Percentage - Haese Mathematics · amount. A percentage is the comparison of a portion with the whole amount. ... f 3:027 g 25 16 h 17 i 54 5 j 0:7 ... To find a percentage of a quantity,

Contents:

Percentage

A Percentage

B

C Finding a percentage of a quantity

D The unitary method in percentage

E Percentage increase and decrease

F Finding the original amount

G Simple interest

H Compound interest (Extension)

3Chapter 3

Expressing one quantity as a percentage of another

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44

Opening problem

PERCENTAGE (Chapter 3)

A manufacturer of breakfast cereals has increased the

contents of their boxes by 20%. The new boxes now

contain 1:5 kg of cereal.

How much cereal did the old cereal boxes contain?

In this chapter we will explore some everyday applications of percentages, including percentage

change, simple interest, and compound interest.

The word percent means “out of every hundred”. We therefore say that there is 100% in a whole

amount.

A percentage is the comparison of a portion with the whole amount.

We use the symbol % to show a percentage.

For example, 10% means 10 out of every 100 or10

100, and 25% means

25

100.

So, x% =x

100.

CONVERTING FRACTIONS AND DECIMALS TO PERCENTAGES

To convert a fraction or a decimal to a percentage, we multiply by 100%. Since 100% = 1, this

does not change the value of the number.

Self Tutor

Convert to a percentage: a5

8b 0:065 c 1:35

a5

8

=5

8£ 100%

= 62:5%

b 0:065

= 0:065 £ 100%

= 6:5%

c 1:35

= 1:35 £ 100%

= 135%

EXERCISE 3A.1

1 Write these fractions as percentages:

a1

2b

2

5c

13

20d

17

25e

23

100f

14

10

g31

50h

21

40i

3

8j

5

16k

101

125l 2

1

8

PERCENTAGEA

Example 1

20% EXTRA!

1.5 kg1.5 kg

Wheaty

Crunch

Wheaty

Crunch

Wheaty

CrunchWheaty

Crunch

20%

EXTRA!

Shifting the

decimal point 2

places to the right

multiplies by 100.


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PERCENTAGE (Chapter 3) 45

2 Write these decimals as percentages:

a 0:13 b 0:35 c 0:03 d 0:9 e 0:06 f 2:7

g 0:2 h 3:25 i 1:05 j 3:41 k 0:075 l 1:225

3 Write as a percentage:

a 0:3 b 1 c 0:0004 d3

2e

5

6

f 3:027 g25

16h 17 i 5

4

5j 0:7

CONVERTING PERCENTAGES TO FRACTIONS AND DECIMALS

To convert a percentage to a fraction, we write the percentage as a fraction with denominator 100,

then express the fraction in simplest form.

Self Tutor

Express as a fraction in simplest form: a 150% b 0:5%

a 150%

=150

100

=150¥ 50

100¥ 50

=3

2

= 11

2

b 0:5%

=0:5

100

=0:5£ 2

100£ 2

=1

200

To convert a percentage to a decimal, we shift the decimal point 2 places to the left.

Self Tutor

Express as a decimal:

a 43% b 321

2%

a 43%

=43

100

= 0:43

b 321

2%

=32:5

100

= 0:325

EXERCISE 3A.2

1 Copy and complete:

a 7% =¤

100b 87% =

¤

100c 45% =

¤

100d 33% =

¤

100

Example 3

Example 2

43.

Shifting the

decimal point 2

places to the left

divides by 100.


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46 PERCENTAGE (Chapter 3)

2 Express as a fraction in simplest form:

a 20% b 150% c 75% d 50% e 310% f 8%

g 100% h 2% i 121

2% j 1:5% k 87

1

2% l 33

1

3%

3 Express as a decimal:

a 44% b 39% c 95% d 60% e 125% f 400%

g 50% h 225% i 0:7% j 1:02% k 6:75% l 501

4%

To express one quantity as a percentage of another, we first make sure they are written with the

same units. We then write the first quantity as a fraction of the second, and multiply by 100%.

Self Tutor

Express as a percentage: Simon drank 200 mL of a 1:25 L bottle of soft drink.

200 mL out of 1:25 L = 200 mL out of 1250 mL fwrite with the same unitsg

=200

1250£ 100%

= 16%

So, Simon drank 16% of the soft drink.

EXERCISE 3B

1 Express as a percentage:

a 15 cm out of 20 cm b 11 minutes out of 25 minutes

c 38 marks out of 40 marks d 18 kg out of 30 kg.

2 Write as a percentage:

a 220 mL out of 2 L b 24 minutes out of 1 hour

c 35 cm out of 1:4 m d 18 months out of 21

2years.

