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Reading Description:

Forster, I. & Thomson, S. (2001). Staging multiple events. In Access to general maths : HSC (pp. 101-132). Port Melbourne, Vic. : Pearson Australia.

Forster, I. & Thomson, S. (2001). Depreciating assets. In Access to general maths : HSC (pp. 253-272). Port Melbourne, Vic. : Pearson Australia.

Forster, I. & Thomson, S. (2001). Extracts from Modelling relationships. In Access to general maths : HSC (pp. 393-399). Port Melbourne, Vic. : Pearson Australia.

Forster, I. & Thomson, S. (2001). Answers to selected chapters. In Access to general maths : HSC (pp. 431-433, 440-441). Port Melbourne, Vic. : Pearson Australia.

Reading Description Disclaimer:

(This reference information is provided as a guide only, and may not conform to the required referencing standards for your subject)

!itaging multiple events

The number of ways the first three horses in a race can finish, the number of possible hands in a card game and the number of

combinations of lotto balls regularly occur in gambling games .

In this chapter you will learn some systematic counting techniques and you will apply the techniques in determining the probability of an event.

!igllabus skills, knowledge and l.lllderstanding ProbabUit;y, PB3: Multi-stage events Constructing and using tree diagrams

I)etepnunng the number of outcomes for a multi-stage event Bstablishing the number of different ways n items can be arranged

Bstablish.ing the number of ways an ordered selection can be made

Establishing the number of unordered selections that can be made from

a group of different items

Using the formula for the probability of an. event to calculate the

. probability of a given selection occurring

101

1Di! AccESS TO GENERAL MATHS : HSC

Before you start

Complete this background quiz

1 Lucy is doing a multiple choice test. She has to choose whether to circle answer A, B, C or D for each question. When she doesn't know an answer she randomly guesses. What is the probability Lucy answers correctly when she randomly guesses an answer?

2 When a normal, six-sided die is rolled, which is the more likely?

A Rolling an even number

B Rolling a number greater than 4

Explain your answer.

3 Match the probabilities of the events in a to e with their positions i to v on the number line.

i ii

0

iii

1 2

iv V t

a Choosing a red card, when 1 card is randomly selected from a normal pack of 52 playing cards .

b Tossing a 'head' with a two-headed coin.

c Randomly picking a red marble from a bag containing 8 red and 2 blue marbles .

d Winning the lottery when you only have 1 ticket .

e Rolling a normal pair of dice and obtaining a sum of 1 3 .

4 Three p eople, Ari, Georgia and Jay, are waiting in a shop to be served.

a List all the possible orders in which they can be served.

b In how many different orders can they be served?

c If they are served at random, what is the probability they will be served in the same order in which they entered the shop?

5 Jake has to choose 3 names from a possible 5 , then arrange them in order. Which two of the following calculations could be used to determine the number of different arrangements he can make?

a 5x4x3 b 3x2xl c 5p3

If you are unsure of the meaning or solution of any of the questions in this background quiz, check with your teacher before proceeding with the

remainder of this chapter.

STAGING MULTIPLE EVENTS 103

Tree diagrams and systematic counting Have you ever wanted to know how many ways something can be done? Writing out a long list of possibilities and counting them is one strategy, but it can often be very time consuming. In the first part of this chapter you will learn some shortcuts for counting possibilities .

Example I When Yassif travels home from school he can cross one of 3 different bridges, then he can walk down one of 2 paths leading to his home . How many different routes can Yassif take between school and home?

!iolut:ion 1

h ome

One way to calculate the number of routes Yassif can take is to draw a tree diagram. � P, 81P1 8,

p2 81P2 � P, 82P, 82

p2 82P2 � P, 83P, 83

p2 83P2

Another way to calculate the number of possibilities is to multiply the number of choices Yassif has at each stage of his trip home.

He can cross "'--.. / He can c hoose 3 br idges. � / one of 2 paths.

[2lxw

Example i!

6 possibilities

The theme of a party Lauren is going to is 'look like a nerd'. Lauren has found 4 shirts, 2 hats and 3 pairs of pants she could wear to the party. How many different sets of 'nerd clothes' can she make?

!iolut:ion i!! Lauren could use a tree diagram to calculate the number of sets of clothes she could make, but multiplying the number of choices she has for each item of clothing would be much easier!

2 h ats

4 • h i rt• � \ / 3 P'iffi of "'""

wxwx� 24 different sets of clothes

104 ACCESS TO GENERAL MATHS: HSC

Workshee'l: 4 : I

4:1 The language of probability: card match activity

1 This tree diagram shows the outcomes when a die and a coin are tossed together.

a List all the possible outcomes when a die and a coin are tossed together.

b How many outcomes are there when a die and a coin are tossed together?

c How can you use the 6 possible outcomes when a die is tossed and the 2 possible outcomes when a coin is tossed to calculate the total number of possibilities when a die and a coin are tossed together?

1 � : 2 � : 3 � : 4�: 5 � : 6 � :

2 Richard and Susie are planning to go out on Friday night. They are planning to go to one of their three favourite restaurants for dinner and then to see one of two different movies .

a By placing appropriate numbers in these boxes, D x 0, calculate the total number of possible combinations of dinners and movies from which they can choose .

b Draw a tree diagram and list all the possible outcomes, to check the accuracy of your answer to part a.

3 For breakfast this morning Kevin is going to eat cereal and then a cooked meal. He has the choice of four different cereals and two different cooked meals .

a Draw a tree diagram then systematically write out a list of the possible breakfasts Kevin could have .

Hint:

Give each of the 4 cereals and 2 cooked meals 'a name' (i . e . Cp C2, C3, C4 and Mp M2) to make i t easier to draw the tree .

b Calculate the number o f possible breakfasts Kevin can have by placing the appropriate numbers in these boxes : D x D.

Did gou know?

Most bugs taste things with their mouth but flies, butterflies and bees can also taste things with their feet. When a fly lands on your breakfast, its feet immediately tell it that your food is good for it to eat!

STAGING MULTIPLE EVENTS

4 At the bank where Maria works the staff all wear a corporate uniform. The women's uniform consists of 2 different shirts (tops) and 3 different bottoms (2 skirts and a pair of pants) . Each day the female employees must wear one of the corporate 'tops' and one of the 'bottoms'.

a Draw a tree diagram and use it to systematically list the number of different uniform combinations available for women at Maria's bank.

105

b Calculate the number of possible 'tops' and 'bottoms' combinations by placing

the appropriate numbers in these boxes: D x D. 5 The corporate uniform for the men who work at Maria's bank includes the choice of

2 shirts, 2 pairs of pants and 2 ties . The men must wear one of each of these items every day. How many different corporate uniform combinations are possible for the men who work at Maria's bank?

6 The lock on a briefcase has 3 dials . Each dial has 8 different settings on it. How many different settings are possible on the lock?

7 Lucy is doing a 'true/false' test. How many different answer patterns are there when there are:

a 4 questions in the test?

b 5 questions in the test?

Tlte f,rd one r alwotp frve, then fa (se, then two frves to breRk the pattern ...

Ea�'! /

8 To determine which prize they win, contestants in a TV gameshow choose one of 5 boxes, then they select one of 3 envelopes from inside the box. How many different envelopes are there?

9 One system of car number plates has 3 letters followed by 3 digits . How many different number plates are possible using this system? (Letters and digits can be used more than once .)

1 0 Hassan works as a quality control officer in a factory that makes pens for businesses . Eight pens are packed into a box and 10 boxes are packed into a carton. How many pens are in each carton?

1 1 When Lachlan travels home from work he can catch any one of 4 different buses then walk along 2 different roads to his house. How many different ways can Lachlan travel home from work?

1 2 Todd is playing a computer game. In the game he has to choose different paths to follow. He has to choose between 3 different paths at 4 stages . How many different paths can he choose?

106 ACCESS TO GENERAL MATHS : HSC

13 At Tristan's school all students are required to study 3 different electives. They must choose one language (either Greek, Arabic or French) , one social science (History, Geography, Legal Studies or Business Studies) and one technical subject (Computing, Design or Technology) . How many different electives combinations are available to students at Tristan's school?

14 A new car is available in 3 different exterior colours and 2 different interiors. Both air conditioning and automatic transmission are optional. How many different combinations of colours, interiors and options are available for this car?

15 How many 4-digit odd numbers can be made using the digits 2, 5, 8 and 9?

Hint:

When you fill the boxes in D x D x D x D , the first step is to write the number of available odd digits in the box on the end, then fill the other boxes.

16 a How many 3-digit numbers can be made from the digits 2, 3, 5, 8 and 9 using each digit only once?

b How many of these 3-digit numbers will be even?

c How many of the 3-digit numbers will be greater than 700?

Review: Permutations When the order in which items are arranged is important, the arrangement is called a permutation.

Example 3 Five men, Bill, Jay, Luke, Tony and Scott, are competing in a crossword competition.

a In how many different ways can the first two places in the competition be filled?

b What is the probability Luke will win and Scott come second?

!iolut:ion 3

a One way to find the number of possibilities is to multiply the number of possibilities at each stage : 5 people could come first, which would leave 4 people to come second;

[[] x [I], which means 20 possibilities.

Alternatively, use 5P2 (5 men to place 2 in order) to calculate the possibilities.

b There is only 1 way in which Luke will win and Scott come second. The probability of this happening is 210 •

4: � Graphic:§ c:alc:ulator in!itruc:tion!i: permutation§


Worksheet 4 : i!

1 a In how many different arrangements can 5 people stand in a line?

107

b When 5 people randomly stand in a straight line, what is the probability they will be arranged from tallest to shortest, with the tallest person on the left?

2 a Six athletes entered a race. In how many different ways can first and second place be filled, assuming no ties?

b What is the probability that the oldest and the youngest runners in the race will fill the first two positions?

3 The student council has 6 members . In how many ways can the positions of president and vice president be filled from the 6 members?

4 a Four people are standing for election to the local council . In how many different ways can their names be listed in order on the ballot sheet?

b If the names of the 4 people are arranged at random, what is the probability they will be in alphabetical order?

5 When a gambler places a 'trifecta bet' they bet that they can pick the first three place getters in a race, in the correct order. How many possible trifectas are there for a 1 0-horse race?

6 Les is a very popular maths teacher. When he gives a class test he gives a different prize to each of the students who come first, second and third in the class . In how many different ways can the 20 boys in his class finish in the first three places in one of his class tests?

7 On Australia Day 1 2 different ethnic community groups are going to take part in a march through the city. The positions of the groups in the march are going to be determined by lucky draw. What is the probability the Irish group will lead the procession and the Arabic group will be second?

8 Con, the fruit and vegetable shop owner, likes to arrange his produce in attractive arrangements .

a In how many different ways can he arrange his asparagus, beans, tomatoes, squashes and peas?

b If he arranges these 5 items at random, what is the probability he will place the asparagus on the right hand end of the display?

Did you know?

