4 Applications of

percentages
LESSON SEQUENCE
4.1 Overview ................................................................................................................................................................. 128
4.2 Percentages ...........................................................................................................................................................130
4.3 Finding percentages of an amount ............................................................................................................... 136
4.4 Discount .................................................................................................................................................................. 142
4.5 Profit and loss ....................................................................................................................................................... 148
4.6 Goods and Services Tax (GST) and Income Tax ..................................................................................... 155
4.7 Review ..................................................................................................................................................................... 162
LESSON
4.1 Overview
Why learn this?

information and even have their own symbol: %. One per cent
Percentages are used to describe many different aspects of

means one-hundredth; therefore 1% means one per hundred,

10% means ten per hundred and 50% means 50 per hundred.
Percentages can be used as an alternative to decimals and
fractions. We can write one-half as a decimal (0.5), a fraction
1
( )
and a percentage (50%).
2
Why do we have so many ways of writing the same number?
Depending on the context, it may be easier to use a certain form.
Percentages are commonly used in finance and shopping. It is
1
easier to express an interest rate as 5% rather than 0.05 or , and
20

. When you see an interest rate of 5% (5 per hundred), you can

easier to say that items are discounted by 70% rather than by 0.7 or
7

easily calculate that for every $100 you will earn $5 in interest.
10

You will see percentages used for discounts at shops, interest rates
for bank accounts and loans, rates of property growth or loss,
statistics for sports matches, data used in the media, and company
statements about profit and loss. Understanding percentages
will help you deal with your own finances and make decisions
regarding your income once you are working.

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128 Jacaranda Maths Quest 8

Exercise 4.1 Pre-test
1. Calculate 5% of $150.

2. MC Which of the following is the correct simplified fraction of 35%?

7 35 7 35
A. B. C. D.
10 100 20 1

, 22%, , 0.31, 111%.

3 1
3. Arrange the following numbers in ascending order:
5 4

4. MC Select the correct percentage of 22g in 1 kg.

A. 222% B. 22% C. 2.2% D. 0.22%

5. Calculate 23% of $80.

original price of $90. Calculate the new sale price.

6. Tennis equipment at a sports shop is reduced by 15% for an end-of-financial-year sale. A racket has an

7. A cricket bat is reduced from $400 to $380. Calculate the percentage discount.

8. MC To calculate an 8% increase of an amount, what number do you multiply the original amount by?
A. 108 B. 8 C. 1.8 D. 1.08

9. William is 55 years old and was born in Scotland. He lived in England for 45% of his life and in
Australia for 11 years, and the rest of his life was spent in Scotland. Determine how long he lived in
Scotland for. Write the answer in years and months.

10. MC When the original price of an item is multiplied by 0.78, what percentage has the item increased or
decreased by?
A. Increased by 78% B. Decreased by 78%
C. Decreased by 0.22% D. Decreased by 22%

11. In an auction, an apartment originally priced at $2 750 000 sells for $2 820 000. Calculate the
percentage profit made on the sale. Write the answer to 2 decimal places.

The cost of a sofa, including GST, is $890. What would be the cost of the sofa before GST?
A. $801.10 B. $809.09 C. $801 D. $809.10
12. MC

13. An item is reduced by 10%, and then increased by 11.1%. This takes the item back to its original price
(to the nearest cent). True or False?

14. The price of a car is reduced by 10% three weeks in a row. Calculate the percentage drop in price by the
end of the third week. Write the answer to the nearest whole number.

can be exchanged for 0.92 Australian dollars. A Toyota Yaris (excluding GST/VAT) costs £9400,
15. The UK pound (£) can be exchanged for 1.6 Australian dollars (A$). The New Zealand dollar (NZ$)

NZ$16 000 and A$13 900. VAT (the UK equivalent of GST) is 20%. GST in New Zealand is 15%. GST
in Australia is 10%. In which of the countries is Toyota Yaris the cheapest, including GST/VAT?

TOPIC 4 Applications of percentages 129

 LESSON
4.2 Percentages
LEARNING INTENTIONS
At the end of this lesson you should be able to:
• convert percentages into fractions and decimals
• calculate percentage increases and decreases
• calculate percentage error.

4.2.1 Writing percentages in different ways

eles-3618

• The symbol for percentage is %. For example, 60% (60 per cent) means 60 parts out of 100.
• The term per cent means ‘per hundred’.

• A quantity can be expressed in different ways using percentages, fractions and decimals.
For example:

60% = = 0.60
60
100

Expressing percentages as fractions or decimals

To convert a percentage to a fraction or a decimal, divide by 100.

• There are a number of common percentages, and their fraction and decimal equivalents, with which you
should be familiar.

Percentage Fraction Decimal

1
10% 0.1
10
1
25% 0.25

0.3̇
4
33 %
1 1
3 3
1
50% 0.5
2
100% 1 1

WORKED EXAMPLE 1 Converting percentages to fractions and decimals

Convert the following percentages to fractions and then decimals.

a. 67% b. 55%

THINK WRITE

a. 67% =
67
a. 1. To convert to a fraction, write the percentage, then
change it to a fraction with a denominator of 100. 100

130 Jacaranda Maths Quest 8

 2. To convert 67% to a decimal, think of it as 67.0%, 67% = 0.67
then divide it by 100 by moving the decimal point
2 places to the left.

b. 55% =
55
b. 1. To convert 55% to a fraction, write the percentage,
then change it to a fraction by adding a denominator 100
of 100.
55% = =
✚55
✚ 11
2. The fraction is not in simplest form, so cancel by
✟
100 20
55% = 0.55
dividing the numerator and the denominator by 5. ✟
3. To convert 55% to a decimal, think of it as 55.0%, then
divide it by 100 by moving the decimal point 2 places to
the left.

• When converting a fraction or decimal to a percentage, do the inverse of dividing by 100; that is, multiply
by 100.

Converting fractions or decimals to percentages

To express a fraction or decimal as a percentage, multiply by 100.

= × 100%
1 1
For example,

= 50%
2 2

Digital technology
Scientific calculators have a % button which can be utilised to
compute calculations involving percentages.
Percentages can be converted into decimals and fractions.

Decimals and fractions can be converted into percentages.

• The easiest method of comparing percentages, fractions and decimals is to convert all of them to their
decimal form and use place values to compare them.

TOPIC 4 Applications of percentages 131

 WORKED EXAMPLE 2 Comparing fractions, decimals and percentages

Place the following quantities in ascending order, and then place them on a number line.

, 0.36, 80%, 2 , 110%, 1.54

7 1
45%,
10 2

THINK WRITE
1. Convert all of the quantities into their 0.45, 0.7, 0.36, 0.80, 2.5, 1.10, 1.54
decimal equivalents.
2. Place them in ascending order. 0.36, 0.45, 0.7, 0.80, 1.10, 1.54, 2.5

, 80%, 110%, 1.54, 2

7 1
3. Place them in ascending order in their 0.36, 45%,
original form. 10 2

4. Draw a number line from 0 to 3, with

increments of 0.25. 0 0.5 1 1.5 2 2.5 3

5. Place the numbers on the number line. 0.70

0.80
0.45 1.10 2.50
0.36 1.54

0 0.5 1 1.5 2 2.5 3

4.2.2 Percentage increases and decreases

eles-3619
• Percentage increases and decreases can be used to calculate and compare prices, markups, discounts,
population changes, company profits and many other quantities.
• To calculate a percentage increase or decrease, calculate the net increase or decrease and then express it as
a percentage of the initial value.
• Note: Percentage increases of more than 100% are possible; for example, the increase from 3 to 7.5 is an
increase of 150%.

Calculating a percentage change

percentage change = ×
increase or decrease in quantity 100
original quantity 1

WORKED EXAMPLE 3 Calculating percentage increase

Calculate the percentage increase when a shop owner marks up a $50 item to $70.

Increase = $70 − $50

THINK WRITE

difference between $50 and $70. = $20

1. The quantity has increased, so calculate the

Percentage increase = × 100

20
2. The percentage increase can be calculated by

= 40
creating the fraction 20 out of 50 and then 50
multiplying by 100.
3. Write the answer. The percentage increase is 40%.

132 Jacaranda Maths Quest 8

 WORKED EXAMPLE 4 Calculating percentage decrease

Calculate the percentage decrease, rounded to 2 decimal places, when the population of a town falls
from 62 000 people to 48 000 people.

Decrease = 62 000 − 48 000

THINK WRITE

= 14 000
1. The difference between 62 000 and 48 000 is
14 000.

Percentage decrease = × 100

14 000
2. The percentage decrease can be calculated by

= 22.58
creating the fraction 14 000 out of 62 000 and 62 000
then multiplying by 100.
3. Write the answer. The percentage decrease is 22.58%.

4.2.3 Percentage error

eles-3620
• Percentage error is used to compare the difference between an estimate and the actual value of a quantity.
• The closer the percentage error is to zero, the better the estimate.