3 On the tennis court, Maria hit 180 of 300 serves ‘in’. Write this as a percentage.

4 Lily scored 76 marks out of 80 for her Physics test. What was her percentage?

5 Declan was late for home group 18 days out of 50 last term. What percentage of the days

was he on time?

6 During the soccer season, Mark conceded 25 goals out of 150.

What percentage of goals did he: a concede b save?

EXPRESSING ONE QUANTITY AS APERCENTAGE OF ANOTHER

B

Example 4


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7 In a chess tournament, Len won 14 matches and lost

2. What percentage of matches did he:

a win b lose?

8 Pete had $5 in his pocket and spent 65 cents on a

postage stamp. What percentage of his money did he

spend?

9 Leila drank 450 mL of a 1:5 L bottle of water. What percentage of the water did she drink?

10 On his way to work, Ben travels 12:5 km by train, and the remaining 750 m on foot. What

percentage of the total distance is travelled on foot?

To find a percentage of a quantity, we convert the percentage to a decimal, and then multiply.

FINDING A PERCENTAGE OF A QUANTITY

Self Tutor

Find:

a 1:5% of $2200 b 108% of 5000 kg (in tonnes)

a 1:5% of $2200

= 0:015 £ $2200

= $33

b 108% of 5000 kg

= 1:08 £ 5 tonnes

= 5:4 tonnes

EXERCISE 3C

2 Jillian scored 70% in her test out of 60 marks. How many marks did she score?

3 In his last snooker match, Neil successfully potted 96% of his long shots. If he attempted

75 long shots, how many did he get in?

4 In a baseball season, Derek hit the ball 25:5% of the

time. If he came up to bat 196 times, how many hits

did he have?

5 A petroleum company claims that cars using their

new premium unleaded fuel will travel 112% of the

distance travelled on regular unleaded. If Geoff can

travel 584 km on a tank of regular fuel, how far should

he be able to travel using the premium unleaded?

FINDING A PERCENTAGE OF A QUANTITYC

Example 5

The word ‘of’

indicates that we

should multiply

the numbers.

1 Find:

a 55% of $8 b 20% of $70 c 15% of 12 L d 250% of 40 kg

e 3% of 3 tonnes f 371

2% of 900 m g 8:5% of $3500 h 2:4% of 65 g

i 31

2% of 375 mL j 71:5% of 18 m/s k 36% of 2

1

2hours (in minutes)


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6 Nicki is part of a syndicate that owns racehorses. He owns 28% of Peter Pan, who last week

won $18 600 in the derby. How much should Nicki receive from this win?

7 A restaurant has a 10% surcharge on the bill on public holidays. If Jack and Jill buy food and

wine to the value of $54 on a public holiday, how much will they have to pay:

a for the surcharge b in total?

Michelle and Brigette own a business. Brigette receives 25% of the profits each month. Last

month, Brigette received $2080. How can she work out the total profit made by the business last

month?

The unitary method can be used to solve this problem. We first find 1%, then multiply by 100to find the whole amount.

Self Tutor

Find 100% of a sum of money if 25% is $2080.

25% of the amount is $2080

) 1% of the amount is $2080 ¥ 25 = $83:20

) 100% of the amount is $83:20 £ 100

) the whole amount is $8320

The unitary method can also be used to find other percentages of a quantity.

Self Tutor

18% of the students at a school are in Year 8, and 16% of the students are

in Year 9. If the school has 126 Year 8 students, determine the number of

Year 9 students.

18% is 126 students

) 1% is126

18= 7 students

) 16% is 7 £ 16 = 112 students

So, the school has 112 Year 9 students.

EXERCISE 3D

1 Find 100% if:

a 10% is 40 mL b 45% is 225 g c 6% is $72

d 70% is 49 kg e 53% is $159 f 95% is 38 minutes

2 22% of students at a school ride their bikes to school. If 132 students ride their bikes to

school, how many students attend the school?

THE UNITARY METHOD IN PERCENTAGED

Example 7

Example 6


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3 a Find 40% of an injection if 5% of the injection is 7 mL.

b Find 84% of a packet of nuts if 14% of the packet is 21 g.

c Find 72% of the contents of a bottle if 9% of the contents is 80 mL.

d Find 18% of a wage if 54% of the wage is $630.

4 An alloy contains 15% manganese and 85% iron. If 37:5 kg of manganese is used to make

the alloy, how much:

a alloy is produced b iron is used?

5 Approximately 75 000 people in the Northern Territory

are of indigenous origin, making up 32:5% of the

Northern Territory population. Estimate, to the nearest

thousand, the population of the Northern Territory.

6 When Vivian bakes cookies, she always burns 20% of

them. How many cookies does Vivian need to bake so

that she finishes with 28 unburnt cookies?