After some people (approximately 4 in every 1 0) eat asparagus their body quickly produces a chemical that makes their urine smell like rotten cabbage, but not everyone can smell the difference! Just because the body makes the change, doesn't mean the person can smell it . Both of these characteristics are inherited and run in families .


[ombinations The tennis coach needs to choose 2 people to play together as a doubles team in a competition. He plans to choose the pair at random from his 4-member talented players' squad. The 4 players are Marie, Gemma, Alice and Donna.

There are 6 different doubles teams he could choose :

Marie and Gemma

Marie and Alice

Gemma and Alice

Gemma and Donna

1\Jote

Alice and Donna

Marie and Donna

The doubles team 'Marie and Alice' is exactly the same as 'Alice and Marie' .

When the order in which items are arranged is not important, it is called a combination.

Calculating the number of combinations

There are two common methods for calculating the number of combinations.

Method 1 In the tennis doubles problem there are 4 players who could be selected first. This leaves 3 players to select second, giving

W x [3] doubles pairs

But this procedure counts each doubles pair twice. (Remember, 'Marie and Alice' is exactly the same team as 'Alice and Marie' . )

To complete the calculation, the expression [±] x [3] must be divided by the number of

times each doubles pair has been counted.

This means, number of doubles pairs = 4 ; 3 = 6

Method 2

Most calculators have a 'combinations' button, ncr. In this notation n = the total number and r = the number being chosen.

In the tennis example, the number of possible players is 4 (n = 4) and 2 are being selected (r = 2) .

Evaluating 4C2 on your calculator gives 6.

1\Jote If necessary, ask your teacher to show you how to use the ncr button on your calculator.

Remember

When order is important, use permutations np r·

When order is not important, use ncr.

4:3a Eiraphic!!i calculator in!!it:ruction!!i: combinat:ion!!i


Example 4 The social club has 20 executive members . A committee of 4 of the executive members is going to be formed to organise the New Year's Eve party. How many different committees are possible?

!!iolution 4

Method 1 Selection of 4 members from 20 gives

1201 x 0QJ x [!}8 x @] committees

But identical committees have been counted twice (ABCD, ABDC, AGED, etc . ) .

So number of committees is

12o1 x [JQJ x [!}8 x @] + (number of times each committee has been counted)

Number of times each committee has been counted is the number of ways the 4 members of a 4-person group can be arranged:

G]x[[] x0 x [JJ @2lx[JQ]x@Jx@]

This means, number of committees = = 4845 G]x[}]x0x[JJ Method 2

The ncr method is much easier!

There are 20 members (n = 20) and 4 are to be chosen (r = 4) .

So number of committees = 2°C4 = 4845

BarKslleet 4 : 9

1 Use the ncr function on your calculator to find the value of each of these expressions .

a 6C5 b IOC3 c 9C2 d 8C4

2 A committee of 5 people is to be selected from 12 people.

a Explain why the calculation 12C5 will give the total possible number of different committees.

b How many committees are possible?

3 In a gambling game, players choose 6 numbers from a possible 10 numbers . How many different combinations of numbers are possible in this game?

4 Phil always cooks 3 different types of vegetables with his evening meal . For tonight's dinner he has the choice of peas, potatoes, carrots, tomatoes and pumpkin. How many different vegetable combinations can he cook tonight?


5 A representative netball training squad contains 1 0 players who play in defence . Three of these are going to be selected at random to play the next match. How many different groups of 3 players can be made from the 1 0 players available?

6 When Bob and Joyce eat at a Chinese restaurant they always share 2 different main dishes . The menu at their local Chinese restaurant contains 1 2 main dishes. How many different main meal combinations are available for Bob and Joyce to choose between?

7 Six girls and 4 boys are camping.

a A group of 4 girls is required to collect wood for a fire . How many different groups can be chosen?

b A group of 4 (2 boys and 2 girls) are going bushwalking. How many different groups are possible?

8 How many different 4-card hands can be dealt from a normal pack of 52 cards?

9 Six friends are going to play singles tennis . How many games will be required for each of the friends to play each of the other 5?

10 The languages French, German, Italian, Japanese, Indonesian and Spanish are all available for students to study at the school Joseph is going to attend next year. Joseph wants to study 2 languages.

a How many possible language combinations of 2 languages are available for Joseph?

b If Joseph chooses his languages at random from the 6 available languages, what is the probability he will choose Spanish and Japanese?

1 1 Grandma likes to display photographs of the family on her wall .

a She is going to choose 4 of her photographs to hang in a row on her loungeroom wall . In how many different arrangements can she hang 4 photographs?

b Grandma has 1 2 different photographs . How many different groups of 4 photographs can she select from them?

D i d you know? According to a Sunday newspaper report in January 2000, the chance of an asteroid hitting and destroying the Earth

this century is 50100 .


1 2 Match each of the situations in parts a to g with the appropriate calculation in i to vii. In how many ways can:

a a committee of 3 people be selected from a group of 1 0 people?

b

c

d

e

f

g

Summary

a hand of 5 cards be dealt from a ii 52p5 pack of 52 cards?

a set of 3 bottles of wine be selected iii szc 5 from a box containing 1 2 bottles?

a dealer place 5 cards on a table in order iv 10p3 from a pack of 52 cards?

3 single people be accommodated in 3 V 10C3 separate motel rooms?

a president, secretary and treasurer vi 12p3 be selected from a group of 1 0 people?

3 books be selected from 1 2 books and vii 3p3

put in order on a shelf?

4:3b Permutation§ and r:ombination!!i: review work!!iheet

Did you know? The misuse of probability is often the basis of jokes . Moore, the actor who plays the character Arthur in the British TV show Minder, suffers from Progressive, Supernuclear Palsy, or PSP, a condition that affects 1 in 1 00 000 people . When he announced he was suffering from the condition he claimed: 'There are 1 00 000 members of the screen Actors Guild. It is very considerate of me to have taken this disease for myself, thus protecting the remaining 99 999 members! '

Systematically counting the number of possibilities

There are five different ways to systematically determine the number of outcomes of a multi-stage event. These are : • Write a list of all possibilities and count them. • Draw a tree diagram to help you systematically list possibilities . • Multiply the number of possibilities at each stage of the multi-stage event. • Use 'combinations', and the ncr function on your calculator, when you are choosing

part of a group and order is not important. • Use 'permutations', and the npr function on your calculator, when you are choosing

part of a group and order is important.

As you work through the questions in the next worksheet you will be practising each of these different approaches to counting the number of possibilities!

11i! ACCESS TO GENERAL MATHS: HSC

ExamplE! 5 Peter is guessing the positions of 3 security switches, because he has forgotten which ones should be 'left' and which ones 'right' .

a In how many different ways can the switches b e arranged?

b What is the probability Peter will position exactly one switch correctly?

!iolution 5 a There are several different ways to calculate the number of ways in which the switches

can be arranged.

• You could write out all the possibilities and count them:

LLL LLR LRL LRR RLL RLR RRL RRR

• You could multiply the number of possibilities at each stage :

m X m X m This method also gives 8 ways .

• You could construct a tree diagram:

1st 2nd 3rd L

L< L � R

R � L

R L

R< L � R

R � L

R

8 ways

b To calculate the probability that Peter will get exactly 1 switch in the correct position, you need to work out the number of ways in which he can get exactly 1 switch in the correct position. There are several ways to do this too ! The easiest way is to write out the possibilities :

.lXX Xv'X xxv' 3 ways

The probability he will get exactly 1 switch in the correct position = �

Worksheet 4 : 4

Remember

Use permutations when order is important and combinations when order is not important.

1 To stop unauthorised employees from having access to the payroll computer, a

special sequence of the keys .. .. - .. .. must be entered into the computer before it can be operated. How many different sequences (arrangements) of these 5 keys are possible?


2 Three members of a 1 2-person singing group are going to be chosen to sing a song in their coming concert.

a How many different groups containing 3 people can be chosen from the 1 2 members of the singing group?

b What is the probability that Mike and Jo will both be in the seiected group?

3 Jake has 1 2 different-coloured handkerchiefs in his suitcase. He is going to take out 2 of the handkerchiefs without looking. How many different combinations of colours are possible?

4 Pier knows his bank PIN (personal identification number) has four digits and he knows the digits are 2, 6, 8 and 9 but he can't remember the order.

a How many different four-digit PINs use all of the digits 2, 6, 8 and 9?

b What is the probability Pier will be able to correctly guess his PIN?

5 When a remote control garage door system was being installed the tradesperson explained why the remote control opened only the door of this garage and not the door of the neighbour's garage. 'Ten switches are used on the control unit,' he said . 'Each one of the switches is in either the "On" or "Off' position. For security, I randomly decide which switches to have in what settings . '

a How many different remote control settings are possible with this ten-switch system?

b What is the probability that a random choice of switch positions will result in a remote control setting that will open the neighbour's garage door?

6 A security firm employs 1 0 security guards . The firm has been requested to supply 4 security guards for a coming political rally.

a How many different groups of 4 guards can be made from the 1 0 guards employed by the firm?

b Jason works as a security guard for the company. What is the probability he will be included in the group selected to attend the political rally?

Hint

If J as on is included, in how many ways can the remaining 3 guards be selected from the 9 other guards the company employs?

7 The menu at a wedding reception includes a choice of 2 entrees, 4 main meals and 3 desserts . How many different meal combinations are possible at the wedding?

8 The manager of a hairdressing chain that employes 42 people wants to form a committee of 5 people to advise her on future publicity ideas .

a How many different committees of 5 people can she make from 42 employees?

b Jayne, who works for the hairdressing chain, is interested in publicity. What is the probability Jayne will be included in the randomly selected committee?


9 To turn off her home security system, Kate has to enter the letters A, B, C, D and E, in the correct order, into the computer panel at her front door. If the security sequence is incorrecly entered on 4 consecutive tries, the computer locks the door and it is impossible to open the door for 1 2 hours .

a How many different arrangements of the letters A, B, C, D and E are possible using each of the letters exactly once?

b How many different arrangements are possible if the letters can be repeated?

c A burglar knows that the 5 letters, A to E, in the code for Kate's front door are all different. What is the probability he will be able to correctly guess the sequence in 4 or fewer tries?

10 Four friends, Effie, George, Anna and Freida, are going to the movies and they are going to sit together in a row.

a In how many different ways can the 4 friends sit in a row?

b In how many of these ways will Effie and George be sitting next to each other?

c If the friends seat themselves at random, what is the probability Effie and George will not be sitting together?

11 A fast food restaurant has 7 managers and 1 6 crew workers . For a busy shift on Thursday nights, 2 managers and 6 crew workers are rostered to work.

a How many different groups of 2 managers and 6 crew workers are possible to fill this shift?

b Jon is a manager and his girlfriend Moira is a crew worker. What is the probability that neither of them is rostered to work on Thursday night?

1 2 Four people can sit at the small square tables in a sidewalk cafe, with one person on each side . Sandy, Kaitlin and Emma sat down at one of the tables .

a In how many different arrangements can 3 people sit at the table?

b In how many of these ways will Sandy be sitting opposite Emma?

c If Sandy, Kaitlin and Emma sit at random, what is the probability Sandy will not be sitting opposite Emma?