Calculating percentage error

• If the approximate value (or estimate) is greater than the exact value, then:

approximate value − exact value

percentage error = ×
100
exact value 1
• If the approximate value (or estimate) is less than the exact value, then:

exact value − approximate value

percentage error = ×
100
exact value 1

WORKED EXAMPLE 5 Calculating percentage error

a. The estimated weight of a newborn baby was 3500 grams, but the baby’s actual weight was
4860 grams. Calculate the percentage error.
b. The estimated distance between two towns was 70 km, but the actual distance was 65.4 km.
Calculate the percentage error.

exact value − approximate value

THINK WRITE

a. 1. The estimated weight was less a. Percentage error = × 100

exact value

4860 − 3500
than the actual weight.

2. Calculate the percentage error. Percentage error = × 100

= 27.98%
4860

approximate value − exact value

3 Write the answer. The percentage error is 27.98%.

b. 1. The estimated distance was b. Percentage error = × 100

greater than the actual distance. exact value

TOPIC 4 Applications of percentages 133

 70 − 65.4
2. Calculate the percentage error. Percentage error = × 100
65.4
= 7.03%
3. Write the answer. The percentage error is 7.03%.

Resources
Resourceseses
eWorkbook Topic 4 Workbook (worksheets, code puzzle and project) (ewbk-1935)
Video eLesson Decimals, fractions and percentages (eles-1868)
Interactivities Individual pathway interactivity: Percentages, fractions and decimals (int-4419)
Percentages, fractions and decimals (int-3741)
Percentage increase and decrease (int-3742)

Exercise 4.2 Percentages

4.2 Quick quiz 4.2 Exercise

Individual pathways
PRACTISE CONSOLIDATE MASTER
1, 3, 5, 8, 10, 14, 17 2, 4, 7, 9, 15, 18 6, 11, 12, 13, 16, 19

Fluency
1. WE1 Convert the following percentages to fractions and then to decimals.
a. 24% b. 13% c. 1.5% d. 250%

2. Convert the following percentages to fractions and then to decimals.

a. 47% b. 6.6% c. 109.8% d. 10.02%

3. Express the following percentages as fractions in simplest form.

a. 20% b. 35% c. 61% d. 105%

4. Express the following percentages as fractions in simplest form.

a. 11% b. 82% c. 12.5% d. 202%

5. Express the following decimals as percentages.

a. 0.15 b. 0.85 c. 3.10 d. 0.024

6. Express the following fractions as percentages. Round your answer to 2 decimal places where appropriate.

7 3 5 1
a. b. c. d. 2
8 5 6 3

Understanding
7. WE2 For the following sets of numbers, place the numbers in ascending order and then on a number line.

, 75%, 10%, 3 , 2.4 b. 3 , 330%, 4.5%, 150%, 3, 2 , 2.8

7 1 4 1
a. 1.6, 25%,
8 2 5 3

134 Jacaranda Maths Quest 8

 8. WE3 Calculate the percentage increase when 250 increases to 325.
9. WE4 Calculate the percentage decrease, rounded to 2 decimal places, when the population of fish in a pond

decreases from 1500 to 650.

10. Express $120 as a percentage of $400.

11. In a library, there are 24 children, 36 women and 42 men. Calculate the
percentage of women visiting the library.
Give your answer rounded to 2 decimal places.
12. During a sale, a jacket originally priced at $79.99 is decreased in price
to $55.99. Calculate the percentage decrease.

a. The estimated grocery bill budgeted for the week was $250, but
13. WE5 Answer the following questions.

the actual bill was $262.20. Calculate the percentage error.

b. A long-distance runner estimated that her run took 120 minutes, but the official time recorded was
118.3 minutes. Calculate the percentage error.

Reasoning
14. A group of students was practising their basketball free throws. Each student had four shots and the results
are displayed in the table.

Free throw results Number of students Percentage of students

No shots in 3
One shot in, three misses 11
Two shots in, two misses 10
Three shots in, one miss 4
All shots in 2

a. Identify how many students participated in the game.

b. Complete the table to show the percentage of students for each result.
c. Calculate how many students made exactly 25% of their shots.
d. Calcuate what percentage of students made less than 50% of their shots.
15. WE5 In supermarkets, potatoes are frequently sold in 2 kg bags. As potatoes are discrete objects, the bags

rarely weigh exactly 2 kg. For reasons relating to both customer satisfaction and profit, the warehouse
supervisor knows that a percentage error of more than 10% is unacceptable.
Two bags of potatoes are chosen at random and weighed. Bag A weighs 2.21 kg and bag B weighs 1.88 kg.
Calculate the percentage error for each of these bags and determine if either or both will pass the inspection.
16. The price of entry into a theme park has increased by 10% every year

cost to $8.80, explain how to determine the price of a ticket 2 years

since the theme park opened. If the latest price rise increased the ticket

ago. Show your calculations in your explanation.

Problem solving
17. Survey your classmates on the brand of mobile phone that they have.
Present your results in a table showing each brand of phone as a
percentage, fraction and decimal of the total number of phones.

TOPIC 4 Applications of percentages 135

 18. The table shows the percentage of households with 0 to 5 children.
Number of Percentage
Calculate:
children (%)
a. the percentage of households that have 6 or more children
0 56
b. the percentage of households that have fewer than 2 children
c. the fraction of households that have no children 1 16
d. the fraction of households that have 1, 2 or 3 children. 2 19
3 6
19. Use the bunch of flowers shown to answer these questions.
4 2
5 1

a. Calculate the percentage of the flowers that are yellow.

b. What fraction of the flowers are pink?
c. Write two of your own questions and swap with a classmate.

LESSON
4.3 Finding percentages of an amount
LEARNING INTENTIONS
At the end of this lesson you should be able to:
• calculate percentages of an amount
• increase or decrease a value by a percentage.

4.3.1 Calculating percentages of an amount

eles-3621
• As percentages can’t be used directly in calculations, they must be converted into fractions or decimals.
• Percentages of an amount can be determined using calculations with either fractions or decimals.

Using decimals
• To calculate a percentage of an amount using decimals, follow these steps:

2. Change ‘of’ to × (multiplication).

1. Write the percentage as a decimal.

3. Multiply.

136 Jacaranda Maths Quest 8

 WORKED EXAMPLE 6 Calculating the percentage of an amount using decimals

Of the 250 students selected at random to complete a survey, 16% were in Year 11. Calculate how
many of the students were in Year 11.
THINK WRITE
1. Decide what percentage of the total is required. 16% of 250

2. Write the percentage as a decimal. Change ‘of’ to ×. = 0.16 × 250

Write an expression to find the percentage of the total.

3. Multiply. = 40
4. Answer the question by writing a sentence. 40 of the 250 students were in Year 11.

Using fractions
• To calculate a percentage of an amount, follow these steps:

Change ‘of’ to ×.
1. Write the percentage as a fraction with a denominator of 100.
2.
3. Write the amount as a fraction over 1 if it is not already a fraction.
4. Cancel.
5. Perform the multiplication.
6. Simplify.

WORKED EXAMPLE 7 Calculating the percentage of an amount using fractions

Calculate 20% of 35.

THINK WRITE
1. Write the question. 20% of 35

= ×
7
20 35
✚
✚
2. Write the percentage as a fraction with a denominator
✟20
100
✟ 1
of 100, change ‘of’ to ‘×’, write the amount as a
fraction over 1 and cancel.

= ×
1
20
✚
✚ 7
3. Cancel again. 1
20
✚
✚ 1

=
7
4. Multiply numerators and multiply denominators.

=7
1
5. Simplify by dividing the numerator by the denominator.
6. Answer the question. 20% of 35 is 7.

Digital technology
The percentage button and the multiplication symbol can be used to
help determine percentages of an amount.

TOPIC 4 Applications of percentages 137

 COLLABORATIVE TASK: Nutritional information
Look at the nutritional information panels on a variety of different foods.
Find the recommended percentage daily values for fat, carbohydrates and
energy, and prepare a one-day diet that ensures that you do not exceed the
recommended daily values for any of these.

4.3.2 Increasing or decreasing a quantity by x%

• To increase a quantity by x%, multiply it by (100 + x)%.
eles-3622

• To decrease a quantity by x%, multiply it by (100 − x)%.

Note: Convert the percentage to a decimal or fraction before multiplying.

WORKED EXAMPLE 8 Increasing and decreasing a quantity by a percentage

a. A newborn baby weighed 3.5 kg. After 1 month the baby’s weight had increased by 20%. Calculate
the weight of the baby after 1 month.
b. Carlos went for a run on Tuesday evening and ran for 10.2 km. When he next went for a run on
Thursday evening, he ran 15% less than he did on Tuesday. Calculate how far he ran on Thursday.

a. 100% + 20% = 120%

THINK WRITE
a. 1. Add the percentage increase to 100%.