7 When churning cream to make butter, 48% of the cream comes out as butter, and the remainder

as buttermilk.

a How much butter will you get from 3:5 kg of cream?

b Amy needs 120 g of butter for a recipe. How much cream would she need to churn?

Flavour % of sales

Chocolate 36%

Vanilla 31%

Strawberry 22%

Mint 7:5%

Banana 3:5%

Total 100%

8 Maddie operates an ice cream van. The table alongside shows

the sales for each flavour in the last week, expressed as a

percentage of the total number sold.

a What percentage of ice creams sold were strawberry

flavoured?

b Given that Maddie sold 124 vanilla ice creams, determine

how many:

i chocolate ice creams were sold

ii mint ice creams were sold.

c Determine the total number of ice creams sold.

9 The children at a camp each had to choose one of canoeing, hiking, or swimming for

their afternoon activity. 45% of the children chose canoeing, 18 children chose hiking, and

15 children chose swimming. How many children were at the camp?

10 Many food and drink packages come with labels like the one below.

For example, this food contains 4:5 g of protein, which is 9% of the recommended Daily

Intake (DI) of protein.

Use this information to find the recommended Daily Intake of:

a energy b protein c saturated fat d sodium.

ENERGY870 kJ

DI*10%

PROTEIN4.5 g

DI*9%

FAT0.7 g

DI*1%

SAT FAT0.25 g

DI*1%

CARBS27.9 g

DI*9%

SUGARS9.5 g

DI*11%

SODIUM115 mg

DI*5%


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Changes in quantities are often expressed as a percentage

of the original quantity.

For example:

² Class sizes have increased by 10%.

² House prices have dropped by 12%.

² Carbon emissions need to fall by 25% by the year

2020.

There are two different methods which can be used to solve percentage increase or decrease

problems.

METHOD 1: WITH TWO STEPS

Step 1: Find the size of the increase or decrease.

Step 2: Apply the change to the original quantity by addition or subtraction.

Self Tutor

A fruit grower picked 1720 kg of apples last year. This year she expects her crop to be 20%bigger. How many kilograms of apples does she expect to pick this year?

Step 1: size of increase

= 20% of 1720 kg

= 0:2 £ 1720 kg

= 344 kg

Step 2: new amount

= 1720 + 344 kg

= 2064 kg

The fruit grower expects to pick 2064 kg.

EXERCISE 3E.1

1 Perform these operations using two steps:

a increase $30 by 10% b decrease 50 kg by 20%

c increase 60 m by 25% d decrease 400 L by 1%

e increase 50 000 people by 2:3% f decrease 45 minutes by 4%.

2 Katie grows spinach in her garden. Last year she harvested 6 kg. This year her harvest

increased by 30%.

a How much extra spinach did she harvest this year?

b Find the total weight of Katie’s harvest this year.

3 Last year there were 2075 South Australian students enrolled in Year 12 Physics. Enrolments

decreased by 4% this year.

a Find the decrease in enrolments. b How many students are enrolled this year?

PERCENTAGE INCREASE AND DECREASEE

Example 8


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METHOD 2: WITH ONE STEP USING A MULTIPLIER

To increase an amount by 15%, we start with 100% of the amount, then add 15% of the amount

to it. We now have 100% + 15% = 115% of the amount.

So, to increase an amount by 15% in one step, we multiply the amount by 115% or 1:15 . The

value 1:15 is called the multiplier.

To decrease an amount by 15%, we start with 100% of the amount, then subtract 15% of the

amount from it. We now have 100% ¡ 15% = 85% of the amount.

So, to decrease an amount by 15% in one step, we multiply the amount by 85% or 0:85 . The

value 0:85 is called the multiplier.

Self Tutor

a Increase 312 kg by 22%, giving your answer to 3 significant figures.

b Decrease $183 by 7%.

a To increase by 22%, we multiply by

100% + 22% = 122%, which is 1:22 .

) the new amount = 1:22 £ 312 kg

¼ 381 kg

b To decrease by 7%, we multiply by

100% ¡ 7% = 93%, which is 0:93 .

) the new amount = 0:93 £ $183

= $170:19

EXERCISE 3E.2

1 Find the multiplier corresponding to:

a an increase of 25% b a decrease of 10% c a decrease of 19%

d an increase of 8:2% e a decrease of 71

2% f an increase of 150%.

2 Use a multiplier to perform the following, giving answers to 3 significant figures where

appropriate:

a increase $50:40 by 20% b decrease 46 cm by 13%

c decrease 230 kg by 55% d increase 35 minutes by 7:5%

e decrease 81 L by 4:5% f increase $67 by 250%.