Did you know? Medical experts define a drug as 'any substance that alters a body's function either physically or psychologically' . This definition includes alcohol, tobacco, prescribed medications and other substances . Current statistics show that the probability

of developing mental illness as a result of taking drugs is 2� •


13 There are 8 teams in an indoor soccer competition. During the season each team will play each other team twice .

a How many games will be required to allow each team to play each other team twice during the season?

b Assuming no ties, in how many different ways can the teams be placed first, second and third at the end of the season?

c Two of the 8 teams are going to be chosen at random to play in a charity match. How many different pairs of 2 teams are possible?

d What is the probability the teams selected for the charity match will be the 2 teams that came last and second-last in the competition?

14 There are n teams in a competition. Match each of the situations in a to e with the algebraic representation of the number of possibilities in i to iv. (Use one algebraic expression twice . )

a The number of games required for each n(n- 1 ) team to play every other team

b The number of games required for each ii n (n- 1 )(n- 2)(n - 3) team to play each other team twice

c The number of ways in which the first iii � n(n- 1 ) 4 places can b e filled in order

d The number of possible combinations of teams to fill the top 4 positions

e The number of possible ways in which teams can be placed last and second last

Modelling challenge

iv n(n-1 )(n-2)(n-3) 4x3x2x1

Thomas and Alexander are both good squash players . When they play each other they

both have a probability of � of winning the game. Alexander thinks he is a better player

than Thomas . To prove it he challenged Thomas to a competition.

'We'll play a series of games, ' Alexander said. 'We'll stop playing when you've won two games or I've won three. I'll concede that you're the better player if you can win two games before I can win three! '

Thomas wasn't sure whether to accept the challenge or not. It sounded like he would certainly win, but was he being conned?

What you have to do • Calculate the number of possible outcomes of this competition and then determine

the probability that Thomas will win the series. Then work out a way to simulate the playing of the competition and check whether your answer seems reasonable .

• Write a short note to Thomas giving him some advice on whether to accept the challenge or not. Remember to specify his probability of winning.


( _______ M_i_d_·c_h_a_p_ t_e_r_m __ o_d_e_ll_in_g __ a_c _t _iv_ i _ty _______ ) ��--------------------------------------------------------------�

4:4a and 4:4b Biased dice (1) and [2): spread!iheet!i

Investigating biased dice When normal dice are tossed they show each of the numbers from 1 to 6 as frequently as they show each of the other numbers. A biased die consistently shows one of the numbers more frequently than each of the other numbers.

Part A Spreadsheet 4 : 4a simulates the tossing of a biased die. In this simulation the number 6 shows three times more frequently than any of the other numbers.

What you have to do

• Simulate the tossing of the die 240 times and record how many times each of the numbers 1 to 6 occurs.

• From your results, determine the experimental probability of tossing each of the numbers 1 to 6 on this die.

• The theoretical probability of tossing each of the numbers from 1 to 5 on this die is

� (or 0·125) and the probability of tossing 6 is � (or 0·375). Investigate how closely

your simulation results match the theoretical values .

• The spreadsheet created this simulation by generating random numbers from 1 to 8.

The numbers from 1 to 5 were recorded as they were, but every outcome of 6, 7 or 8 was recorded as a 6. Write a sentence to explain how this simulation method makes

the probability of each of the numbers from 1 to 5 equal to � and the probability of

the number 6 equal to � .

Part 8 Spreadsheet 4 : 4b simulates the rolling of a different biased die.

What you have to do

• Use the spreadsheet to simulate tossing the second die 240 times. Then use your results to determine: - which number is the biased number - the experimental probability of tossing each of the six numbers on this biased die

• Extrapolate from your results, to determine the theoretical probability of each of the numbers occurring when this die is tossed.


Part C A chaUenge for the high fliers! A third die is biased so that the number 5 shows 'n times as often' as each of the other numbers. For this die, write an algebraic expression for the probability of rolling:

a a 1 b a 5

When this die is rolled 360 times, theoretically it will show a 5 on 160 rolls.

c What is the value of n?

Part D Austin has a pair of biased dice. Each die shows a 1 twice as often as each of the other numbers on the die. He constructed this grid to help him calculate the probability of obtaining different totals when he rolls these dice together.

+ 1 1 2 3 4 5 6 1 1 2 3 4 5 6

a What is the probability of obtaining a sum of 4 when these dice are rolled?

b Is 'rolling a double' (i.e. the same number on each die) more or less likely with Austin's dice compared to a pair of normal dice? Write a sentence to explain your answer.

Part E a Gavin rolled a normal die 48 times and he calculated the average of the numbers the

die showed. Calculate the value of this average.

b On a particular biased die, the number 5 occurs three times as often as each of the other numbers. Calculate the average of the numbers showing when this die is rolled 48 times.

c One of the numbers on another biased die occurs three times as frequently as each of the other numbers. When this die was rolled 48 times, the average of the numbers that showed was close to 3·52. Which number was biased?


Using probability trees to calculate probability

Tree diagrams showing all possibilities can quickly become large and hard to follow. Using probabilities on each branch of the tree diagram can simplify the situation.

Example 6 In a bag there are three blue discs and two white discs .

Two discs are selected at random without replacement.

What is the probability:

a both discs are white?

b the discs are white and blue?

!iolut:ion 6 Method 1

1st disc

There are 20 equally likely outcomes.

a P(both white) = 2

20

or 110

2nd disc

b PC different) =

P (white then blue or blue then white)

1 2 3 = 20

or 5

On the next page there is an easier way to draw the tree diagram for this question.


Method 2: a better solution • The first disc selected can be blue or white .

• In the probability tree in method 1 , 3 branches out of 5 lead to blue .

This means that the probability of selecting blue is � . • This probability can be written on a single branch leading to 'blue' .

• Similarly, the probability � can be written on a single branch leading to 'white '

(to represent the 2

branches out of 5 on the method 1 tree) .

Method 1 1st disc Method 2

B

B

B

w

w

• The second disc also can be blue or white .

1st disc

<(' 5 w

119

• If a blue disc was selected the first time, there will be 2

blue and 2

white discs left in

the bag. Both the branches from 'blue' will have � written on them.

• If the first disc selected was white, there will be 3 blue and 1 white disc left in the bag.

The probability � will be written on the branch leading to 'blue' and � on the branch

leading to 'white' .

1st disc 2nd disc 2

�B <B�

2 w

w � '

1_ w 4

a To find P(both white) , follow the first branch to 'white' then the second to 'white' .

The probabilities on the branches are � followed by � . P (both white) = � x � = 2

20 or /0 which is the same answer as before .

To move out along branches, multiply the probabilities on the branches together.

b To find P(different) the complete branches 'blue then white' and 'white then blue' have to be combined.

The expressions � x � and � x � must be added together to give �� or � . To combine branches, find the probability for each complete branch by multiplying, then add these probabilities together.

li!O ACCESS TO GENERAL MATHS: HSC

Remember

Multiply probabilities to move out along the branches and add probabilities to combine branches .

Example 7 In a box of chocolates there are 8 soft-centred and 1 0 hard-centred chocolates . Grant selected 1 chocolate at random and ate it. Then he ate another chocolate selected at random.

a What is the probability that the first chocolate he ate had a soft centre?

b What is the probability that both chocolates had a soft centre?

c What is the probability that he ate a soft-centred and a hard-centred chocolate?

Solution 7

1st chocolate 2nd chocolate

7

a soft __!!----soft

� hard 17

8 4 a P(first soft) =

1 8 or 9

8 7 b P(both soft) =

1 8 x

1 7

56 28 =

306 or

1 53

hard 8 i7 soft

9 17 hard

c P(soft and hard) = P(soft then hard or hard then soft)

8 1 0 1 0 8 =

1 8x

1 7+

1 8x

1 7

1 60 80 =

306 or

1 53

Dependent events

• Two events are dependent if what happens in the second event depends on what happened in the first event.

In the chocolates example above, the probability the second chocolate is soft depends on whether the first chocolate was hard or soft. Thus they are dependent events .

• Independent events are events that do not influence each other.

STAGING MULTIPLE EVENTS 1i!1

Example 8 Alan tossed a normal die three times. What is the probability he tossed: a 3 sixes? b no sixes?

!!iolut:ion 8 When a die is tossed, any of the numbers 1 to 6 is possible, but in this question the only relevant outcomes are '6 ' and 'not 6 ' .

a P(3 sixes) = ! x! x! 6 6 6

1 2 1 6

b P( . ) 5 5 5

no SIXeS = 6 X 6 X 6

1 25 2 1 6

Not:e

1 6 1st toss

6

not 6

2nd toss

�6 6 �not6

<,6 6 not 6

3rd toss

!_ 6 �not6

6

!_ 6 �not6 6 ! 6 �not6 6

!_ 6 �not6 6

Tossing a 6 on the first toss and tossing a 6 on the second toss are examples of independent events .

1 Ian drives through two sets of traffic lights on his way to work. The probability of

the first lights being red is � and the probability of the second lights being red is � . a What is the probability:

i the first lights will not be red?

ii both lights will be red?

iii the first will be red and the second won't be red?

b Are the events 'first lights red' and 'second lights red' dependent or independent events?

2 Rachel is a keen pistol shooter. On average, during competitions, she hits the target four times for every five shots she fires .

a What is the probability Rachel will hit the target with one shot in a competition?

b In a competition, Rachel is to have two shots at a target. What is the probability she will hit the target twice?

c In a competition practice session, Rachel had 1 20 shots at the target. Approximately how many times would she expect to miss the target?

1i!i!

3 Dentist phobia is a fear of going to the dentist . In Australia, 7% of adults have a dentist phobia.

A journalist chose two adults at random to interview about health issues. What is the probability that both of the adults have a dentist phobia?

ACCESS TO GENERAL MATHS: HSC

4 Helen has two coins : a biased coin, which has a probability of � of showing a head

and � of showing a tail, and a normal coin. When she tosses the two coins together,

what is the probability she tosses :

a two heads? b two tails? c a head and a tail?

Hint: When two coins are tossed together, consider one falling before the other. This will help you draw the probability tree . It doesn't matter which one you consider falls first!

5 Agoraphobia is a fear of open spaces which affects 5 % of Australian adults. Two randomly selected Australian adults were asked to complete a survey about fears and phobias. What is the probability exactly one of the two adults surveyed suffers from agoraphobia?

6 Red-green colour blindness occurs in 8% of Australian males .

a What is the probability that an Australian male selected at random does not suffer from red-green colour blindness?

b Two Australian men are selected at random. What is the probability at least one of them suffers from red-green colour blindness?

7 The blood of a person with haemophilia lacks one of several factors that enable blood to clot . Research has shown that 8 5 % of Australian haemophiliacs lack a factor known as 'factor 8' in their blood. If two Australian haemophiliacs are selected to have surgery on the same day, what is the probability they both lack 'factor 8 ' in their blood?