× 3.5 = × 1
✟ 12 7
120 120
✟ 35
✚
✚
2. Express the percentage as a fraction and
100 ✟ 20
multiply by the amount to be increased. 100
✟ 10
✚
✚

= ×
3
12
✚
✚ 7
5 1
20
✚

=
✚
21

= 4.2 kg
5

3. Write the answer. The weight of the baby after 1 month is 4.2 kg.

b. 1. Subtract the percentage decrease from 100%. b. 100% − 15% = 85%

138 Jacaranda Maths Quest 8

 × 10.2 = ×
17✟ 51
85 85
✚
✚ 102
✟
2. Express the percentage as a fraction and 2
100 ✟50
100 10
✚

=
multiply by the amount to be increased. ✟ ✚
867

= 8.67 km
100

3. Write the answer. Carlos ran 8.67 km on Tuesday.

Resources
Resourceseses
eWorkbook Topic 4 Workbook (worksheets, code puzzle and project) (ewbk-1935)
Video eLesson Percentages of an amount (eles-1882)
Interactivities Individual pathway interactivity: Finding percentages of an amount (int-4420)
Percentage of an amount (int-3743)

Exercise 4.3 Finding percentages of an amount

4.3 Quick quiz 4.3 Exercise

Individual pathways
PRACTISE CONSOLIDATE MASTER
1, 3, 5, 8, 9, 12, 14, 18, 20, 23, 2, 4, 6, 10, 15, 16, 19, 21, 26, 29 7, 11, 13, 17, 22, 24, 27, 30
25, 28

Fluency
1. WE6&7 Calculate the following.
a. 50% of 20 b. 20% of 80 c. 5% of 60 d. 10% of 30

2. Calculate the following.

a. 31% of 300 b. 40% of 15 c. 12% of 50 d. 35% of 80

3. Calculate the following.

a. 70% of 110 b. 52% of 75 c. 90% of 70 d. 80% of 5000

4. Calculate the following.

a. 44% of 150 b. 68% of 25 c. 24% of 175 d. 38% of 250

5. Calculate the following.

a. 95% of 200 b. 110% of 50 c. 150% of 8 d. 125% of 20

6. Calculate the following.

a. 66% of 20 b. 2% of 95 c. 55% of 45 d. 15% of 74

TOPIC 4 Applications of percentages 139

 7. Calculate the following.
a. 95% of 62 b. 32% of 65 c. 18% of 80 d. 82% of 120

8. MC 60% of 30 is:
4 31
A. 19 B. C. 186 D. 18
5 5

a. 1% of $268 b. 1% of $713 c. 1% of $573 d. 1% of $604

9. Calculate the following, rounding answers to the nearest 5 cents.

a. 1% of $19.89 b. 1% of $429.50 c. 1% of $4.25 d. 1% of $6.49

10. Calculate the following, rounding answers to the nearest 5 cents.

a. 1% of $9.99 b. 1% of $0.24 c. 1% of $0.77 d. 1% of $1264.37

11. Calculate the following, rounding answers to the nearest 5 cents.

a. 22% of $10 b. 13% of $14 c. 35% of $210 d. 12% of $150

12. Calculate the following, rounding answers to the nearest 5 cents.

a. 2% of $53 b. 7% of $29 c. 45% of $71.50 d. 33% of $14.50

13. Calculate the following, rounding answers to the nearest 5 cents.

Understanding
14. Thirty per cent of residents in the suburb Hunters Hill are over the age of 65. If there are 180 000 residents,
calculate how many are over the age of 65.
15. In a survey, 40 people were asked if they liked or disliked
Vegemite. Of the people surveyed, 5% said they disliked
Vegemite. Calculate how many people:
a. disliked Vegemite
b. liked Vegemite.

16. WE8 The grocery bill for Mika’s shopping was $250. The

following week, Mika spent 7% more on his groceries. How

much did he spend in the following week?

17. A long-distance runner completed a 15-kilometre run in

120 minutes. The next time she ran 15 kilometres, she reduced
her time by 5%. How fast did she complete the 15 kilometres
on the second occasion?

at $950, but if Maria pays cash, the shop will take 10% off the
18. Maria is buying a new set of golf clubs. The clubs are marked

marked price. How much will the clubs cost if Maria

pays cash?
19. When you multiply a quantity by 0.77, determine by what
percentage you are decreasing the quantity.
20. Increase the following quantities by the given percentages.
a. 33 kg by 10%
b. 50 lb by 20%
c. 83 cm by 100%

140 Jacaranda Maths Quest 8

21. Decrease the following quantities by the given percentages.
a. 25 kg by 10%

c. $96 by 90%
b. 40 km by 20%

22. Ninety per cent of students at a school were present for school photographs. If the school has 1100 students,
calculate how many were absent on the day the photographs were taken.
23. Jim can swim 50 m in 31 seconds. If he improves his time by 10%, calculate Jim’s new time.

24. Thirty-two thousand four hundred people went to the SCG to watch a Sydney versus Collingwood football
match. Of the crowd, 42% went to the game by car and 55% caught public transport.
Calculate how many people:
a. arrived by car
b. caught public transport.

Reasoning
25. When I am 5% older than I am now, I will be 21 years old.
Calculate how old I am now.

ago. If a loaf of bread costs $2.00 now, determine how much it

26. The price of bread has increased to 250% of its price 20 years

would have cost 20 years ago.

27. My mother is four times older than I am. My sister is 75% of my
age, and 10% of my grandfather’s age. My father is 50, 2 years
older than my mother. Determine the ages of my sister and my
grandfather.

Problem solving
28. In a Maths competition, the top 8% of students across the state
achieve a score of 40 or more out of a possible 50.
a. In a school where 175 students have entered the Maths
competition, calculate how many scores higher than 40 you
would expect.
b. In one school, there were 17 scores of 40 or more, and
204 scores that were less than 40. Compare the results to
determine whether the students performed better than the
state average.
29. Broadcasting regulations specify that 55% of television programs shown between 6 pm and midnight must
be Australian content and that, between 6 pm and midnight, there should be no more than 13 minutes per
hour of advertising.
Calculate:
a. how many minutes of advertising are allowed between 6 pm and midnight
b. for how many minutes programs are screened between 6 pm and midnight
c. the maximum percentage of time spent screening advertising
d. how many minutes of Australian content must be screened between 6 pm and midnight.
30. I am 27 years old and have lived in Australia for 12 years. If I continue to live in Australia, calculate how old
I will be when the number of years I have lived here is 75% of my age.

TOPIC 4 Applications of percentages 141

 LESSON
4.4 Discount
LEARNING INTENTIONS
At the end of this lesson you should be able to:
• understand the concept of discount
• calculate the cost of a discounted product
• calculate the discount from an initial price and sale price.

4.4.1 Applying discount

eles-3623
• A discount is a reduction in price, commonly
used by businesses aiming to clear out old stock
or attract new customers.
• There are two types of discounts:
• A fixed price discount is a set amount (in
dollars) that a product is discounted by.
• A percentage discount is a discount that is a
set percentage of the product’s price.

Calculating discount
In general, if an r% discount is applied:

discount = × original price

r
100

Calculating selling price of a discounted item

• Method 1
Use the percentage remaining after the percentage discounted has been subtracted from 100%; that is, if an
item for sale has a 10% discount, then the price must be 90% of the marked price.

WORKED EXAMPLE 9 Calculating the price of a discounted item

Calculate the sale price on a pair of shoes marked $95 if a

10% discount is given.

100% − 10% = 90%

THINK WRITE
1. Determine the percentage of the marked price
that is paid, by subtracting the percentage

90% of $95 = 0.9 × $95

discount from 100%.

= $85.50
2. Calculate the sale price of the shoes.

3. Write the answer in a sentence. The sale price of the shoes is $85.50.

142 Jacaranda Maths Quest 8

 • Method 2
The new sale price of the item can be solved by calculating the amount of the discount, then subtracting the
discount from the marked price.
Alternative solution to Worked example 1:

Discount = 10% of $95.00

= $9.50
Sale price = marked price − discount
= $95.00 − $9.50
= $85.50

WORKED EXAMPLE 10 Calculating discount and sale price

Peddles is a bicycle store that has offered a discount of 15%

on all goods.
Determine:
a. the cash discount allowed on a bicycle costing $260
b. the sale price of the bicycle.

a. Discount = 15% of $260

THINK WRITE

= 0.15 × $260
a. Calculate the discount, which is 15% of the marked price.

= $39
The cash discount allowed is $39.
b. Sale price = marked price − discount
= $260 − $39
b. 1. To calculate the sale price, subtract the

= $221
discount from the marked price.

2. Write the answer in a sentence. The sale price of the bicycle is $221.

Calculating the percentage discount

• When given the original and the discounted prices, the percentage discount can be determined.

Calculating percentage discount

To calculate the percentage discount, write the discounted
amount as a percentage of the original price.

percentage discount = × 100%

discounted amount
original price

TOPIC 4 Applications of percentages 143

 WORKED EXAMPLE 11 Calculating percentage discount

At Peddles, the price of a bicycle is reduced from $260 to $200. Calculate the percentage discount.

Discount = $260 − $200

THINK WRITE

= $60
1. Calculate the amount of the discount.

Percentage discount = × 100%

60
2. Write the discount as a percentage of the

= 23.0769...%
original price. 260

≈ 23%
3. Write the answer in a sentence. The percentage discount is about 23%.

COLLABORATIVE TASK: Let’s go shopping!