3 In 1996, the total population of Tasmanian devils

in the wild was estimated at 130 000. However,

the transmissible Devil Facial Tumour Disease has

caused a 70% decrease in numbers.

Estimate the population of Tasmanian devils in the

wild today.

4 Ed is 160 cm tall, and his sister Peggy is 8:5%taller than he is. How tall is Peggy?

5 In the year 2000, it was estimated that there were

361 million Internet users worldwide. In the

next ten years, that number increased by 445%.

Estimate the number of Internet users worldwide

in 2010.

Example 9


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A multiplier

greater than 1

shows an increase.

A multiplier less

than 1 shows a

decrease.


6 Jessi’s rent increases by 5% at the end of each year. If she currently pays $220 per week,

how much will Jessi be paying two years from now?

7 On Monday, the value of Rijken Industries shares were $7:50 . The price increased by 6%

on Tuesday, decreased by 1:5% on Wednesday, and decreased by 2% on Thursday. Find the

value of each share when trade opened on Friday.

FINDING A PERCENTAGE CHANGE

If we know that the amount of a quantity changes, the multiplier for the change is calculated by

multiplier =new amount

original amount.

We can then use the multiplier to determine the percentage change.

Self Tutor

Determine the percentage change when:

a 50 kg is increased to 70 kg b $160 is decreased to $120.

a multiplier =new amount

original amount

=70 kg

50 kg

= 1:4

This corresponds to a 40% increase.

b multiplier =new amount

original amount

=$120

$160

= 0:75

This corresponds to a 25% decrease.

EXERCISE 3E.3

1 Find the percentage change when:

a $20 is increased to $22

b 80 mL is decreased to 68 mL

c 45 g is decreased to 27 g

d 90 cm is increased to 1:35 m.

2 Describe the percentage change in the following situations:

a The price of a haircut last month was $30. It has since risen to $34:50.

b 1500 people attended the Christmas carols last year. It rained this year, so only 1080attended.

c Arthur bought a house two years ago for $320 000. It is now worth $360 000.

d At her school’s sports day, Casey threw the javelin 56:33 m, beating the previous school

record of 52:40 m.

e John completed a half-marathon in 1 hour and 52 minutes, improving on his previous

best time of 2 hours and 8 minutes.

Example 10


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3 Harriot really loves snakes. For her birthday in 2009,

she was given a pet carpet python by her uncle. She

measured its length on her birthday each year:

Year 2009 2010 2011 2012

Length 58 cm 79 cm 94 cm 1:07 m

a Calculate the percentage by which the snake’s

length increased from:

i 2009 to 2010 ii 2010 to 2011 iii 2011 to 2012.

b For the 3-year period, calculate the overall percentage increase.

4 The following table shows the number of road crash deaths in Australian states and territories

in May 2010 and May 2011:

NSW QLD SA TAS VIC WA ACT NT

May 2010 38 22 12 5 31 22 3 3

May 2011 26 25 11 2 30 9 1 2

a Calculate the percentage change in road deaths from May 2010 to May 2011 for each

state and territory.

b Which state or territory had the:

i largest percentage increase ii largest percentage decrease in road deaths?

c Find the percentage change in road deaths for the whole of Australia.

Given an original amount and a certain percentage change, we have seen how to obtain a new

amount.

It is often useful to be able to solve the reverse problem. If we know the percentage change and

the new amount, we want to calculate the original amount.

We multiplied the original amount by the multiplier to obtain the new amount. Reversing this

process, we divide the new amount by the multiplier to obtain the original amount.

Self Tutor

An electrical goods store buys a TV set at a wholesale price. The price of the TV is increased

by 25% for sale by the store. Its selling price is now $550.

For what price did the store buy the TV set?

cost price £ multiplier = selling price

) cost price £ 1:25 = $550 f100% + 25% = 125% = 1:25g

) cost price =$550

1:25= $440

So, the television set cost the store $440.

FINDING THE ORIGINAL AMOUNTF

Example 11

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54

Discussion


EXERCISE 3F

1 Find the original amount given that:

a after an increase of 20%, the length was 24 cm

b after a decrease of 15%, the mass was 51 kg

c after an increase of 3:6%, the amount was $129:50

d after an increase of 130%, the capacity was 9200 L

e after a decrease of 0:8%, the attendance was 49 600 people.

2 In 2011, the Adelaide Fringe Festival had 747 events. The festival director said this was an

increase of 66% since 2006. How many events were at the 2006 Adelaide Fringe Festival?

3 A clothing store is having a 15% off sale. The price

of a coat has been reduced to $119. How much does

the coat usually cost?

