. for 15 yearsd $4.5 million that depreciates at 40% p.a. for 10 yearse $250 000 that depreciates at 33 % p.a. for 4 years.

17 A company buys a new bus for $600 000. The company keeps buses for 10 years then trades them in on a new bus. The estimate is that the value of the bus will depreciate by 12% p.a. Calculate:a the amount the owner can expect as a

trade-in for the bus in 10 yearsb the amount by which the bus will

depreciate in 10 years.

18 A company has office equipment that is valued at $100 000. The value of the equipment can be depreciated at $10 000 each year or by 15% p.a.a Draw a table that will show the

salvage value of the office equipment for the first ten years.

b Draw a graph of the depreciating value of the equipment under both methods of depreciation.

2C

2D

2D

2D

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2D

2E



19 A personal computer is purchased for $4500. A tax deduction for depreciation of the computer is allowed at the rate of 33% p.a. When the value of the computer falls below $1000, the computer can be written off. Copy and complete the table below.

20 Use the table of future values of $1 on page 90 to calculate the future value of an annuity of $4000 deposited per year at 7% p.a. for 8 years, with interest compounded annually.

21 Use the table of future values of $1 to calculate the future value of the following annuities.a $750 invested per year for 5 years at 8% p.a., with interest compounded annuallyb $3500 invested every six months for 4 years at 12% p.a., with interest compounded

six-monthlyc $200 invested every 3 months for 2 years at 16% p.a., with interest compounded

quarterlyd $1250 invested every month for 3 years at 10% p.a., with interest compounded

six-monthly

22 Use the table of present values of $1 on page 92 to calculate the present value of an annuity of $500 invested per year for 6 years at 9% p.a., with interest compoundedannually.

23 Use the table of present values to calculate the present value of each of the following annuities.a $400 invested per year for 5 years at 10% p.a., with interest compounded annuallyb $2000 invested every six months for 5 years at 14% p.a., with interest compounded

six-monthlyc $500 invested every three months for 2 years at 16% p.a., with interest compounded

quarterlyd $300 invested every month for 4 years at 12% p.a., with interest compounded half-yearly

Year Salvage value ($) Tax deduction ($)

1

2

3

4

5

2E

2F

2F

2F

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CHAPTERyyourselfourself

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Year 12 Maths A Textbook - Chapter 2

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