Equipment: sales catalogues from nearby shops, paper, pen, calculator

Part A
1. As a class, brainstorm percentage discounts that you see advertised
in sales. Pick three common ones.
2. Each person should think of an item they want to buy and its
current price. A volunteer might like to draw a table on the board
with the column headings ‘Item’ and then the three common
percentage discounts.
3. Each person should then calculate the new prices of the selected
item, assuming the discount shown on the board. Repeat this
process for the item listed underneath yours.
4. As a class, fill in the table and discuss the results.

Part B
1. Work in groups of three or four. Select a page from one of the sales
catalogues and calculate the percentage discount on five items.
2. Discuss the results as a class. How would you calculate the
average percentage discount shown on the items in the catalogue?

Resources
Resourceseses
eWorkbook Topic 4 Workbook (worksheets, code puzzle and project) (ewbk-1935)
Interactivities Individual pathway interactivity: Discount (int-4421)
Selling price (int-3745)
Discount (int-3744)

144 Jacaranda Maths Quest 8

Exercise 4.4 Discount
4.4 Quick quiz 4.4 Exercise

Individual pathways

PRACTISE CONSOLIDATE MASTER

1, 3, 5, 8, 10, 13, 17, 19, 23, 27 2, 4, 6, 11, 12, 14, 18, 21, 24, 7, 9, 15, 16, 20, 22, 26, 29
25, 28

Fluency
1. Calculate the discount on each of the items in the table, using the percentage
shown.

$210
Item Marked price Discount

$185
a. Smart watch 20%
b. Skateboard 25%

2. Calculate the discount on each of the items in the table, using the percentage
shown.

$330
Item Marked price Discount

$190
a. Mobile phone 15%
b. Tennis racquet 40%

3. Without using a calculator, calculate the percentage discount of the following.

$100 $10
Marked price Discount

$250 $125
a.
b.

4. Without using a calculator, calculate the percentage discount of the following.

$90 $30
Marked price Discount

$80 $20
a.
b.

5. WE9 Calculate the sale price of each item with the following marked prices and percentage discounts.

$1000
Marked price Discount

$250
a. 15%
b. 20%

6. Calculate the sale price of each item with the following marked prices and percentage discounts.

$95
Marked price Discount
a. 12%
$156 33 %
1
b.
3

TOPIC 4 Applications of percentages 145

 7. Calculate the sale price of each item with the following marked prices and percentage discounts.

Marked price Discount

$69.95 7 %
1
a.

$345
2
b. 30%

8. Determine the percentage discount given on the items shown in the table. Round to the nearest per cent.

$25 $15
Original price Selling price

$100 $72
a.
b.

9. Determine the percentage discount given on the items shown in the table. Round to the nearest per cent.

$69 $50
Original price Selling price

$89.95 $70
a.
b.

Understanding

a. $50 by 10% b. $90 by 50% c. $45 by 20%

10. Decrease the following amounts by the percentages given.

11. A tablet computer that usually sells for $599 was advertised with a saving of $148. Calculate the percentage
discount being offered. Round to the nearest per cent.
12. The following items are all discounted.

$380 $450 $260 $600

33 % discount
1
25% discount 20% discount 15% discount
3
a. Compare the values of the discounts to decide which item had the largest dollar discount.
b. Identify which items have the same dollar discount.

d. If the surfboard had a discount of 20%, would $470 be enough to buy it?
c. Calculate the difference between the largest and the smallest dollar discounts.

a. the cash discount allowed on a $350 sound system

13. WE10 A sale discount of 20% was offered by the music store Solid Sound. Calculate:

b. the sale price of the system.

14. Fitness trackers are advertised at $69.95, less 10% discount. Calculate the sale price.

146 Jacaranda Maths Quest 8

15. A store-wide clearance sale advertised 15% off everything.

at $49.
a. Determine the selling price of a pair of jeans marked

b. If a camera marked at $189 was sold for $160.65,

determine whether the correct percentage was deducted.
16. T-shirts are advertised at $15.95 less 5% discount. Calculate
the cost of five T-shirts.
Calculators were advertised at $20, discounted from
$25. What percentage discount was given?
17. WE11

18. CDs normally selling for $28.95 were cleared for $23.95. Calculate the percentage discount given (correct to
1 decimal place).
19. At a sale, Ann bought a $120 jacket for $48. What percentage of the original price did she save?

20. Kevin bought a mobile phone priced $199.95 and signed up for a 1-year plan. He received a 10% discount on
the telephone and a 15% discount on the $75 connection fee.
How much did Kevin pay altogether (correct to the nearest 5 cents)?
21. Alannah bought two hairdryers for $128 each. She sold
one at a loss of 5% and the other for a profit of 10%.
a. Determine the selling price of each.
b. Will she have made a profit or a loss?

22. MC Kristen’s car insurance was $670, but she had a ‘no claim

bonus’ discount of 12%. Which of the following will not give

A. First calculate 12% of $670 and add your answer to $670.

the amount she must pay?

B. Calculate (88 ÷ 100) × 670.

C. Find 88% of $670.
D. First calculate 12% of $670, and subtract your answer from $670.

Reasoning
23. Is there a difference between 75% off $200 and 75% of $200? Explain.

24. Concession movie tickets sell for $12.00 each, but if you buy 4 or more you get $1.00 off each ticket. What
percentage discount is this (correct to 2 decimal places)? Show your working.
25. Henry buys a computer priced at $1060, but with a 10% discount. Sancha finds the same computer selling at
$840 plus a tax of 18%.
Who has the better price? Explain.
26. You are in a surf shop and you hear ‘For today only: take fifty percent off the original price and then a
further forty percent off that.’ You hear a customer say ‘This is fantastic! You get ninety percent off the
original price!’
Is this statement correct? Explain why.

Problem solving
27. What would you multiply the original prices of items by to get their new prices with:
a. a 35% discount b. an 11% increase c. a 6% discount d. a 100% increase?

TOPIC 4 Applications of percentages 147

 28. A student was completing a discount problem where she needed to calculate a 25% discount on $79. She
misread the question and calculated a 20% discount to get $63.20.
She then realised her mistake and took a further 5% from $63.20. Is this the same as taking 25% off $79?
Use calculations to support your answer.
29. a. At the local market there is a ‘buy two, get one free’ offer on handmade soaps. Explain what percentage
discount this is equivalent to.
b. At a rival market there is a ‘buy one, get another half price’ offer on soaps.
Explain whether this deal is the same, better or worse than the discount offered in part a.

LESSON
4.5 Profit and loss
LEARNING INTENTIONS
At the end of this lesson you should be able to:
• calculate profit from cost price and selling price
• calculate the selling price of an item from cost price and profit/loss
• calculate the cost price of an item from selling price and profit/loss.

4.5.1 Cost prices and selling prices

eles-3624
• Overhead costs are not directly linked to a specific product, but
are required to sell products. These include staff wages, rent,
store improvements, electricity and advertising.
• The cost price of a product is the total price that a business
pays for the product including overhead costs.
• The selling price is the price that a customer buys a product for.
• Profit is the amount of money made on a sale. It is the
difference between the total of the retailer’s costs (cost price)
and the price for which the goods actually sell (selling price).

The profit equation

profit = selling price − cost price

Note: If the profit is negative, it’s said that a loss has been made.

148 Jacaranda Maths Quest 8

Calculating the selling price
• The selling price of an item can be calculated by using the information of percentage profit or loss.

Calculating the selling price from percentage profit or loss

The following equations can be used to determine the selling price of an

selling price = (100% + percentage profit) × cost price

item, given the cost price and the percentage profit or loss.

selling price = (100% − percentage loss) × cost price

WORKED EXAMPLE 12 Calculating selling price given percentage profit

Ronan operates a sports store at a fixed profit margin of 65%. Calculate

how much he would sell a pair of running shoes for, if they cost him $40.

Selling price = (100 + 65)% of $40

THINK WRITE

= 165% of $40
1. Determine the selling price by first adding the

= 1.65 × $40
percentage profit to 100%, then determining

= $66
this percentage of the cost price.

2. Write the answer in a sentence. The running shoes would sell for $66.

WORKED EXAMPLE 13 Calculating selling price given percentage loss

David bought a surfboard for $300 and sold it at a 20% loss a year later.
Calculate the selling price.

Selling price = (100 − 20)% of $300

THINK WRITE

= 80% of $300
1. Determine the selling price by first subtracting

= 0.80 × $300
the percentage loss from 100%, then determining

= $240
this percentage of the cost price.

2. Write the answer in a sentence. David sold the surfboard for $240.

TOPIC 4 Applications of percentages 149

 • Profit or loss is usually calculated as a percentage of the cost price.

Percentage profit on cost price

percentage profit = × 100%

profit
cost

percentage loss = × 100%

loss
cost

WORKED EXAMPLE 14 Calculating profit as a percentage of the cost price

A music store buys records at $15 each and sells

them for $28.95 each. Calculate the percentage
profit made on the sale of a record.

Profit = $28.95 − $15

THINK WRITE

profit = selling price − cost price = $13.95

1. Calculate the profit on each record:

× 100% Percentage profit = × 100%

profit 13.95
2. Calculate the percentage profit:

= 93%
cost 15

3. Write the answer in a sentence, rounding to the The profit is 93% of the cost price.
nearest per cent if applicable.

• Modern accounting practice favours calculating profit or loss as a percentage of the selling price. This is
because commissions, discounts, taxes and other items of expense are commonly based on the selling price.