4 Answer the Opening Problem on page 44.

5 From July to September, the number of people reading

the Melbourne newspaper ‘The Age’ dropped by 5:3%to 650 000. Estimate, to the nearest thousand, the

number reading ‘The Age’ in July.

6 Joan has just received an electricity bill of $569:75 . This is 32:5% more than her previous

bill. How much was Joan’s previous bill?

² If an amount is increased by 18%, and the resulting amount is then decreased by 18%,

do we return to the original amount?

² A new model car is 18% more expensive than the model it replaces. What percentage

discount must a salesman give in order to sell the new model car at the old model price?

When a person borrows money from a lending institution such as a bank or a finance company,

the borrower must repay the loan in full, and also pay an additional interest payment. This is a

charge for using the institution’s money.

Similarly, when money is invested in a bank, the bank pays interest on any deposits.

The amount of interest paid is usually given as a percentage of the amount which is borrowed or

invested.

SIMPLE INTERESTG

We divide the new

amount by the

multiplier to find

the original

amount.


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SIMPLE INTEREST

Simple interest is interest that is calculated each year as a fixed percentage of the original

amount of money borrowed or invested.

The fixed percentage is called the interest rate, and is usually given per annum which means

“per year”.

Suppose $8000 is invested for 5 years at 10% per annum simple interest.

The simple interest paid for 1 year = 10% of $8000

= 0:1 £ $8000

= $800

The simple interest paid for 2 years = 10% of $8000 £ 2

= 0:1 £ $8000 £ 2

= $1600...

The simple interest paid for 5 years = 10% of $8000 £ 5

= 0:1 £ $8000 £ 5

= $4000

These observations lead to the simple interest formula.

SIMPLE INTEREST FORMULA

The simple interest I can be calculated using the formula:

I = Pin where P is the principal, or initial amount borrowed or invested,

i is the flat rate of interest per annum (p.a.),

n is the time or duration of the loan in years.

Self Tutor

Find the simple interest payable on an investment of $20 000 at 12% p.a.

over a period of 4 years.

P = 20 000

i = 12% = 12 ¥ 100 = 0:12

n = 4

Now I = Pin

) I = 20 000 £ 0:12 £ 4

) I = 9600

) the simple interest is $9600.

In some areas of finance, sums of money may be invested over a period of months or days.

However, the interest rate is still normally quoted per annum, so the time period n in the formula

must be converted to years.

Example 12 Write the interest

rate as a decimal.


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Self Tutor

Calculate the simple interest payable on an investment of $15 000 at

8% p.a. over 9 months.

P = 15000

i = 8% = 8 ¥ 100 = 0:08

n =9

12= 0:75

Now I = Pin

) I = 15 000 £ 0:08 £ 0:75

) I = 900

) the simple interest is $900.

EXERCISE 3G

1 Find the simple interest payable on an investment of:

a $3000 at 11% p.a. for 2 years b $5500 at 71

2% p.a. for 5 years

c $15 000 at 5:75% p.a. for 41

2years d $25 000 at 6:2% p.a. for 10 years.

2 Find the simple interest payable on a loan of:

a $2000 at 1:4% p.a. over 9 months

b $8500 at 8% p.a. over 3 months

c $2 750 000 at 7% p.a. over 18 months

d $60 000 at 3:25% p.a. over 10 months.

Self Tutor

Determine the simple interest payable on an investment of $100 000 at 15% p.a. from

April 28th to July 4th.

From April 28th there are 2 days left in April.

) 2 days left in April we exclude the first day (April 28)

31 days in May

30 days in June

4 days in July we include the last day (July 4)

67 days

P = 100 000

i = 15% = 15 ¥ 100 = 0:15

n =67

365

Now I = Pin

) I = 100 000 £ 0:15 £67

365

) I = 2753:42 fto the nearest centg

So, the interest payable is $2753:42 .

3 Gemma deposits $7800 in a special investment account on May 6th. The account pays

8:75% p.a. simple interest. If Gemma withdraws the money on September 9th, how much

will her investment have earned over this time?

Example 14

Example 13

Remember to

convert the time

period to years.

We divide the

number of days by

365 to give the

period in years.


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4 Warren Barr deposited $7500 on July 1st in a special investment account which earns

11% p.a. simple interest. On September 19th he deposited another $4900 in the account.

He closed the account on December 20th by withdrawing the total balance.

a In total, how much interest has his investment earned?

b Find the total amount Warren has withdrawn.