8 The chance that Nicole will be late

for class on any day is � . What is the

probability she will be late for class:

a 2 days in a row?

b exactly once in 2 days?


9 The probability Sarah will pass next week's assessment item is 90% and the probability Carl will pass is 60%. What is the probability:

a they will both pass?

b Carl will pass and Sarah will fail?

c at least one of them will pass?

1 0 Recent TV commercials report that the probability of being tested by a random breath-testing unit late on Friday night is � . On three Friday nights Darren went drinking and then drove home.

a What is the probability he was not tested by a random breath-testing unit?

b What is the probability he was tested once by a random breath-testing unit?

1i!3

1 1 The butcher has a large jar of jelly beans to give to children. He knows that 60% of the jelly beans in the jar are red. The butcher lets Sam choose two jelly beans at random. What is the probability:

a the first jelly bean Sam chooses is red?

b both the jelly beans Sam chooses are red?

c at least one of the jelly beans Sam chooses is red?

Hint

When probabilities are given as a percentage or ratio and you don't know the total number of items, you can assume that the total number of items is large. The probabilities you are given will stay the same even after a few items are removed from the total number.

1 2 In a country hospital on average equal numbers of babies are born each day of the week and an equal number of boys and girls are born. Tracey is expecting a baby soon. What is the probability she will have a baby girl on a Thursday?

1 3 Ben is going on a four-wheel-drive outback holiday. He plans to be driving in the desert for 3 days . At the time he is planning to go, the probability of temperatures

in excess of 40°C on any day in the desert is 190 •

a What is the probability he will have 3 days with temperatures over 40°C?

b What is the probability he will have 2 or more days with temperatures over 40°C?

Did you know?

Sturt's Stony Desert, in South Australia, is well named. It is covered in stones and very little else from one horizon to the other!

li!4 ACCESS TO GENERAL MATHS: HSC

14 Influenza is raging throughout the city. In the southern suburbs � of the people have

influenza . In the northern suburbs � of the people have influenza.

One person from the city is to be selected to serve on a jury. A random choice will first be made to choose either a northern or southern suburb. Then a person will be selected at random from that area. What is the probability the person selected is from a northern suburb and is suffering from influenza?

15 Charles regularly competes in 1 0-pin bowling competitions . With his first bowl he has a 7 5 % chance of knocking over all 1 0 pins . If he doesn't knock over all 1 0 pins he has a 50% chanc� of knocking over the remaining pins with his second bowl . Charles has 1 more set of 1 0 pins left in a competition. What is the probability he:

a will not knock over all 1 0 pins with his first shot?

b will knock over all 1 0 pins using 1 or 2 bowls?

16 Moira likes to play 'Kick Boxer' on her computer. When she tries to kick her opponent she has a 0 ·4 chance of hitting and eliminating him. She has 3 chances to eliminate him. What is the probability she can:

a eliminate him with her second kick?

b eliminate him with one of her 3 kicks?

17 'Ping Pong' is a game similar to tennis . I t i s played on a table. Players have a second serve only if their first serve does not go in. Kim's first serve has a probability of 0 · 3 of going in and her second serve has a probability of 0 ·75 of going in. What is the probability:

a neither of K.im's serves goes in?

b one of Kim's serves goes in?

18 Richard has trouble remembering things ! He has a � probability of bringing

his calculator to class and the probability he will remember his textbook is � . On any

particular day, what is the probability he will:

a remember to bring both his calculator and his textbook to class?

b forget both his textbook and his calculator?

c bring either his textbook or his calculator but not both?

1 9 Roslyn has three keys on her key ring. At random she is going to try a key in her front door until she finds the one that opens the lock. If the key doesn't open the door she will take it off her key ring and put it in her pocket.

a What is the probability the first key she uses will not open the door?

b What is the probability she will have to try all three keys before she can open the door?


Questions including the expressions lat least' or lat most'

Example g In a bag there are 7 blue and 3 black discs. Two discs

are selected at the same time. What is the probability:

a both of the discs are black?

b at least one of the discs is blue?

Hin�

1i!S

When a question requires you to select two items at the same time, consider selecting the first one a fraction of a second before selecting the second (without replacing the first) . This will make drawing the probability tree easy!

Solution 9 3 2

a P(both black) = 10 x 9 1

15

6

7 blue ___.!-----� � � bl.ok� 9

blue

black blue

black

b There are two ways to calculate the probability of 'at least 1 blue' .

First way

P(at least 1 blue) = P(blue then black or black then blue or blue then blue)

7 3 3 7 7 6 =- X - +- X - +- X -10 9 10 9 10 9

14

15

Second way

Concentrate on the opposite to what the question is asking for! 'At least 1 blue' is the

opposite to 'both black', which has a probability of 115 • There are no possibilities other

than 'both black' or 'at least 1 blue', so these two probabilities add to 1 .

The easy way to calculate the probability o f 'at least 1 blue' i s t o subtract the probability of 'both black' from 1 .

P(at least 1 blue) = 1 - P(both black)

= 1 - _!_ = 14

15 15

Did gou know?

Many people know that by counting the rings in a tree trunk you can tell how old the tree is. Tree rings are also a record of the weather! Wide rings correspond to years that were very warm or wet. Thin rings mean the weather was cold or dry.

1i!6 ACCESS TO GENERAL MATHS: HSC

Example 10 Five men and 2 women have volunteered to work on a 3-person committee which is going to be selected at random. What is the probability the committee will consist of:

a all men?

b all women?

c at most 2 men?

!iolution 10

5 4 3 a P(all men) = 7 x 6 x S

�M�: w 5

1 l! W 5 M

2 7

b P(all women) = 0 (There aren't enough women volunteers!)

c P(at most 2 men) means the probability of no more than 2 men.

P(at most 2 men) = 1 - P(3 men)

1 - � 7

5 = -7

Hint: It's usually easier to use a ' 1 - the opposite' technique when the words 'at least' or 'at most' appear in questions than it is to add all the appropriate probabilities together.

1 Julie is going to toss a normal coin twice . What is the probability she will toss:

a 2 heads? b at least 1 tail?

2 In a large urban area, 60% of adults have influenza. Three adults from the area are required to give evidence in a court case . What is the probability:

a all 3 people have influenza?

h none of them has influenza?

c at least 1 of them has influenza?

STAGING MULTIPLE EVENTS 1i!7

3 A medical research scientist reported that 65% of migraine headache sufferers have attacks triggered by oranges. Three migraine sufferers were waiting to see the research scientist. Based on the report, what is the probability that:

a none of them has attacks triggered by oranges?

b all 3 have attacks triggered by oranges?

c at most 1 of them has attacks triggered by oranges?

4 The weather bureau has predicted that there is a 25% probability of rain on each of the next 3 days . Assuming that the report is accurate, what is the probability it will rain on:

a all of the next 3 days?

b none of the next 3 days?

c at least 1 of the next 3 days?

d at most 1 of the next 3 days?

Did you know? Weather forecasting is an ancient practice . Clay tablets in the British Museum, dating from around 650 BC, show how the Babylonians used the sky to predict the weather. A dark halo surrounding the moon indicated rain within the next month and a small halo around the sun meant rain tomorrow!

5 The ability to smell cyanide is sex linked. Only 30% of women can smell cyanide and no males can smell it. Four men and 3 women are working in a laboratory. If there is a cyanide leak, what is the probability that at least one of the workers will be able to smell it?

6 The record books show that 10% of all athletes who competed in the 1996 Olympic Games suffered from exercise-induced asthma. A TV reporter chose 3 of the Olympic athletes at random to interview. What is the probability that at most 2 of these athletes suffered from exercise-induced asthma?

7 Karin is having trouble with the vegetables in her vegetable garden. Her vegetables have

a � chance of being attacked by caterpillars

and a � chance of being partially eaten by

rabbits . What is the probability that a randomly selected lettuce from Karin's garden will have been attacked by at least one caterpillar or rabbit?

.�e· ...

4:6 U5ing probability tree§: 5upplementary work5heet

li!B ACCESS TO GENERAL MATHS: HSC

Probabil ity word hunt

What word or term used in this chapter matches each of these descriptions?

1 Selecting an object without making a deliberate choice

2 More than 1 die

3 A special type of drawing used to systematically find all possible outcomes

4 A type of coin or die which is not fair

5 The list of all possible outcomes

Always, sometimes, never

Which of the words 'always', 'sometimes' and 'never' is missing in each of the following statements?

6 The probability of an event is _____ greater than 1 .

7 In probability, all the outcomes are _____ equally likely.

8 To go out along the branches in a probability tree you _____ multiply the probabilities on each branch.

9 In a probability situation the sum of the probabilities of every possible outcome ______ equals 1 .

1 0 The probability of an event happening is _____ equal to the probability of the event not happening.

1 1 The order in which events happen is _____ important in a 'combination'.

12 To calculate the number of total possibilities in a multi-stage event you ______ add the number of possibilities at each stage together.

Understanding the skills

Rate each of these events as impossible, unlikely, 50/50, likely or certain.

13 Correctly guessing the answer to a true/false question

14 Tossing a sum of 1 3 with a pair of normal dice

15 Winning a raffle in which there are 1 00 tickets and you have 2 tickets

16 Choosing a red frog from a jar containing 1 2 red and 4 green frogs


Applying the skil ls 1 7 Use your calculator to evaluate the following .

a 7p3 b sc4

1i!9

18 The local icecream parlour sells 20 different flavours of icecream. Miffy is going to buy 3 different scoops of icecream.

a How many different flavour combinations can she buy?

b Miffy is going to have the 3 different flavours placed one on top of the other in an icecream cone. How many different arrangements can she make from the 20 flavours available?

19 There are 6 different cans of soft drink in Kim's refrigerator . She is going to choose 2 cans at random to serve with lunch. How many different pairs of drinks can she serve?

20 Steve has a packet containing 1 2 chocolates: 5 white chocolates and 7 dark chocolates . Steve chose 1 chocolate at random, ate it, then chose another chocolate to eat at random.

a Is this an example of dependent or independent events? Explain your answer.

b Copy and complete this tree diagram to show the probabilities involved when Steve ate the chocolates.

1 st chocolate

5 i2

D

white

dark

c Calculate the probability Steve ate:

i 2 white chocolates

2nd chocolate

ii a white chocolate followed by a dark chocolate

iii a white and a dark chocolate

iv at least 1 white chocolate

21 Musgrave is going to travel from Woodville to Newcastle. There are 3 different roads from Woodville to Maidand and 2 different roads from Maitland to Newcastle. How many different routes can he travel from Woodville to Newcastle?

22 The CD player in Richard's car plays tracks from CDs in a random order. Richard inserted a CD with 1 2 tracks into the player. What is the probability that his 2 favourite songs on the CD will be the first 2 items selected by the CD player?


23 The 'Bright Bulb' factory makes torches. Each torch has one battery and one bulb . From experience the manager knows that )s% of the batteries will be defective and 6% of the bulbs will be defective. \ 0

The manager chooses a torch a t random. What is the probability:

a both the battery and the bulb are defective?

b at least one of the battery or bulb is defective?