Percentage profit on selling price

percentage profit = × 100%

profit
selling price

percentage loss = × 100%

loss
selling price

150 Jacaranda Maths Quest 8

Calculating the cost price
• If you are given the selling price and the percentage profit or loss, you can work backwards to calculate the
cost price.

Cost price
cost price = selling price − profit = selling price + loss

WORKED EXAMPLE 15 Calculating cost price

A fashion store sells a pair of jeans for $180. If they made a

percentage profit of 80% of the selling price, determine the cost
price of the pair of jeans.

THINK WRITE

Percentage profit = × 100%

profit
1. Enter the given information into the percentage
selling price

80% = × 100%
profit selling price formula.
profit
180

=
80 profit
2. Rearrange the formula to make profit the subject.
100 180

× 180 = profit
80
100

Profit = × 180
80

Profit = $144
100
3. Complete the calculation to determine the profit.

Cost price = $180 − $144

= $36
4. Subtract the profit from the selling price to
determine the cost price.

Resources
Resourceseses
eWorkbook Topic 4 Workbook (worksheets, code puzzle and project) (ewbk-1935)
Interactivities Individual pathway interactivity: Profit and loss (int-4422)
Profit and loss (int-3746)

TOPIC 4 Applications of percentages 151

Exercise 4.5 Profit and loss
4.5 Quick quiz 4.5 Exercise

Individual pathways

PRACTISE CONSOLIDATE MASTER

1, 3, 6, 8, 11, 13, 16, 18, 21 2, 4, 7, 9, 12, 14, 17, 19, 22 5, 10, 15, 20, 23, 24

Assume percentage profit or loss is calculated on the cost price unless otherwise stated.

Fluency
1. Calculate the profit or loss for each of the following.

$15 $20
Cost price Selling price

$40 $50
a.
b.

2. Calculate the profit or loss for each of the following.

$52 $89.90
Cost price Selling price

$38.50 $29.95
a.
b.

3. WE12&13 Calculate the selling price of each of the following.

$18
Cost price % Profit/loss

$116
a. 40% profit
b. 25% loss

4. Calculate the selling price of each of the following.

$1300
Cost price % Profit/loss

$213
a. 30% profit
b. 75% loss

5. Calculate the selling price of each of the following.

Cost price % Profit/loss

$699
1
a. 33 profit
$5140
3
b. 7% loss

6. WE15 Calculate the cost price of the following.

Selling price Percentage profit of

$80
selling price

$125
a. 55%
b. 90%

152 Jacaranda Maths Quest 8

 7. Calculate the cost price of the following.

Selling price Percentage profit of

$3500
selling price

$499.95
a. 24%
b. 35%

Understanding
8. WE14 A restored motorbike was bought for $350 and later sold for $895.
a. Calculate the profit.
b. Calculate the percentage profit. Give your answer correct to the nearest
whole number.
9. A music store sold a drum kit for $480. If they made a percentage profit of
75% of the selling price, determine the cost price of the drum kit.

10. James’s Secondhand Bookshop buys secondhand books for $4.80 and sells them for $6.00.
a. What is the ratio of the profit to the cost price?
b. What is the percentage profit on the cost price?
c. What is the ratio of the profit to the selling price?
d. What is the percentage profit on the selling price?
e. Discuss how the answers to parts a and b are related.
11. A retailer bought a laptop for $1200 and advertised it for $1525.
a. Calculate the profit.
b. Calculate the percentage profit (to the nearest whole number)
on the cost price.
c. Calculate the percentage profit (to the nearest whole number)
on the selling price.
d. Compare the differences between the answers to
parts b and c.

12. Rollerblades bought for $139.95 were sold after six months
for $60.
a. Calculate the loss.
b. Calculate the percentage loss. Give your answer to the
nearest whole number.

a. Jeans costing $20 are sold with a profit margin of 95%.

13. Calculate the selling price for each item.

b. A soccer ball costing $15 is sold with a profit margin of 80%.

c. A sound system costing $499 is sold at a loss of 45%.
d. A skateboard costing $30 is sold with a profit margin of 120%.

a. A diamond ring sold for $2400 with a percentage profit of 60% of the selling price
14. Determine the cost price for the following items.

b. A cricket bat sold for $69 with a percentage profit of 25% of the selling price
c. A 3-seater sofa sold for $1055 with a percentage profit of 35% of the selling price

TOPIC 4 Applications of percentages 153

15. A fruit-and-vegetable shop bought 500 kg of tomatoes for $900
and sold them for $2.80 per kg.
a. What is the profit per kilogram?
b. Calculate the profit as a percentage of the cost price (round to
1 decimal place).
c. Calculate the profit as a percentage of the selling price (round
to 1 decimal place).
d. Compare the answers to parts b and c.

16. Sonja bought an old bike for $20. She spent $47 on parts and paint and renovated it. She then sold it for
$115 through her local newspaper. The advertisement cost $10.
a. What were her total costs?
b. What percentage profit (to the nearest whole number) did she make on costs?
c. What percentage profit (to the nearest whole number) did she make on the selling price?

A clothing store operates on a profit margin of 150%. The selling price of an article bought for $p is:
$151p
17. MC

$150p
A.

$2.5p
B.

$1.5p
C.
D.

Reasoning
18. A fruit-and-vegetable retailer buys potatoes by the tonne (1 tonne is 1000 kg) for $180 and sells them in
5-kg bags for $2.45. What percentage profit does he make (to the nearest whole number)? Show your
working.
19. What discount can a retailer offer on her marked price of $100 so that she ends up selling at no profit and no
loss, if she had initially marked her goods up by $50? Justify your answer.
20. Two business partners bought a business for $158 000 and sold it for $213 000. The profit was to be shared
between the two business partners in the ratio of 3 ∶ 2.
What percentage share does each person receive?
How much does each receive?

Problem solving
21. To produce a set of crockery consisting of a dinner plate, soup

$0.98, $0.89, $0.72 and $0.69 respectively.

bowl, bread plate and coffee mug, the costs per item are

These items are packaged in boxes of 4 sets and sell for $39.
If a company sells 4000 boxes in a month, what is its total profit?
22. Copy and complete the table below.

$4.55 $7.99
Cost per item Items sold Sale price Total profit

$20.00 $40.00 $8040.00

504

$6.06 $225 123.50

$89.95 $28 425.60
64 321
672

154 Jacaranda Maths Quest 8

 23. The method used to calculate profits can make a difference when comparing different profits.

Cost = $20.00 Cost = $26 500.00 Cost = $1.00 (homemade)

Price = $120.00 Price = $32 000.00 Price = $3.50
a. i. Describe the profits on each of the items above as a raw amount.
ii. List the items from largest profit to smallest profit.
iii. Discuss whether this is a fair method of comparing the profits.
b. i. Express the profit on each of the items as a percentage of its cost.
ii. List the items from largest profit to smallest profit.
iii. Discuss whether this is a fair method of comparing the profits.
c. i. Express the profit on each of the items as a percentage of its price.
ii. List the items from largest profit to smallest profit.
iii. Discuss whether this is a fair method of comparing
the profits.
24. Max bought a car for $6000.00. He sold it to Janine for 80% of the
price he paid for it. Janine sold it to Jennifer at a 10% loss.
Jennifer then sold it to James for 75% of the price she paid. What
did James pay for the car?
What was the total percentage loss on the car from Max to James?

LESSON
4.6 Goods and Services Tax (GST) and Income Tax
LEARNING INTENTIONS
At the end of this lesson you should be able to:
• understand what GST is
• calculate prices before and after GST
• calculate Income Tax.

4.6.1 Investigating GST

eles-3625
• GST is a tax imposed by the Australian federal government on
goods and services. (As with all taxes, there are exemptions, but
these will not be considered here.)
• Goods: A tax of 10% is added to new items that are
purchased, such as petrol, clothes and some foods.
• Services: A tax of 10% is added to services that are paid
for, such as work performed by plumbers, painters
and accountants.

TOPIC 4 Applications of percentages 155

 Calculating the amount of GST
The amount of GST on an item can be determined by dividing by 10 if
the price is pre-GST, or by dividing by 11 if the price is inclusive of GST.

÷ 10
Price before GST

Amount of GST

Price, including GST

÷ 11

WORKED EXAMPLE 16 Calculating the amount of GST

A packet of potato chips costs $1.84 before GST.

Calculate:
a. the GST charged on the packet of chips
b. the total price the customer has to pay, if paying with cash.

$1.84
THINK WRITE

a. 1. GST is 10%. Calculate 10% of 1.84. 10% of $1.84 = or $0.184

10

$0.18 (rounded).
2. Write the answer in a sentence. The GST charged on the packet of chips is

$1.84 + $0.18 = $2.02

The total price the customer has to pay is $2.00
b. 1. Total equals pre-GST price plus GST.
2. Write the answer in a sentence.
(rounded down by the seller).

• To calculate the pre-GST cost, when the total you are given includes GST, divide the GST-inclusive amount
by 110 and multiply by 100. This is equivalent to dividing by 1.1.