Self Tutor

Todd borrows $6000 for 5 years at 11% p.a. simple interest.

a Calculate the total amount he must repay.

b If Todd repays the loan in equal monthly repayments over the 5 years, find the amount

he must repay each month.

a P = 6000

i = 11% = 11 ¥ 100 = 0:11

n = 5

Now I = Pin

= 6000 £ 0:11 £ 5

= 3300

The total amount to be repaid = initial amount + interest

= $6000 + $3300

= $9300

b Todd repays the loan over 5 years = 60 months.

) each monthly repayment =$9300

60

= $155

5 $4000 is borrowed under simple interest terms. Find the total amount to be repaid after:

a 10 years at 7% p.a. b 6 years at 91

2% p.a. c 8 months at 13% p.a.

6 Alice borrows $4700 from a finance company to buy her first car. The rate of simple interest

is 17% and she borrows the money over a 5 year period. Find:

a the total amount Alice must repay the finance company

b her equal monthly repayments.

7 Cameron borrows $15 000 from a bank to renovate

his house. He borrows the money at 9% p.a. simple

interest over 8 years. What are his monthly

repayments?

8 An electric guitar with all attachments is advertised

at $2400. Ben can afford to pay a deposit of $600,

but he then has to borrow the remainder at 12% p.a.

simple interest over 3 years. What are his monthly

repayments?

Example 15


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A more common method for calculating interest is

compound interest.

If you leave your money in the bank for a period

of time, the interest is automatically added to your

account.

With compound interest, any interest that is added to

your account will also earn interest in the next time

period.

Compound interest is calculated as a percentage of the total amount at the end of the previous

compounding period.

In this course we will only consider interest which compounds each year.

Suppose $1000 is placed in an account earning interest at a rate of 10% p.a. The interest is allowed

to compound itself for three years. We say it is earning 10% p.a. compound interest.

We can show this in a table:

Year Amount at beginning of year Compound interest Amount at end of year

1 $1000 10% of $1000 = $100 $1000 + $100 = $1100

2 $1100 10% of $1100 = $110 $1100 + $110 = $1210

3 $1210 10% of $1210 = $121 $1210 + $121 = $1331

After 3 years there is a total of $1331 in the account. We have earned $331 in compound interest.

If we construct a similar table for $1000 in an account earning 10% p.a. simple interest for 3years, we can compare the two different types of interest.

Year Amount at beginning of year Simple interest Amount at end of year

1 $1000 10% of $1000 = $100 $1000 + $100 = $1100

2 $1100 10% of $1000 = $100 $1100 + $100 = $1200

3 $1200 10% of $1000 = $100 $1200 + $100 = $1300

After 3 years there is a total of $1300 in the account, so we have earned $300 in simple interest.

Comparing the two:

Year Compound interest Simple interest

1 $100 $100

2 $110 $100

3 $121 $100

COMPOUND INTEREST (EXTENSION)H

Compound interest

allows you to earn

interest on interest!

Simple interest

remains constant.

Compound interest

increases over

time.


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Self Tutor

How much interest will I earn if I invest $10 000 for 3 years at:

a 15% p.a. simple interest b 15% p.a. compound interest?

a We use the simple interest formula with P = 10 000, i = 0:15, n = 3.

Now I = Pin

) I = 10 000 £ 0:15 £ 3 = 4500

Thus, the simple interest is $4500.

b Year Initial amount Interest Final amount

1 $10 000 15% of $10 000 = $1500:00 $11 500:00

2 $11 500 15% of $11 500 = $1725:00 $13 225:00

3 $13 225 15% of $13 225 = $1983:75 $15 208:75

The compound interest = final amount ¡ initial amount

= $15 208:75 ¡ $10 000

= $5208:75

EXERCISE 3H.1

1 $25 000 is invested at 6% p.a. compound interest. Use a table to find:

a the final amount after 3 years

b how much interest was earned in the 3 year period.

2 Calculate the interest earned on an investment of $2000 for 3 years at:

a 5% p.a. simple interest b 5% p.a. compound interest.

3 Use a table to determine the interest earned for the following investments:

a $4000 at 7% p.a. compound interest for 2 years

b $500 at 2:5% p.a. compound interest for 3 years

c $11 000 at 5% p.a. compound interest for 4 years.

A spreadsheet allows us to construct a compound interest table very rapidly.

What to do:

1 Suppose $5000 is invested at 4% p.a. compound interest for 10 years.

Open a new spreadsheet and enter the following:

Investigation Compound interest spreadsheet

Example 16

SPREADSHEET


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Highlight the formulae in row 6. Fill down to row 14 for the 10th year of the investment.

a How much interest is paid in: i year 1 ii year 10?

b How much is in the account after: i 5 years ii 10 years?

2 Suppose $15 000 is invested at 6% p.a. compound interest for 10 years. Enter 15 000 in

B1 and 0:06 in B2.

a How much interest is paid in: i year 1 ii year 10?

b How much is in the account after: i 5 years ii 10 years?