24 A company employs 20 people in its workshop . In how many ways can a group of 6 people be selected from the 20 workshop employees to attend a group interview on safe workplace practices?

25 Seven athletes of equal ability and experience have entered a triathlon.

a In how many ways can the 7 athletes finish (assuming no ties) ?

b In how many ways can the first 2

places be filled?

c Anna and Karen have both entered the race . What is the probability:

i Anna will win and Karen will come second?

ii one of Anna and Karen will win and the other come second?

26 A tennis player gets a second s erve only if the first serve does not go in. Todd's first serve has a probability of 0 .6 of going in, and his second serve has a probability of 0 .9 of going in.

a Copy and complete this tree diagram, showing the probability on each branch.

1 st serve 2nd serve (in 0

out

b What is the probability Todd serves a double fault? (Remember, a double fault occurs when both the first and the second serve do not go in.)

c What is the probability one of his serves goes in?

2 7 Alcoholism is becoming a serious social problem. Medical experts claim that 5% of all males and 2 % of all females will develop alcoholism during their lifetime.

In any group of 3 male teenage friends, what is the probability:

a none of them will develop alcoholism?

b at least 1 of them will develop alcoholism?


28 a In how many different ways can a sum of 4 be tossed using:

i 2 dice? ii 3 dice?

131

b Which is more likely? A sum of 4 using 2 dice or 3 dice? Explain your answer!

29 There are 40 red, 50 green and 1 0 black jelly beans in a jar. Kylie ate all the black jelly beans and then she chose another one at random to eat. What is the probability this jelly bean is green?

30 The college principal is going to choose a student at random to interview about changes to the school exam timetable . He is going to toss a coin to choose either

Year 1 1 or Year 1 2 and then he is going to choose a name at random from that year.

School population

Year 1 1 1 20 girls 1 80 boys

Year 1 2 1 20 girls 1 50 boys

What is the probability he chooses :

a a Year 1 2 student? b a boy in Year 1 1 ? c a girl?

3 1 When Ian went on holidays he wrote a letter to each of his 3 girlfriends. By mistake he put each letter into an envelope addressed to the wrong girl . What is the probability of putting 3 letters into 3 incorrect envelopes? (Ian had no girlfriends when he returned from holidays !)

32 Jane, Kate and Julie are going to the movies and they plan to sit together in a row. What is the probability Kate won't be seated in the middle?

33 At a trivia night to raise money for charity, 1 00 tickets, numbered 1 to 1 00, were sold for 50 cents each. Different quantities of cans of soft drinks were the prizes .

Each ticket that was a multiple o f 1 0 won 1 can o f drink.

Each ticket that was a multiple of 2 5 won 3 cans of drink.

Each ticket that had two digits the same won 2 cans of drink.

a How many tickets won a prize?

b What was the probability of winning a prize in this competition?

c The organisers of the trivia night paid 80 cents for each can of drink. How much profit did they make from the competition?

34 A bag contains 1 2 marbles : 4 red, 4 white and 4 blue . A handful of marbles can be drawn from the bag at the same time. How many marbles would you need to draw (at random) to be certain of getting:

a at least 2 of the same colour?

b at least 1 of each colour?

c at least 2 white marbles?

13i! ACCESS TO GENERAL MATHS : HSC

Working backward!i A challenge exercise for the high fliers!

The challenge is:

Human blood can contain an antigen known as the 'Rh factor' . 85 % of Australians have the 'Rh factor' in their blood and their blood is said to be 'Rh positive' . People without the 'Rh factor' in their blood are said to be 'Rh negative' .

The ' Rh factor' i s important during pregnancy. It is possible for a mother and baby's blood to be 'Rh factor' incompatible, a situation that is potentially fatal for the baby. This can only happen when the mother is 'Rh negative ' , the father 'Rh positive ' and the baby 'Rh positive' .

When the mother is 'Rh negative' and the father is 'Rh positive' there is a 5 0% chance the baby will be 'Rh positive' .

Provided the mother's doctor is aware of the possibility of an 'Rh compatibility' problem, the baby's life can usually be saved.

Write two probability questions based on this information. The answer to the first question is 0 ·0637 5 and the answer to the second question is 0 ·8725 .

[hapter

Depreciating B!i!iet!i

Most items lose value as they become older. This is known as depreciation. Depreciation can be an allowable deduction for taxation purposes .

In this chapter you will learn about the two different ways of calculating depreciation allowed by the Australian Taxation Department and you will

be making financial decisions based on the calculations you have made.

Syllabus &kills, knowledge and under&tanding Financial mathematics, FM6: Depreciation

Using graphs, tables and functions to model depreciation

Using formulas for depreciation : • the straight line method • the declining balance method

Preparing tables of values and graphs of the current value of an asset over time, for different rates of depreciation

Calculating tax deductions based on depreciation of assets

i!54 ACCESS TO GENERAL MATHS: HSC

Before you start

Complete this background quiz

1 Write these percentages as decimals .

a 1 7 % b 3 %

2 s 300

250

200

1 50

1 00

50

2 3 4 a For graph Y, when n = 5 , what is the value of S?

c 1 2 � %

5 n

b For graph X, when S = 220, what is the value of n? Answer to the nearest whole number.

c When n = 3, how much more is the value of S on the X graph than the Y graph?

3 For the graph at right:

a when n = 1 what is the value of S?

b what is the gradient of the line?

1 2 3 4 n 4 In the equation y = b - mx, find the value of m when y = 30, b = 40 and x = 5 .

/

5 In the equation y = A ( 1 - x) n :

a find the value of y when A = 40, x = 0 · 3 and n = 5 .

b when y = 40, x = 0 · 1 and n = 1 2, what is the value of A, to the nearest whole number?

If you are unsure of the meaning or solution of �i§f, th� �� background quiz, check with your teacher bef'� p,t��:tb-�• -- • , . -

remainder of this 4'ap't�·: i : : - .· _ . : · <S > . , �,-�;'·. ;., _.�

DEPRECIATING ASSETS i!55

Most motor vehicles, computers, office equipment, tools, etc . depreciate in value with age. This means they lose value or are worth less than their original purchase price . If you own any of these items and use them in your work you can claim the depreciation (loss of value) as a tax deduction when you are completing your tax return.

There are two methods of calculating depreciation: the straight line method and the declining balance method. Both methods will be discussed in this chapter.

§traight line method of calculating depreciation In this method the value of the item decreases by the same amount each year which is usually a percentage of the original value of the item. When the values of the item (as it gets older) are shown on a graph they lie on a straight line . The straight line method is sometimes called the 'flat rate' method because its calculation is similar to flat rate interest calculations .

The formula you will be given in the examination Formula Sheet is :

Salvage value � -� �Vo - �n- ��-- - --�--.___ (current value) Origir{�l value Amo�nt of

Number of periods

after n periods (purchase depreciation price) per period

l\lote The depreciated value of an item after n periods can be called many things, including: current value, book value, resale value, depreciated value and salvage value.

The Australian Taxation Office supplies a tax guide indicating the maximum rate of depreciation that can be claimed on any item. The taxpayer may decide to spread the depreciation over a longer period and can choose to use a lower rate of depreciation than the one given in the tax guide . This table contains examples of the depreciation rates on some commonly used items . For more detailed information concerning the law and depreciation it is recommended you see an accountant.

Item

Air-conditioner

Photocopier

Video game machine

Depreciation rates

Straight l ine Decl in ing balance

1 3 % 20%

1 7 % 2 5 %

2 7 % 40%

Did you know? The Australian Taxation Office does not use the terms 'straight line' rate of depreciation and 'declining balance' rate of depreciation. It uses 'prime cost' and 'diminishing value' rate of depreciation.

i!S& ACCESS TO GENERAL MATHS: HSC

Example 1 A filing cabinet with a purchase price of $420 depreciated at $42 per year.

a What is the salvage (current) value of the filing cabinet after 3 years?

b After how many years will the filing cabinet be valued at $0?

!iolution 1 a S = V0 - Dn

= 420 - 42 X 3

= $294

V0 (original value) = 420

D (amount of depreciation) = 42

n (number of periods) = 3

After 3 years the filing cabinet is valued at $294.

b The question is asking what the value of n is when S = 0 .

S = V0 - Dn S = 0, V0 = 420, D = 42, n = ?

0 = 420 - 42 x n

42n = 420

n = 1 0

The filing cabinet will be 'written off' (worth nothing) after 1 0 years .

8 : 1 a liraphics calculator in!il:ructions: !il:raight line depreciation

Example i! Rodney said his new computer cost him $3200 and it will depreciate 22% of its original value each year.

Use the straight line method of depreciation to calculate :

a the amount it will depreciate each year

b the amount Rodney can claim as a tax deduction each year

c the value of the computer after 3 years

Solution i! a Depreciation (D) = 22% of $3200

= 0·22 X 3200

= $704

The computer will depreciate $704 each year.

b Rodney can claim a tax deduction of $704 each year for the value his computer depreciates .

c S = V0 - Dn

= 3200 - 704 X 3

= 1 088

The salvage value after 3 years is $ 1 088.

8 : 1 b Working with formulas: review worksheet

DEPRECIATING ASSETS

Work&beet 8 : 1

1 Use the formula S = V0 - Dn to find the value of S when:

a V0 = 5000, D = 650 and n = 4

b n = 7, V0 = 9600 and D = 970

2 Use the formula S = V0 - Dn to find the value of:

a V0 when S = 3640, n = 3 and D = 920

b n when V0 = 2900, S = 590 and D = 330

c D when S = 94, V0 = 550 and n = 8

i!57

3 A fibreglass spa at Mittagong Squash Courts cost $ 1 25 000. Use the straight line method of depreciation and a depreciation rate of 5% per year to answer the following questions about the value of the spa.

a By how much does its value drop each year?

b What is the salvage (current) value of the spa after 8 years?

c After how many years will the spa be valued at $50 000?

d When the spa has a salvage value of about $ 1 2 000 it will be considered to be no longer safe to use and will be scrapped. After how many years will its salvage value be approximately $ 1 2 000?

4 Norm needs his tractor for the heavy work around his farm. He bought the tractor for $ 1 42 500 and when he prepares his taxation return he claims 1 5 % per year depreciation, using the straight line depreciation method.

a How much can he claim each year on his taxation return for the depreciation on his tractor?

b What is the salvage value of the tractor after 4 years?

c After how many years will the salvage value of the tractor be zero?

5 Two years ago the Aseerberg Accounting Company purchased a computer for $4500.

a With a straight line depreciation rate of 20%, how much can the company claim as a taxation deduction each year?

b What is the salvage (current) value of the computer after 2 years?

c The company plans to sell the computer when its resale value reaches $ 1 800. After how many years will the computer have a resale value of $ 1 800?