Finding the cost without GST

cost without GST =

cost with GST
1.1

156 Jacaranda Maths Quest 8

 WORKED EXAMPLE 17 Calculating prices before GST

A plumber’s hourly charge includes GST. If she worked for 5 hours and the
total bill including GST was $580, calculate her hourly price before GST.

$580
THINK WRITE

1. Calculate the hourly price including GST by = $116

dividing the total bill by the total number of 5

110% of pre-GST hourly rate = $116

hours (5).
2. Calculate the hourly price excluding GST.

$116
Pre-GST hourly rate =

= $105.45
1.1

3. Write the answer. The plumber’s hourly rate is $105.45 before GST.

4.6.2 Income tax

Taxation is a means by which state and federal governments raise revenue for public services, welfare and
community needs by imposing charges on citizens, organisations and businesses.

Tax file numbers

A tax file number (TFN) is a personal reference number for every tax-paying individual, company, funds and
trusts. Tax file numbers are valid for life and are issued by the Australian Taxation Office (ATO).

Income tax
Income tax is a tax levied on people’s financial income. It is deducted from each fortnightly or monthly pay.
The amount of income tax is based upon total income and tax deductions, which determines a worker’s
taxable income.

Formula to calculate taxable income

taxable income = total income − tax deductions

The calculation of income tax is based upon an income tax table. The income tax table at the time of writing is:
Taxable income Tax on this income

$18 201–$45 000 19c for each $1 over $18 200

0–$18 200 Nil

$45 001–$120 000 $5092 plus 32.5c for each $1 over $45 000
$120 001–$180 000 $29 467 plus 37c for each $1 over $120 000
$180 001 and over $51 667 plus 45c for each over $180 000
Note: The income tax table is subject to change.

TOPIC 4 Applications of percentages 157

Tax deductions
Workers who spend their own money for work-related expenses are entitled to claim the amount spent as tax
deductions. Tax deductions are recorded in the end-of-financial-year tax return. The deductions are subtracted
from the taxable income, which lowers the amount of money earned and hence reduces the amount of tax to
be paid.

WORKED EXAMPLE 18 Calculating income tax

A worker earned a salary of $82 500 for the year and had $2400 worth of deductions.
a. Determine their taxable income.
b. Determine the taxable income bracket their income falls into.
c. From the income bracket in part b, determine the percentage tax that needs to be paid in this
bracket.
d. Calculate the tax required to be paid.
e. Calculate the percentage of their taxable income that is paid as tax, to 1 decimal place.

a. Taxable income = total income − tax deductions

THINK WRITE

= 82 500 − 2400
a. Taxable income = total income – tax

= $80 100
deductions.

Taxable income = $80 100

b. With a taxable income of $80 100, this fits
Write the answer.

in the bracket of $45 001 and $120 000. $45 001–$120 000.
b. The salary falls in the tax bracket of

Write the answer. $45 001–$120 000

c. In the tax bracket of $45 001–$120 000, the c. Percentage tax = 32.5 cents per $1
= × 100
tax paid is 32.5 cents for each dollar. 32.5

= 32.5%
100

Write the answer. 32.5%

d. Tax paid = (80 100 − 45 000) × + 5092

32.5
32.5 cents for each dollar over $45 000 and
d. Determine the tax bracket. Calculate

add $5092. = 35 100 × + 5092

100
32.5

= 11 407.5 + 5092
100

= $16 499.50

Write the answer. $16 499.50

e. Percentage tax paid = × 100

16 499.50
e. Calculate by dividing the tax paid by the

= 20.6%
taxable income and multiply by 100 to 80 100
convert to a percentage.
Write the answer. 20.6%

COLLABORATIVE TASK: Interpreting receipts

Collect some receipts from a variety of different shops. Look for the section of the receipt that details the GST
information. Are any items on your receipts exempt from GST? As a class, collate your findings and determine
any similarities in the GST-exempt items/services.

158 Jacaranda Maths Quest 8

 Resources
Resourceseses
eWorkbook Topic 4 Workbook (worksheets, code puzzle and project) (ewbk-1935)
Interactivities Individual pathway interactivity: Goods and Services Tax (GST) (int-4423)
Goods and Services Tax (int-3748)

Exercise 4.6 Goods and Services Tax (GST) and Income Tax
4.6 Quick quiz 4.6 Exercise

Individual pathways

PRACTISE CONSOLIDATE MASTER

1, 3, 6, 8, 10, 14, 17, 20, 23 2, 4, 7, 9, 11, 15, 18, 21, 24 5, 12, 13, 16, 19, 22, 25

Fluency
1. Explain GST in your own words.

2. Does GST apply below? Answer yes or no for each example.

a. Petrol b. A lawyer’s fee c. Hotel accommodation
d. Lounge room carpet e. Floor tiling f. Wages at a fast-food restaurant

3. WE16 The pre-GST price of a packet of laundry powder is $4.50.

a. Calculate the GST on the laundry powder.
b. Calculate the total price including GST.

4. The pre-GST price of a tin of peaches is $2.12.

a. Calculate the GST on the tin of peaches.
b. Calculate the total price including GST.

5. The pre-GST price of 1 kg of jellybeans is $3.85.

a. Calculate the GST on the jellybeans.

b. Calculate the total price including GST.

6. WE17 The prices of the following items are inclusive of GST.

a. 1 kg of apples at $3.85
Calculate the pre-GST price of each.

b. A basketball that costs $41.80

7. The prices of the following items are inclusive of GST. Calculate

a. 5 kg of potatoes at $6.50
the pre-GST price of each.

b. A couch that costs $730

8. Millie buys a pack of batteries and pays 25 cents GST. How much did she pay in total for the batteries?

9. A new bicycle costs $450, including GST. How much is the GST?

10. WE18 A worker earned $67 240 and accumulated $1890 of tax deductions. Calculate their taxable income.

TOPIC 4 Applications of percentages 159

Understanding
11. The telephone company Ringtel charges home customers $42.50 per month plus $0.24 per local call.
Determine the monthly phone bill, including GST, if a customer makes 51 local calls in a month.

$44 per day plus $0.47 per kilometre travelled. A customer

12. All car rental agencies use similar charging plans. Drivo charges

wishes to rent a car for four days and travels 1600 km.
Calculate the customer’s total bill, including GST.

car repair business. It charges $85 per hour plus a flat $40
13. Expresso is a company that operates in the ‘we-visit-you’

visiting fee.
a. Set up an expression, which includes GST, for the cost
of a repair that takes t hours.
b. If the repair takes 3 hours and 30 minutes, determine the
final cost.
14. A company that installs floor tiles charges $35 per square metre for the actual tiles, and a fee of $100 plus
$10 per square metre to install the tiles in a home.
Let the area of the floor to be tiled be x m2 .
a. Determine an expression, including GST, that represents the total cost of tiling in terms of x.
b. What would be the total cost for a 20 m2 floor?

15. To buy my new super-dooper mobile phone outright I must pay $30 per month, including GST, for 3 years.
How much GST will I pay?

or VAT, is levied at 20%. If I paid £67 for a jumper purchased in

16. In the United Kingdom a similar tax, called the value-added tax

a shop on Bond Street, London:

a. how much VAT did I pay
b. what was the pre-VAT price of the jumper?

17. In New Zealand GST is levied at 15% of the purchase price of

goods.
If I buy a pair of jeans and pay NZ$12 in GST, calculate the total
price I paid for the jeans in NZ dollars.

18. An employee earns an income of $92 375 per year with tax deductions of $3258.
a. Calculate the taxable income.
b. Calculate the amount of tax required to be paid, to the nearest cent.
b. Calculate the percentage of their taxable income is paid in tax, to one decimal place.

19. A worker was paid $128 425 for the year and was able to claim $6850 worth of tax deductions.
a. Determine their taxable income.
b. What taxable income bracket does their income fall into?
c. From the income bracket in part b, what percentage tax is needed to be paid in this bracket?
d. Calculate the tax required to be paid.
e. Calculate the percentage of their taxable income that is paid as tax to one decimal place.

Reasoning
20. Explain what the terms inclusive of GST and exclusive of GST mean.

21. Explain why the amount of GST on an item is not equivalent to 10% of the GST-inclusive price.

160 Jacaranda Maths Quest 8

22. Explain why the pre-GST price of an item is not equivalent to 10% off the GST-inclusive price.

Problem solving
23. The GST rate is 10%. This means that when a business sells
1
something or provides a service, it must charge an extra of
10
the price/cost. That extra money then must be sent to the tax

For example, an item that would otherwise be worth $100 now

office.

has GST of $10 added, so the price tag will show $110.
The business will then send that $10 to the tax office, along with
all the other GST it has collected on behalf of the government.
a. Suppose a shopkeeper made sales totalling $15 400.
Determine how much of that amount is GST.
b. Explain whether there is a number they can quickly divide by
to calculate the GST.

a. the GST payable on an item whose pre-GST price is $P, and the price payable
24. Taking GST to be 10%, calculate:

b. the pre-GST price of an item that costs $A, and how much GST would need to be paid.