3 How long would it take for $8000 invested at 5% p.a. compound interest to double in

value?

Hint: Enter 8000 in B1, 0:05 in B2, and fill down further.

4 What compound interest rate is needed for $12 000 to double in value after 6 years?

Hint: Enter 12 000 in B1 and repeatedly change the interest rate in B2.

THE COMPOUND INTEREST FORMULA

Suppose you invest $1000 in the bank for 3 years, earning 10% p.a. compound interest.

Your investment increases in value by 10% each year, so its value at the end of the year is

100% + 10% = 110% of the value at the start of the year. This corresponds to a multiplier of

1:1 .

After one year your investment is worth $1000 £ 1:1 = $1100

After two years your investment is worth $1000 £ 1:1 £ 1:1

= $1000 £ (1:1)2 = $1210

After three years your investment is worth $1000 £ (1:1)2 £ 1:1

= $1000 £ (1:1)3 = $1331

This suggests that if the money was left in your account for n years, it would amount to

$1000 £ (1:1)n.

This leads us to the compound interest formula:

Fv = Pv(1 + i)n where Fv is the future value

Pv is the present value or original amount

i is the annual interest rate as a decimal

(1 + i) is the multiplier

n is the number of years of the investment.

Notice that the formula for Fv above gives the total future value, which is the original amount

plus interest.

To find the interest only we use:

Compound interest = Fv ¡ Pv


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Self Tutor

a What will $5000 invested at 8% p.a. compound interest amount to after 2 years?

b How much interest is earned?

a An interest rate of 8% indicates that i = 0:08 .

For 2 years, n = 2 and so Fv = Pv(1 + i)n

= $5000 £ (1:08)2

= $5832

b Interest earned

= $5832 ¡ $5000

= $832

EXERCISE 3H.2

1 $6000 is invested for 4 years at 8% p.a. What will this investment amount to if the interest is

calculated as:

a simple interest b compound interest?

2 $7500 is borrowed for 10 years at 11% p.a. How much will have to be repaid if the interest

is calculated as:

a simple interest b compound interest?

3 a What will an investment of $3000 at 9% p.a. compound interest amount to after 5 years?

b What part of this is interest?

4 How much compound interest is earned by investing $20 000 for 6 years at 4:5% p.a.?

5 You have $8000 to invest for 3 years. You have been offered two investment options:

Option 1: Invest at 9% p.a. simple interest.

Option 2: Invest at 8% p.a. compound interest.

a Calculate the amount accumulated at the end of the 3 years for both options and decide

which option to take.

b Would you change your decision if you were investing for 5 years?

1 Express as a decimal: a 57% b 375%

2 Write as a percentage: a 0:002 b9

40

3 Express as a percentage:

a 21

2hours out of 10 hours b 88 cm out of 12 metres.

4 Jillian has repaid $700 of a $1500 debt. What

percentage of the debt has been repaid?

5 Terry has memorised 60% of a piece of music. If

the piece of music takes 5 minutes to play, how

long is the section that Terry has memorised?

Review set 3

Example 17


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62

Practice test 3A Multiple Choice


6 Jess needs 30% of a container of sugar for her recipe. The container is presently 75%full, and this is 60 g of sugar. What weight of sugar does Jess need for her recipe?

7 A hospital has 36 doctors, who make up 24% of the total hospital staff. How many people

work at the hospital?

8 On Tuesday, petrol was being sold at 132:5 cents per litre. On Wednesday, the price was

121:9 cents per litre. Describe the percentage change in the price.

9 As a result of a cyclone, the price of bananas rose 85% to $9:99 per kg. How much did

bananas cost before the cyclone?

10 A football stadium has a maximum capacity of 60 000 people. One stand is currently

under renovation however, so the stadium’s capacity has been reduced by 8:5%. How

many people can the stadium currently hold?

Offence Old fine New fine

Speeding $200

Drink driving $840

Not wearing seatbelt $273

Illegal parking $52:50

11 As part of a road safety campaign, fines

for all traffic offences will increase by

5%.

Copy and complete the table alongside,

showing the changes to each fine.

12 Find the simple interest payable on a loan of $4000 at 7% p.a. for 5 years.

13 Raj borrows $5000 from a bank at 81

2% p.a. simple interest. He borrows the money over

a 6 year period.

a Find the total amount that Raj must repay the bank.

b Find Raj’s equal monthly repayments.

14 a What will an investment of $8000 at 6% p.a. compound interest amount to after

4 years?

b What part of this is interest?

Click on the icon to obtain this printable test.

1 Convert 621

2% to: a a decimal b a fraction in simplest form.