6 A university library photocopier has a working life of 5 years (i .e . its value after 5 years is zero) . What straight line rate of depreciation must be applied to the photocopier to make its value zero after 5 years?

i!SB ACCESS TO GENERAL MATHS: HSC

7 A tea trolley used at the Better Baking Company was originally valued at $200 and was expected to last for 20 years (i .e . its salvage value after 20 years would be zero) .

a Use the straight line method of depreciation to explain why the company can claim $ 1 0 per year as depreciation on the trolley.

b What percentage rate of depreciation is $ 1 0 on an item with an original value of $200?

c Calculate the value of the trolley after 1 5 years .

8 The Peecroft Mine accountant expects the company's heavy duty loaders to have a working life of 8 years . What straight line rate of depreciation is this assuming?

9 The cash register purchased by P&S Stationery cost $800. The straight line method of depreciation allows a depreciation of $80 per year.

a Copy and complete this table.

Age in years, n Yearly depreciation, D Salvage val ue, S 0 $0 $800

1 $80 $720

2 $80

3

5

b Copy this number plane and plot the points from the table in part a on it.

Value of the cash register

Join the points plotted together and extend the line until it reaches the n axis. Answer the following questions using the graph.

c After how many years is the salvage value of the cash register:

i $240? ii $0?


d Why is this method called the 'straight line method of depreciation'?

e What straight line method percentage rate of depreciation is being applied to this cash register?

f What is the gradient of this line?

g Write a sentence to explain the relationship between the gradient of the line and the rate of depreciation being applied.

1 0 This graph shows the depreciation in the value of the lighted display case at Reynold's Book Shop.

3200

2800

<h 2400 c: � 2000 eo � 1 600 Cl "' � 1 200 (J)

800

400

2

Value of the display case

3 4 5 6 7 Age of display case i n years

a What was the original value of the lighted display case?

b By how much per year did its value decrease?

c What is the gradient of this line?

8

d What flat percentage rate of depreciation is being applied to the display case?

e When the salvage value of the display case is $ 1 200 the manager of the book shop believes it is time to get a new display case . How many years old is the original lighted display case when its salvage value is $ 1 200?

1 1 A heavy duty jack which is used to lift cars at Alex's Motor Repairs depreciates at $45 per year. It has been used at Alex's for the last 1 1 years and it now has a salvage value of $ 1 80.

a What was its original value?

b What rate of straight line depreciation is being used to calculate the value of the jack?


1 2 After 8 years the resale value of the Port Paradise Rugby League Club's outdoor catering stove and BBQ was $6000. Its original value was $30 000.

a Using straight line depreciation, how much per year did the outdoor catering stove and BBQ depreciate?

b What rate of straight line depreciation was applied to the catering equipment?

1 3 The safety manager and accountant at Bywater Open Cut Mines inspect all machinery at the mine every 3 months and they record any safety issues and the value of each piece of equipment. The accountant uses a straight line method of depreciation with an annual rate of 1 8% when he calculates his 3-monthly valuation of the large coal trucks .

a What 3-monthly straight line rate of depreciation is equivalent to 1 8% p .a . ?

b How much will the value of a new coal truck that cost $400 000 decrease to during 3 months at a straight line depreciation of 1 8 % p .a . ?

c Complete this table of values to show the value of the coal truck from new to 1 8 months.

Age of the truck (months) Value of the truck ($) 0 400 000

3 382 000

6 364 000

9

1 2

1 5

1 8

d Use the formula S = V0 - Dn to calculate the value of the coal truck after 36 months . (Remember: n = the number of periods . )

e i After how many time periods will the salvage value of the truck be $40 000?

ii After how many months will the salvage value of the truck be $40 000?

f The accountant said it didn't make any difference whether he used 3-monthly depreciation or annual depreciation with the straight line method of depreciation. He said the answer was the same both ways . Is he right? Explain why or why not.

��--------------------------------------------------------------� 8:1c Straight line depreciation : spreadsheet and worksheet

Did you know?

Undertakers have found that the human body is not deteriorating as fast as it used to . Some medical expects think this may be caused by the diet people follow these days . The modem diet has so many preservatives in it that

these chemicals may prevent the body from decomposing

at the rate it once did.


Declining balance method of calculating depreciation

In the declining (reducing) balance method the depreciation is calculated as a percentage of the value of the item at the start of each year. The formula is very similar to the compound interest formula.

�------� S = V ( 1 - r)n � -- --- - - �� Salvage value / - � o �- ---------

after n periods Origirial value Inte;;;st rate Number of

periods

Example 3

per period (as a decimal)

After 3 years what is the salvage value of a fax machine that originally cost $540? The declining balance rate of depreciation is 1 5 % .

Solution 3

The rate of depreciation (r) is 1 5 % and, as a decimal, r = 0 · 1 5 .

The number of time periods (n) is n = 3 (years)

S = V0( 1 - r) n

= 540( 1 - 0 · 1 5) 3

= 540 X 0 ·853

= $33 1 · 63

After 3 years the fax machine has a salvage value of $33 1 · 63 .

Example 4 Patricia is a graphic designer who works from home. Her new computer cost $5400 and she uses a declining balance rate of depreciation of 22% .

Use this table to determine how much Patricia can claim as a taxation deduction for the computer when it is :

a 1 year old

Solution 4 b 2 years old

Age in years

0

3

6

9

a Taxation deduction after 1 year = the drop in value in the first year

= $5400 - $42 1 2

= $ 1 1 88

Salvage value

$5400

$42 1 2

$3285

$2563

When the computer is 1 year old Patricia can claim $ 1 1 88 as a taxation deduction.

b Taxation deduction after 2 years = the drop in value in the second year

= $42 1 2 - $3285

= $927

When the computer is 2 years old Patricia can claim $927 as a taxation deduction.

8:2a 6raphic:s c:alc:ulator instruc:tions: dec:lining balanc:e deprec:iation

i!6i! ACCESS TO GENERAL MATHS: H S C

Worksheet B : i!

1 Use the formula S = V0 ( 1 - r)n to find the value of S when:

a V0 = $40 000, r = 0 · 1 and n = 4

b n = 9, V0 = $800 and r = 0 ·22 . Answer correct to the nearest cent.

2 Use the formula S = V0 ( 1 - r)n to find the value of:

a V0 when S = $274 000, n = 6 and r = 0·04. Answer correct to the nearest dollar.

b r when S = $ 1 3 629, V0 = $20 000 and n = 3 . Answer correct to 2 decimal places .

3 Norm uses his tractor on his property. Using the declining balance method of depreciation the rate of depreciation is 22% p.a . The original value of the tractor was $ 1 42 500. What is the value of the tractor after 5 years?

4 Chuck's Gymnasium uses the declining balance method (with a rate of 25% p .a . ) to calculate the salvage value of its weighttraining equipment. The equipment originally cost $ 1 5 000.

a Calculate the value of the equipment after 1 0 years .

b Copy and complete this depreciation schedule for the weight-training equipment.

Age in years Amount of depreciation

in dollars

0 0

1 3750

2 28 1 2 · 5 0

3 i

4 i i i

Salvage value (to the nearest dollar)

1 5 000

1 1 250

8 43 8

i i

iv

5 The declining balance depreciation on a photocopier is 25% p.a . After 5 years the salvage value of the photocopier at Port Paradise Real Estate was $925 . What was the original value of the photocopier? Answer correct to the nearest $ 1 0 .

Did you know?

A study of university students has shown that about 60% of students suffer from some high-frequency hearing loss .

The main cause of this premature 'deafness' is noise. Continual loud noises such as jet aircraft, cars and music

destroy the ears' tiny hair cells .


6 A paging system at the office of Hill's Quarry originally cost $600 and has a declining balance depreciation rate of 1 5 % p.a .

a Complete the missing sections of this depreciation schedule.

Age in years Amount of depreciation Salvage value (to the

in dollars nearest dollar)

0 0 600

1 90 5 1 0

2 77 433

3 i i i

4 i i i iv

6 40 226

1 0 20 1 1 8

b Use the values found in part a to copy and complete this graph.

� c: Q) :::s "' > Q) Cl "' > eo

(/)

600

500

400

300

200

1 00

0

' ...........

-· .. .

l

c Use the graph to find:

2

. .... . . ·· · -- ·

3

Value of the paging system · · -- r•

' · ····- ··· ··- ·- ··�--: i

·· · · · · · ..... ... . . . · · ··· ····· .. ...... · ········· ··· ············ .. .. ... ........ .

·· · · · ·············· ........ ......... . .... . ..

' ) -··· · ·· · ···· . . .... . . ........

. . .. . ...

. . ....

········-

...... . . ... ....... . .

! ····· · ······ · ·

f · 1 ' " i 4 5 6 7 8 9 1 0

N u m ber o f years

the salvage value of the paging system after 5 years

i i the number of years required for the salvage value of the paging system to be $ 1 60

d Complete this sentence. 'The graph representing straight line depreciation is a straight line. The graph representing declining balance depreciation is a

1 A digital video camera and equipment were stolen from Warren's Photography. The camera and equipment were originally valued at $ 1 5 000. The i�surance company used a declining balance depreciation rate of 25% when it valued the camera and equipment at $4750.

Use the estimate and refine (guess and check) method to find the number of years Warren's Photography had been using the camera and equipment.


8 Musgrave River Steel purchased a machine for cutting sheet metal . The Australian Taxation Office allows the owner to use either a straight line depreciation method or a declining balance depreciation method of calculating depreciation and salvage value for taxation purposes.

These graphs show the salvage value of the sheet metal cutting machine with both methods of depreciation.

Value of the sheet metal cutting machine

60 000 I� � 50 000 c: Q) 40 000 ::::J eo > Q) Cl 30 000 "' �

'\." " " ,.,.- St a i g ht l i 1e

.. '\.-. " � ae p rec1at1 on

"-.... " """ ' ( ....... '" "'

(Jl 20 000 Decl i n i ng / """" � baler ce '"" de breciati :>n

1 0 000 1"-----"-....

0 2 3 4 5 6

N u m ber of years

The values on the blue line are calculated using straight line depreciation.

The values on the black line are calculated using declining balance depreciation.

a The salvage values can be read from the graphs and then the amount of depreciation per year calculated. Copy and complete these two tables by using the information on the graphs .

Straight l ine depreciation Decl ining balance depreciation

Age in Salvage Amount of Age in Salvage Amount of years value depreciation years value depreciation

0 $60 000 $0 0 $60 000 $0

1 $ 5 1 000 1

2 $42 000 2

3 3

4 4

5 5

6 6

b What was the original value of the machine?

c Using the straight line method of depreciation, how much depreciation can be claimed each year as a tax deduction for the first 6 years?

DEPRECIATING ASSETS i!&S

d What rate of depreciation is used in the straight line depreciation of the cutting machine?

e Using straight line depreciation:

what is the salvage value of the machine after 7 years?

ii how much could the company claim as a taxation deduction for depreciation on the machine in the 7th year?

iii how much could the company claim as a taxation deduction in the 8th year? f After 1 year, how much more could be claimed as a taxation deduction for

depreciation using the declining balance method rather than the straight line method?

g After how many years does the declining balance method give less depreciation than the straight line method?

h Estimate the rate of depreciation being applied in the declining balance depreciation.