25. In the country Snowdonia, GST is 12.5%. Kira has purchased a new hairdryer that cost 111 kopeks including
GST. There are 100 plens in 1 kopek.

b. If 1 Australian dollar = 2 kopeks, calculate how much GST Kira would have paid if she had purchased the
a. Calculate how much GST Kira paid.

hairdryer in Melbourne, where GST is currently 10%.

TOPIC 4 Applications of percentages 161

LESSON
4.7 Review
4.7.1 Topic summary

Discount
• A discount is a reduction in price.
discount 100%
percentage discount = – × –
original selling price 1
• The new sale price can be obtained either by
subtracting the discount amount from the original
price, or by calculating the remaining percentage.
e.g. A 10% discount means there is 90% remaining
of the original selling price.

Percentages
• The term ‘per cent’ means ‘per hundred’.
• Percentages can be converted into fractions and
APPLICATIONS OF decimals by dividing the percentage by 100.
77
PERCENTAGES e.g. 77% = – = 0.77
100
• To convert fractions and decimals into percentages,
multiply by 100.
• Percentage increase/decrease
amount of increase/decrease 100
= × –%
original amount 1
Profit and loss • To find the percentage of an amount, convert the
percentage to a fraction or decimal and then multiply.
• A profit means the selling price > cost price,
25 48
i.e. money has been made. e.g. 25% of 48 = – × – = 12
100 1
profit = selling price – cost price • To increase a quantity by x%, multiply the quantity by
• A loss means the selling price < cost price, (100 + x)%.
i.e. money has been lost. • To decrease a quantity by x%, multiply the quantity by
loss = cost price – selling price (100 – x)%.
Percentage profit and loss is usually calculated
based on the cost price.
profit or loss 100% Good and Services Tax (GST) and Income Tax
percentage profit/loss = – × –
cost price 1
• GST is the Goods and Services Tax. This is a 10% tax
added by the government to the cost of many items
and services.
÷ 10
Price before GST

Amount of GST

Price including GST

÷ 11
• Taxable income = total income − tax deductions

162 Jacaranda Maths Quest 8

4.7.2 Success criteria
Tick a column to indicate that you have completed the lesson and how well you think you have understood it
using the traffic light system.
(Green: I understand; Yellow: I can do it with help; Red: I do not understand)

Lesson Success criteria

4.2 I can convert percentages into fractions and decimals.

I can calculate percentage increases and decreases.

I can calculate a percentage error.

4.3 I can calculate percentages of an amount.

I can increase or decrease a value by a percentage.

4.4 I can calculate the cost of a discounted product.

I can calculate the discount from an initial price and sale price.

4.5 I can calculate profit from cost price and selling price.

I can calculate the selling price of an item from cost price and profit/loss.

I can calculate the cost price of an item from selling price and profit/loss.

4.6 I understand what GST is.

I can calculate prices before and after GST.

4.7.3 Project
The composition of gold in jewellery
You may be aware that most gold jewellery is not
made of pure gold. The materials used in jewellery
are usually alloys, or mixtures of metals. The finest
gold used in jewellery is 24 carat and is known as
fine gold. Gold in this form is very soft and is easily
scratched. Most metals will form an alloy with
gold; silver, copper and zinc are commonly used
in jewellery making. Other metals may be used to
create coloured gold.

TOPIC 4 Applications of percentages 163

 A table of the composition of some of the common gold alloys used in jewellery is shown below.

Gold name Composition

Gold (24 carat) Gold 100%
Yellow gold (22 carat) Gold 91.67%
Silver 5%
Copper 2%
Zinc 1.33%
Pink gold (18 carat) Gold 75%
Copper 20%
Silver 5%
Rose gold (18 carat) Gold 75%
Copper 22.25%
Silver 2.75%
Red gold (18 carat) Gold 75%
Copper 25%
White gold (18 carat) Gold 75%
Palladium 10%
Nickel 10%
Zinc 5%
Grey-white gold (18 carat) Gold 75%
Iron 17%
Copper 8%
Green gold (18 carat) Gold 75%
Silver 20%
Copper 5%
Blue gold (18 carat) Gold 75%
Iron 25%
Purple gold Gold 80%
Aluminium 20%

Use the table to answer the following questions.

1. Study the table and list the metals used to create the alloys of gold mentioned.
2. A particular rose-gold bracelet weighs 36 grams. Calculate the masses of the various components in the
bracelet.
3. How much more gold would there be in a yellow-gold bracelet
of the same mass? What fraction is this of the mass of the
bracelet?
4. Pink, rose and red gold all contain 75% gold. In addition,
they each contain copper, and pink and rose gold also
contain silver. Describe the effect you feel the composition
of the alloy has on the colour of the gold.
5. Why does white gold not contain any copper?
6. Compare the composition of the alloys in red gold and blue gold.

164 Jacaranda Maths Quest 8

 7. Twenty-four-carat gold is classed as 100% gold. On this basis, an alloy of gold containing 75% gold has a
carat value of 18 carat. Note this fact in the table above. Purple gold is 80% gold. What would its carat
value be?
8. Just as there are various qualities of gold used in jewellery making, the same is true of silver jewellery.
Sterling silver, which is commonly used, is actually not pure silver.
Find out about the composition of silver used in jewellery making. Write a short report on your findings
on a separate sheet of paper.

Resources
Resourceseses
eWorkbook Topic 4 Workbook (worksheets, code puzzle and project) (ewbk-1935)
Interactivities Crossword (int-2626)
Sudoku puzzle (int-3186)

Exercise 4.7 Review questions

Fluency

a. $2.45 + $13.20 + $6.05

1. Calculate these amounts.

b. $304.60 − $126.25
c. $9.65 × 7

2. What is $65.50 ÷ 11? (Round your answer to the nearest 5 cents.)

3. Jill purchased a handbag for $250 and later sold it on eBay for
$330.
a. Calculate the percentage profit on the cost price.
b. Calculate the percentage profit on the selling price.
c. Compare the answers to parts a and b.

haircuts from $26.50 to $29.95. Determine the percentage by which

4. William owns a hairdressing salon and raises the price of men’s

he increased the price of men’s haircuts.

Give your answer correct to the nearest per cent.

5. A discount of 18% on a tennis racquet reduced its price by $16.91. Calculate the sale price.

6. A washing machine bought for $129 was later sold for $85. What percentage loss was made on
the sale?

7. Calculate the percentage profit on a sound bar purchased for $320

and later sold for $350.

8. A 15% discount reduced the price of a basketball by $4.83. What

was the original price?

9. Tim works in a sports shop. He purchased wholesale golf shirts for $55 each. If he made 163% profit,
determine the sale price of the golf shirts.

TOPIC 4 Applications of percentages 165

 10. The profit on a gaming console is $240. If this is 60% of the cost
price, calculate:
a. the cost price
b. the selling price.

11. A music store sells an acoustic guitar for $899. If the cost price of
the guitar is $440, determine the profit the store makes, taking GST

12. A company made a profit of $238 000. This represents a 10% profit increase compared to the previous
into account.

year. Determine last year’s profit.

paid for a sleeping bag that sells for $89.95.

13. A camping goods shop operates on a profit margin of 85%. Calculate how much the shop would have

Problem solving
14. After a 5% discount, a telephone bill is $79.50. Calculate the original amount of the bill.

15. Pablo spent $82.20 at the supermarket. If 15% of this was spent on tomatoes priced at $3 per kilogram,
determine the weight of tomatoes purchased.

16. You buy ten pairs of headphones for a total of $150. Determine the price you should sell six pairs for if
you wish to make a profit of 25% on each pair.

17. An art dealer sold two paintings at an auction. The first painting sold for $7600, making a 22% loss on
its cost. The second painting sold for $5500, making a profit of 44%.
Explain whether the art dealer made an overall profit or loss.

166 Jacaranda Maths Quest 8

 1

suite was $5689 including GST, determine the sale price including GST.
18. Jacques’ furniture shop had a sale withoff the usual price of lounge suites. If the original price of a
3

19. Goods listed at $180 were discounted by 22%.

b. If they had sold for $100, determine what the percentage discount would have been.
a. Calculate the sale price.

20. Steve Smith buys a cricket bat for $85, signs it and donates it for

If it sells for $500, calculate:

an auction.

a. the percentage increase in the bat’s value

b. the dollar value of the signature.

21. Andrew buys a pair of jeans for $59.95. The original price tag
was covered by a 30% sticker but the sign on top of the rack said
‘Additional 15% off already reduced prices.’
a. Calculate the original price of the jeans. Give your answer
correct to the nearest 5 cents.
b. Determine what percentage of the original cost Andrew ended
up saving.

If the final bill is $55.55, determine the original price, taking into
22. Café Noir charges a 1% levy on the bill for trading on Sundays.

account that the levy has been charged and then 10% GST has
been added.

To test your understanding and knowledge of this topic, go to your learnON title at
www.jacplus.com.au and complete the post-test.

TOPIC 4 Applications of percentages 167

Answers c. 11
̇
d. 46.6%
Topic 4 Applications of
16. $7.27
15. Bag B will pass. Bag A will not pass.
percentages
17. Answers will vary. To calculate the percentage, fraction and
4.1 Pre-test
$7.50
decimal of a particular brand, the number of phones of that
1. brand needs to be divided by the total number of phones.
2. C 14 41
18. a. 0% b. 72% c. d.
1 3 25 100
3. 22%, , 0.31, , 111%
4 5 8
19. a. 38% b.