2 Convert 15

16to a percentage.

3 In an ice hockey match, Roberto saved 17 of 20 shots on goal. What percentage of shots

did he save?

4 Kate had $20 in her wallet. While waiting at the airport, she spent $4:50 on a large coffee.

What percentage of her money did she spend?

Practice test 3B Short response

PRINTABLE

TEST


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5 Marco has 300 mL of red wine simmering in a pan. He reduces the volume to 60% to

make a sauce. How much sauce will he get?

6 Nikki scored 82% in her last test. Her sister Bree scored 76%. If Bree scored 38 marks:

a how many marks did Nikki score b how many marks were in the test?

7 An ant is carrying 5000% of its own body weight. If the leaf it is carrying weighs 1 g,

how much does the ant weigh?

8 The inflation rate is currently 3:3%. If a chocolate bar costs $1:60 now, how much will

it cost next year?

9 A souffle increases in height by 125% while in the oven. If the finished height is 13:5 cm,

how tall was the souffle before it was put in the oven?

10 Darren borrows $15 000 from the bank to extend his house. The bank will charge him

simple interest of 6:8% p.a. for a 5 year loan. Find the total amount Darren will need to

repay.

1 The following table shows the number of births in 2008 and in 2009 in Australia:

NSW QLD SA TAS VIC WA ACT NT

2008 93 238 62 072 19 937 6683 69 976 31 418 4718 3903

2009 89 911 64 829 19 452 6531 70 820 30 377 5605 3715

a Find, correct to one decimal place, the percentage change in births for each state and

territory.

b Which state or territory had the:

i largest percentage increase ii largest percentage decrease?

c Find the percentage change in births for the whole of Australia.

2 A manufacturer sells refrigerators to an electrical store.

a The parts and labour cost the manufacturer $500. If the manufacturer wishes to add

a 40% profit, how much does the electrical store pay for the refrigerator?

b The electrical store sells the refrigerator for $910 excluding GST.

i How much profit does the store make?

ii If 10% GST must be added to the price of the refrigerator, how much does a

consumer pay?

3 Al borrows $10 000 from the bank to buy solar panels for his house. He borrows the

money at 7% p.a. simple interest for 5 years.

a Calculate the interest payable on Al’s loan.

b In total, how much money will Al pay back?

c Al wishes to pay back the loan in equal monthly instalments.

i How many repayments will there be?

ii What is the size of each repayment?

Practice test 3C Extended response


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4 Halfway through the 2011 AFL season, the ladder looked as follows:

Position Team P W L D F A %

1 Geelong Cats 12 12 0 0 1226 858 142:89

2 Collingwood 11 10 1 0 1312 736 178:26

3 Carlton 12 9 2 1 1212 874 138:67

4 Hawthorn 12 9 3 0 1215 914 132:93

5 West Coast Eagles 12 8 4 0 1182 959 123:25

6 Sydney Swans 12 7 4 1 991 950 104:32

7 Fremantle 12 6 6 0 1070 1171 91:37

8 Essendon 12 5 6 1 1214 1055 115:07

9 Melbourne 12 5 6 1 1109 1085 102:21

10 Richmond 12 5 6 1 1192 93:49

11 North Melbourne 12 5 7 0 1141 1123 101:60

12 St Kilda 12 4 7 1 948 1035

13 Western Bulldogs 12 4 8 0 1014 1155 87:79

14 Adelaide 12 3 9 0 935 1197 78:11

15 Brisbane Lions 12 2 10 0 946 1223 77:35

16 Port Adelaide 12 2 10 0 1358 69:22

17 Gold Coast Suns 11 2 9 0 792 1471 53:84

a What percentage of matches had Fremantle won?

b What percentage of matches had Hawthorn lost?

c The % column shows the points scored for a team, expressed as a percentage of the

points scored against the team. Find:

i St Kilda’s percentage

ii the total number of points scored for Port Adelaide

iii the total number of points scored against Richmond.

Drink % of sales

Cola 35:9%

Diet Cola 15:2%

Orange 12:6%

Lemonade

Raspberry 8:5%

Total 100%

5 A fast food restaurant sells soft drinks. The percentage of

each type sold in one week is shown in the table.

a What percentage of drinks sold were lemonade?

b If the store sold 4000 drinks, how many were:

i cola ii raspberry?

c If 697 raspberry drinks were sold several weeks later,

estimate the number of diet cola drinks sold.

d What have you assumed in finding your answer to c?


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Documents

Percentage - Haese Mathematics · amount. A percentage is the comparison of a portion with the whole amount. ... f 3:027 g 25 16 h 17 i 54 5 j 0:7 ... To find a percentage of a quantity,