If you were the company accountant, which of the two methods of calculating depreciation would you choose? Write a paragraph to explain your answer. ·

9 The Palm Tree Cove Tyre Service has a pneumatic tyre remover which originally cost $2600. Using the declining balance method of depreciation the depreciation rate is 30% .

Complete this depreciation schedule for the pneumatic tyre remover.

Age in years Amount of depreciation Salvage value (to the

in dollars nearest dollar)

0 i i i

1 i i i i v

2 V vi

3 vi vi i i

1 0 The table on the next page shows the declining balance depreciation of a carpetcleaning machine which originally cost $20 000 . The owner plans to sell the machine after 6 years but he is not sure what rate of depreciation he should apply to value it.

a Complete the table for the resale value using 1 0% depreciation.


Resal e value (to the nearest $100) Age in years 10% depreciation 1 5% depreciation 20% depreciation

0 20 000 20 000 20 000

1 1 8 000 1 7 000 1 6 000

2 1 6 200 1 4 500 1 2 800

3 1 4 600 1 2 300 1 0 200

4 1 3 1 00 1 0 400 8 200

5 i 8 800 6 600

6 i i 7 5 0 0 5 200

b Using 1 5 % depreciation:

i what is the resale value of the machine after 6 years?

ii how much has the resale value decreased by the end of the 1 st year?

c The owner considered selling the machine after 5 years instead of 6 years .

At 20% depreciation, by how much does the resale value drop between 5 years and 6 years?

ii Using 1 5 % depreciation instead of 20% depreciation, how much less does the resale value drop between 5 years and 6 years?

1 1 Andrew owned The Cove Dry Cleaner and was planning to buy a new drycleaning machine. He was interested in the possible resale value of the current drycleaning machine. The three graphs were drawn using declining balance depreciation rates of 1 0%, 20% and 30%.

Resale price of d rycleaning machine

50 000

45 000

40 000 � 35 000 ·= Q) ::l 30 000 "' > Q) 25 000 Cl "' > (ii 20 000 (/)

i

1 5 000

1 0 000 ii 5000

0 2 3 4 5 6 7 8

N u m ber of years


a Which graph, i, ii or iii, represents 30% depreciation?

b What was the original value of the current drycleaning machine?

c After 8 years, what is the difference between the least price and the largest price he can expect to receive when he sells the drycleaning machine?

d Andrew wants to look at the possible resale prices with 1 5 % declining balance depreciation. Where will the 1 5 % graph fit compared to the other 3 graphs?

1 2 Alex and Chris each own a taxi . In 1 999 they bought identical taxis for $60 000 and they are both planning to keep their taxis for 5 years before they replace them. Chris plans to use a straight line method of depreciation at 20% and Alex is going to use a declining balance method at 30%.

The graph shows the value of each of their taxis when they are n years old.

Value of Alex's and Chris's taxis

60 000

w 50 000 c Ql 40 000 ::l "' > Ql 30 000 Cl "' > (ij

(j) 20 000

1 0 000

0 2 3 4 5 N u m be r of years

a Match the graphs X and Y with the taxi owners Alex and Chris .

b When will the salvage value of the two taxis be the same?

c Who will be able to claim the greater total amount of depreciation during the 5 years they each plan to own their taxi? Explain your answer.

Did you know?

In some card-playing groups the king of hearts is known as the 'suicide king' . This is because in every pack of cards, the king of hearts appears to have his sword going through his head. The other three kings have their swords by their sides.

i!68 AcCESS TO GENERAL MATHS: HSC

1 3 Brooke bought a new car for $24 000 in January 1 996 . This table shows how the resale value of her car has changed.

Year Age of car i n Resale value (January) years in dollars

1 99 6 0 24 000

1 997 1 20 640

1 99 8 2 1 7 750

1 999 3 1 5 265

2000 4 1 3 1 2 8

This graph shows how the value of an item originally priced at $24 000 changes at different rates of declining balance depreciation.

a Copy the graph and show on it the value of Brooke's car from January 1 996 to January 2000 .

b Estimate the rate at which the value of Brooke's car has been depreciating.

c Use the formula for declining balance depreciation and your answer to part b to predict the value of Brooke's car in January 2003 (i .e . n = 7) .

d In January 2000 Brooke's car had depreciated to $ 1 3 1 28 . What annual rate of straight line depreciation would have produced the same value?

8 : 2 b [alculating depreciation: review work§heet

Did you know?

Computer viruses were discovered late in the 1 980s. It is estimated that about six new viruses are discovered every day. However, thankfully, most of these do not exist for very long.


14 Joseph's restaurant has just bought new equipment. Joseph's accountant said that for taxation purposes, on major assets such as this equipment, he could choose to depreciate the equipment at either 1 0% using a straight line depreciation method or 1 5 % using a declining balance method.

This graph shows the total amount ( T) of depreciation that the restaurant could claim in the first n years.

Total depreciation for restaurant

1-0 20 000 .: c:: .g 1 6 000 ea ·c:; � Q. 1 2 000 Q) '0 -0 E 8000 � 0 E ea iij 4000 0 1-

0 2 3 4 5

Time in years, n

a During the first 5 years how much more depreciation could be claimed with the declining balance method than with the straight line method?

b At 1 0% straight line depreciation, in how many years is the salvage value zero?

c Use the graph to determine the original value of the new equipment for Joseph's restaurant.

d Which method of depreciation do you recommend J oseph choose to obtain the maximum amount of depreciation for taxation purposes? Explain your answer.

6 7

��--------------------------------------------------------------� B:i2!c Declining balance depreciation: spreadsheet and worksheet

Always, sometimes, never

Which of the terms 'sometimes', 'always' or 'never' is missing from each of the following statements? Explain your choice.

1 Straight line depreciation ------ gives less depreciation per year than declining balance depreciation.

3 A depreciating asset is -"--::-''----- a piece of equipment used by the company owning it.

Applyin g the skil ls

S What is the yearly straight line depreciation of a juice machine that has an expected life of 5 years?

6 The Walls Bros Tool Company uses the straight line method of depreciation to calculate the salvage value of its lathe, which originally cost $ 1 5 000. The taxation office allows the company to claim 1 5% p.a . depreciation.

a How much can Walls Bros claim per year for depreciation for the lathe?

b What is the salvage value of the lathe after 3 years?

c Mter how many years will the salvage value of tp.e lathe be zero?

7 This graph shows the depreciation in value of the boat ramp at the Dolphin Marina.

"' Value of boat ramp

:.,. ____ 1 -''. 60000 I 1 .,

............ I � .5 50 000 Q) :;::) 40000 m > Q) 30 000 0> <tl

�------F--�-- --� i .......... i ........ !'--.

--- � ! -........... "

............ "' ' ''" > m 20000 (/) --- �-=- --··

1 0 000 -

0 i I

2 4 6 8 1 0 1 2 N u m ber of years

a What was the original cost of the boat ramp?

b By how much does it depreciate each year?

c What is the gradient of this line?

' ......... l ............. i ..............

14 1 6 1 8 2 0

'"- ·'' .:'� ;:: "

" c+

/'!• �


d What straight line (flat) rate of depreciation does the owner claim for the ramp?

e After 5 years, what fraction of the original cost is the value of the boat ramp?

8 The owners of the Port Paradise Shopping Mall use the declining balance method of depreciation to assess the current value of the 'people mover' (escalator) in the Mall which originally cost them $75 000. The tax office allows them to claim 1 8% p.a . depreciation.

a Copy and complete this depreciation schedule.

b How much value had the 'people mover' lost after 3 years?

c Use the 'guess and check' method to find after how many years the 'people mover' will be valued at $ 1 8 700. Answer to the nearest year.

Age i n years

0

1

2

3

4

Depreciation Current value

i n dollars i n dollars

- 7 5 000

1 3 500 6 1 500

9 077 4 1 353

9 Bernice and Angie each own a bookstore . They both bought the Allstrong Bookshelf System of bookshelves. Bemice uses the straight line method for calculating depreciation and Angie uses the declining balance method.

Value of bookshelves

6000 � 5000 £ Q) ::I 4000 7ii > E 3000 Q) .... :; 2000 u

1 000

0 2 3 4 5 6 7 8 9

Age i n years

a What was the original cost of the Bookshelf System?

b How much depreciation can Bemice claim each year using straight line depreciation?

c What rate of depreciation is Bemice allowed for the bookshelves?

d After 1 year, how much more depreciation could Angie claim than Bemice?

e After how many years will the value of the two sets of bookshelves be the same?

f They both intend to keep the Bookshelf System for 6 years . Who will have claimed a greater total depreciation after 6 years?

AcCESS TO GENERAL MATHS: HSC

Modelling assignment

Equipmmt needed: Copies of newspapers that advertise cars for sale.

What you have to do • Choose a common make of car which can still be bought new but which has also been

on the road for at least 6 years (e.g. Mitsubishi Lancer) .

• Research the value of the car for different ages and complete this table. You will probably be able to find several values for some years. Record them all.

1 :. I new I year 2 years

• Plot the values you have found on a scatter graph such as this one.

3 years 4 years

Age in years • Draw a •cwve of best fit' through the points on the scatter graph.

• Determine the declining balance rate of depreciation for this make of car.

• Predict the value of the car when it is 8 years old.

5 years

• Are there any 8-year-old cars of the same make advertised for sale? How accurate was your predicted price?

• Write a list of factors that influence the resale value of cars of similar age.


Modell ing a!i!iignment

��·--------------------------------------------------------------� 1 2 :3c: Population of planet: !!iipread!!iiheet!!ii

The population of plan et X The president of the United Cities of planet X is very worried. The population is becoming too large for the planet. Food and other resources on the planet are inadequate for the increasing number of people .

The president has called an emergency meeting of his advisors for 12 noon tomorrow to discuss the implications of two possible solutions he is considering to solve the population problem. His possible solutions are :

Limit every family to a maximum of 2 children.

or

o Allow families to have as many children as they want until a daughter is born. After the birth of a daughter they will not be allowed to have any more children.

Equipment required

Spreadsheet 1 2 : 3c

What you have to do

o Use spreadsheet 1 2 : 3c to simulate the families of 1 00 couples to allow you to investigate the consequences of both of the president's possible solutions . You should consider questions like :

a Will either of the possible solutions produce a population with equal numbers of male and female children?

b With the second possible solution, what is the most likely number of children in a family?

o Use a probability tree to determine the theoretical probability of families comprising:

a 2 girls, 2 boys, or a boy and a girl under the '2 children limit' rule

b 1 child, 2 children, 3 children, 4 children, etc . , assuming each family comprises the maximum number of children, using the ' 1 girl ' rule

o Think about any social, religious or ethical problems related to either or both of the president's possible solutions .

o Write a report suitable for presentation at the president's emergency meeting tomorrow. Your report should include your results from your spreadsheet simulations and your theoretical calculations . You should also describe any social, religious or ethical problems that you think may be associated with either one or both of the president's possible solutions to the population problem. Include any alternative solutions you may have to solve planet X's current problem.

MODELLING RELATIONSHIPS 393

Direct variation

The distance a car travels in an hour i s directly related to its speed. The faster its speed, the further it goes in an hour. When two variables are related in this way they are said to be in direct variation. The distance (D) is directly proportional to the speed (S) . In symbols this is written as D ex: S.

A variation, or proportional statement, can be written as an equation. For example, the proportional statement D ex: S can be written as D = k x S where k is called the constant of variation.

Example 8 The number of metres (m) a ball falls in t seconds after being dropped off a cliff is directly proportional to the square of the number of seconds it has been falling. That is, m ex: t2 .

The ball falls 54 m in the first 3 seconds . Write the variation statement, m oc t2, as m = k x t2, then answer the following questions .

a Find the value of the constant of variation.

b How far does the ball fall in the first 2 · 5 seconds?

c How long will the ball take to fall 1 20 m?

!iolut:ion 8 a In the formula m = k x t2, k is the constant of variation . Substitute the information

m = 45 and t = 3 . m = k x t2

45 = k X 32

= 9k k = 5

The value of the constant of variation is 5 .

b The question asks for the value of m when t = 2 · 5 seconds . m = 5t2

= 5 X 2 · 52

= 3 1 · 25

The ball falls 3 1 · 2 5 m in the first 2 · 5 seconds .

c This part of the question asks for the value of t when m = 1 20 . m = 5 t2

1 20 = 5 t2

t2 = 24

t = J24 = 4 ·9

I t will take 4 · 9 seconds for the ball to fall 1 20 m.


Remember Whenever you are attempting to solve a variation or proportion question, always find the value of k, the constant of variation, before you try to answer the question.

Warksheet 13 : 7 1 The area (A) of a reguJar hexagon is in direct

variation to the square of the length (l) of its sides . That is, A ex: !2 • A regular hexagon with sides of 4 cm has an area of 4 1 · 6 cm2 .

·

a In the equation A = k x !2, what is the value of k? b What is the area of the regular hexagon with sides

of 5 cm? c Calculate the length of the sides of the regular hexagon that has an

area of 1 46 cm2 •

2 In this table of values Y ex: )(2 .

1 0 1 2

400 ii

a Write the direct variation statement Y ex: X2 as Y = k x X2 and find the value of k, the constant of variation.

b Calculate the two missing values in the table .

3 In this table of values M ex: p 3 .

4

3 2

a What is the value of the constant of variation? b Determine the two missing values in the table .

4 The stopping distance (D m) of a racing car is proportional to the square of its speed (S km/h) . That is, D ex: S2 . A racing car travelling at 200 kmlh requires 1 72 m to stop . a Calculate the stopping distance of a racing

car travelling at 1 60 km/h. b To avoid a collision, a racing car had to

stop within 60 m. At what maximum speed could the car have been travelling?

MODELLING RELATIONSHIPS

5 The time (t seconds) it takes for a pendulum to complete one oscillation is directly proportional to the square root of its length (l cm) . That is, t oc Jz . It takes 5 seconds for a 1 6 cm long pendulum to complete one oscillation.

a How long will it take a 36 cm long pendulum to complete one oscillation?

b A pendulum takes 2 � seconds to complete one oscillation. How long is the pendulum?

395

6 The mass of an egg in grams is proportional to the cube of its length in centimetres . A 6-cm�long he{;. egg has a mass o f 70 g. What i s the mass o f a 24-cm-long ostrich egg?

7 Joyce makes delicious sponge cakes. All the cakes she makes are always the same height. In one of Joyce's recipes the number of eggs required varies with the diameter of the tin in which she plans to cook it. When the diameter of the tin is 20 cm she needs 3 eggs . How many eggs will she need when she uses a tin with a diameter of 30 cm?

, . . . -9. -

8 The distance (D km) to the visible horizon is in direct variation to the square root of

the height (h m) above sea level . That is, D oc Jh. . At a height of 1 6 m above sea level, the distance to the horizon is 1 4 ·4 km.

a At what height above sea level is the distance to the horizoq 36 km?

b How much further to the horizon is it at a height of 62 m .above sea level than at 42 m above sea level?

9 The length (D) of the diagonal of a square is directly proportional to the square root of its area (A) . A square with an area of 25 cm2 has a diagonal 7 · 07 cm long.

a Write a rroportional statement relating D and A .

b Calculate the length o f the diagonal of the square that has an area o f 4 2 · 2 5 cm2 .

Did you know? Bacteria that can cause infection and disease multiply rapidly following an exponential model . Even though they double in number every 20 minutes, fortunately they cannot keep reproducing indefinitely. The maximum number of bacteria is limited by the food and air supply. When these essential supplies run out the bacteria begin to die .


Reaction time!i and !iafe driving

A group practical activity

A driver's reaction time is the amount of time it takes the driver to register the need to do something and begin to take action. In the adult population the average driver reaction time is 0 · 7 5 seconds . Are you slower or faster to react than the average driver?

In this practical activity you will measure and calculate your own driving reaction time .

Equipment required:

a 1 m ruler without a handle on it

a chair and a strong table

Method • Place the table close to, but not touching, a

wall .

• Position the chair on top of the table to simulate the driver's seat in a car.

Seat a member of your group on the chair with his/her right foot on the wall, pretending that the wall is the accelerator in the car. Make sure a strong member of your group holds on to the chair to stop it moving.

• Ask another member of your group to hold the metre ruler on the wall 1 0 cm to the left of the driver's foot, in the position of the brake in a car. Line up the zero on the ruler with the ball of the foot.

• When the driver is ready, the person holding the ruler lets the ruler fall . Don 't say 'Go ! ' . In this activity you are measuring the driver's visual/foot reaction time.

• When the driver sees the ruler falling, the right foot should be moved off the wall and onto the ruler.

• Use the number of centimetres showing on the ruler and the formula in question 1 on the next page to calculate the driver's reaction time.

Did gou know?

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In an emergency situation, the reaction time of a driver who has been drinking is directly proportional to their blood alcohol concentration level .


Work&heet 13 : 8

1 In the practical activity, a driver's reaction time is directly proportional to the square root of the number of centimetres the ruler dropped before the driver stopped it with

his/her foot. That is t = Jd where t represents the reaction time in seconds and d is the number of centimetres the ruler fell. When the ruler falls 7 4 centimetres the driver's reaction time is 0 · 3 9 seconds .

a Calculate the value of the constant of proportion for the statement t = Jd . b What is your driving reaction time?

c A driver's reaction time was 0 · 2 5 seconds . How far did the ruler fall?

2 The distance a car travels during the driver's reaction time (i . e . the distance the car travels before the brakes are applied) is directly proportional to the product of the driver's reaction time (t seconds) and the speed of the car (S km/h) . That is, D = t x S. A car travelling at 90 km/h will travel 1 0 m during a driver reaction time of 0 · 4 seconds .

a This direct variation statement can be expressed as D = ktS. Find the value of k .

b A car is travelling at a speed of 1 20 km/h. How far will i t travel during a driver reaction time of 0 ·8 seconds?

c When you are driving at 60 km/h, how far will you travel during your reaction time?

3 The total distance (D m) it takes to stop a car on a dry road is in direct variation to the product of the car's initial speed (S km/h) and the square of the driver's reaction time (t seconds) . That is, D = S x t2 . When a car travelling at a speed of 60 km/h is being driven by a driver with a reaction time of 0 ·36 seconds, it will take 1 0 · 8 m to stop .

a Express D = S x t2 as an equation and find the value of the constant of variation.

b The car is being driven by a driver with a reaction time of 0 · 6 seconds . How much further does it take for the car to stop from a speed of 1 00 km/h than from a speed of 80 km/h?

c A special speed limit of 40 km/h applies to roads near schools . How many metres shorter can you stop from 40 km/h than 60 km/h?

d Why do you think this special speed limit is necessary?

Did you know? A car crashing at 1 00 km/h puts the same force of impact on a human body as hitting the ground after falling from the top of a 1 2-storey building.


lnver!ie variation When 'green frogs' cost 5 cents each, David could buy 20 frogs for $ 1 . When the price of frogs increased to 6 cents each, he could buy only 1 6 frogs for $ 1 . As the price (P) increased, the number (N) of frogs he could buy decreased.

In this example N oc � which means 'N is inversely

proportional to P' or 'N varies inversely with P'.

When two variables are related in this way, one decreases as the other increases . As N decreases, P mcreases .

This relationship also can be written as an equation:

N = k x � or N = � where k is the constant of variation.

E x a m p l e 9 When inflation is less than 25% p . a . , the number of years (N) it takes prices to double is inversely proportional to the rate of inflation (R) . When the rate of inflation is 6% p . a . , it takes 1 2 years for prices to double.

How long will it take prices to double when the rate of inflation is 9% p .a . ?

§olu'tion 9 As the rate of inflation increases, the number of years it takes prices to double decreases.

1 This relationship is inversely proportional . That is, N oc R .

N = k x .!_ or N = !!_ R R

The first step is to find the value of the constant of variation (k) . When N = 1 2 years, R = 6% p .a .

N = !!_ R

1 2 = � 6

6 X 1 2 = k k = 72

The second step is to answer the question!

N = 72 R 72 9

= 8 It takes 8 years for prices to double when the rate of inflation is 9 % p .a .


1 Neil is a scuba diver. He knows that the length of time (t minutes) the air in his scuba tank will last is inversely proportional to the depth (D m) at which he is diving. When Neil is diving at a depth of 30 m the air in his tank will last 20 minutes .

a Show that the equation t = 6�0

represents this information. !_//

b How long will the air in Neil's tank last when he is diving at a depth of 40 m?

c The air in Neil ' s tank lasted 50 minutes . At what depth was he diving?

2 When less than 1 00 people attend the formal, the cost per person ($C) to run the formal varies inversely with the number of people (N) attending. When 60 people attend, the cost is $36 per person.

3

a If only 40 people attend the formal, how much will each person have to pay?

b Kerry said the tickets should cost $24 each. How many people have to attend the formal for the price to be $24 per person?

c Greg said: ' If you double the number of people attending the formal, you halve the cost per person . ' Is Greg correct? Give a reason for your answer.

In this table, t = �. Find the missing values. w

I w I 4 I 2 I b t 5 a 1 · 2 5

4 The cost per person ($C) of hiring a plane for a 30-minute scenic flight i s inversely proportional to the number of people (N) on the flight. When there are 3 people on the flight they each have to pay $30 . -� � � � � a The check-in clerk counted the people who

wanted to go on the flight and told them the cost would be $22 · 5 0 each. How many people did the clerk count?

b The maximum number of passengers the plane can hold is 6 . What is the cheapest price available for this flight?

Did you know? In recent years, inflation was at its worst in the 1 970s when oil producers suddenly increased their prices. These price increases were quickly passed on to consumers as higher prices, which led to rapid increases in inflation.

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