$18.40
4. C 21

$76.50
5.
6.
4.3 Finding percentages of an amount
7. 5% 1. a. 10 b. 16 c. 3 d. 3
8. D 2. a. 93 b. 6 c. 6 d. 28
9. 19 years 3 months 3. a. 77 b. 39 c. 63 d. 4000
10. D 4. a. 66 b. 17 c. 42 d. 95
11. 2.55%
5. a. 190 b. 55 c. 12 d. 25
12. B
6. a. 13.2 b. 1.9 c. 24.75 d. 11.1
13. True
14. 27% 7. a. 58.9 b. 20.8 c. 14.4 d. 98.4

$2.70 $7.15 $5.75 $6.05

15. Australia 8. D

$0.20 $4.30 $0.05 $0.05

9. a. b. c. d.
4.2 Percentages

$0.10 $0.00 $0.00 $12.65

10. a. b. c. d.
1. a. 0.24 b. 0.13 c. 0.015 d. 2.5

$2.20 $1.80 $73.50 $18.00

11. a. b. c. d.
2. a. 0.47 b. 0.066 c. 1.098 d. 0.1002

$1.05 $2.05 $32.20 $4.80

12. a. b. c. d.
1 7 61 21
3. a. b. c. d. 13. a. b. c. d.
5 20 100 20
14. 54 000
11 41 1 101

$267.50
4. a. b. c. d. 15. a. 2 b. 38
100 50 8 50
16.
5. a. 15% b. 85% c. 310% d. 2.4%

$855
17. 114 minutes
6. a. 87.5% b. 60% c. 83.33% d. 233.33% 18.

a. 10%, 25%, 75%, , 1.6, 2.4, 3

7 1 19. 23%
7.
8 2
b. 4.5%, 150%, 2 , 2.8, 3, 330%, 3 $9.60
20. a. 36.3 kg b. 60 lb c. 166 cm
1 4
21. a. 22.5 kg b. 32 km c.
3 5
8. 30% 22. 110
9. 56.67% 23. 27.9 seconds
24. a. 13 608 people b. 17 820 people
10. 30%

$0.80
11. 35.29% 25. 20 years old
12. 30% 26.

13. a. 4.65% b. 1.44% 27. Sister: 9 years old; grandfather: 90 years old
28. a. 14
14. a. 30
b. 7.69% of students achieved a score of 40 or more, which
b.
Number of Percentage is just below the state average.

̇
Free throw results students of students 29. a. 78 minutes b. 282 minutes
No shots in 3 10% c. 21.6% d. 155.1 minutes
One shot in, three 11 36.6% 30. 60 years old
misses
4.4 Discount
$42 $46.25
Two shots in, two 10 33.3%
misses
$49.50 $76
1. a. b.
Three shots in, one 4 13.3% 2. a. b.
miss
3. a. 10% b. 50%
All shots in 2 6.6%

168 Jacaranda Maths Quest 8

 33 % 1∶4
1
4. a. b. 25% 10. a.
3
$850 $200 c. 1 ∶ 5
b. 25%

$83.60 $104
5. a. b.

$64.70 $241.50
6. a. b. d. 20%
7. a. b. e. The ratio of the profit to the cost price as a fraction is the

11. a. $325
same as the percentage profit on the cost price.
8. a. 40% b. 28%

$45 $45 $36

9. a. 28% b. 22% b. 27%
10. a. b. c. c. 21%

$79.95
d. The percentage profit is greater on the cost price.

Mobile phone $95

11. 25%

$8.35 $39 $27 $274.45 $66

12. a. b. Surfboard and bike 12. a. b. 57%

$70 $280
c. d. No
$960 $51.75 $685.75
13. a. b. c. d.

$62.96
13. a. b.

$1.00 profit per kg

14. a. b. c.

15. a. $41.65
14. 15. a.

$75.76
b. Yes b. 55.6%
16. c. 35.7%

$77
17. 20% d. The percentage profit is greater on the cost price.
18. 17.3% 16. a. b. 49% c. 33%

20. $243.70
19. 60% 17. C

21. a. $121.60; $140.80

18. 172%

60%, 40%; $33 000, $22 000

b. Profit 19. 50%

$103 520
22. A 20.

$200 = $150 off the price, so you would pay only $50.
21.
23. Yes, there is a difference in the meanings. 75% off
22.

75% of $200 = $150, i.e. of $200

Cost per
3
$4.55 $7.99 $1733.76
item Items sold Sale price Total profit

$1.00/$12.00 × 100% = 8.33%, so this is an 8.33% discount.

4
$20.00 $40.00 $8040.00
504

Henry pays $954; Sancha pays $991.20. Henry has the

24.

$6.06 $9.56 $225 123.50

402
25.

$47.65 $89.95 $28 425.60

64 321
best buy.
672

cost of $100, a 50% discount = $50 and a 40% discount (on $100, $5500, $2.50
26. No, the statement is not correct. For example, if you have a

that $50) = $20.

23. a. i.

Total discount = $70; this represents a 70% discount, not

ii. Car, shoes, cookies

90%. iii. Not fair; profit should be compared as a proportion of

cost.

95% of $63.20 = $60.05; 75% of $79 = $59.25. The two

27. a. 65% b. 111% c. 94% d. 200%
b. i. 500%, 20.75%, 250%
28.
ii. Shoes, cookies, car
methods calculate percentages of different amounts so they
result in different answers. iii. Fairer than in part a

29. a. 33.33% c. i. 83.3%, 17.2%, 71.4%

b. This deal is worse than the deal offered in part a as it is ii. Shoes, cookies, car

24. James paid $3240. The total percentage loss was 46%.
equivalent to only a 25% discount; however, it should be iii. Not fair; the profit should be calculated on the cost.
used if you only want to buy two soaps.

4.5 Profit and loss

$5 profit $10 profit
4.6 Goods and Services Tax (GST) and
Income Tax
$37.90 profit $8.55 loss
1. a. b.
1. GST is a tax of 10% levied by the Australian federal
$25.20 $87
2. a. b.
government on goods and services.

$1690 $53.25
3. a. b.
2. a–e. Yes

$932 $4780.20
4. a. b.

$0.45 $4.95
f. No

$36.00 $12.50
5. a. b.

$0.21 $2.33
3. a. b.

$2660 $324.95
6. a. b.

$0.39 $4.24
4. a. b.

$545
7. a. b.

$3.50 $38
5. a. b.

$120
8. a. b. 156% 6. a. b.
9.

TOPIC 4 Applications of percentages 169

 $5.91 $663.64
$2.75
4.7 Review questions
$21.70 $178.35 $67.55
7. a. b.

$40.91 $5.95
8. 1. a. b. c.

$65 350
9. 2.

$60.21
10. 3. a. 32%

$1020.80
11. b. 24.24%

a. 1.1 (85t + 40) $371.25

12. c. The percentage profit is greater on the cost price.

$77.03
4. 13%
1.1 (45x + 100) $1100
13. b.
5.

$98.18
14. a. b.
6. 34%
16. a. $11.17 $55.83
15.

$32.20
b.
7. 9.375%

$144.65
8.

18. a. $89 117 $19 430.03

17. NZ$92

a. $400 $640
9.
b.

$377.27
10. b.

$121 575 $120 001–$180 000

c. 21.80%

$34 119.75 $216 364

11.
19. a. b.

$48.62
c. 37% d. 12.

$83.68
e. 28.1% 13.
20. Inclusive includes GST in the total price and exclusive 14.

$112.50
excludes GST from the total price. 15. 4.11 kg

Loss of $463.03
21. GST is equal to 10% of the pre-GST price, which is 16.

if an item costs $100 pre-GST, the GST would be $10

less than 10% of the GST-inclusive price. For example,
$4171.93
17.

and the GST-inclusive price would be $110. 10% of the

GST-inclusive price would be $11, not $10. a. $140.40
18.

$415
19. b. 44.4%

pre-GST price. For example, if an item costs $100 pre-GST, $100.75

22. 10% off the GST-inclusive price is equal to 99% of the 20. a. 488% b.

the GST would be $10 and the GST-inclusive price would

$50
21. a. b. 40.5% saved

be $110. Taking 10% off the GST-inclusive price would be

$99, not $100.
22.

23. a. $1400 b. 11

$ ,$ $ ,$
P 11P 10A A
24. a. b.
10 10 11 11

$4.93 GST; total price ≈ $54.27

25. a. 12 kopeks, 1233 plens
b.

Project
1. Metals used as alloying elements with gold are silver, copper,
zinc, palladium, nickel, iron and aluminium.
2. 27 g gold, 8.01 g copper, 0.99 g silver

1
3. 6 g,
6
4. From pink to rose to red gold, the percentage of silver
decreases, causing the gold alloy to darken in colour. At
the same time, the percentage of copper increases, also
contributing to the darker colour.
5. The copper would colour the gold with its familiar reddish
colour so that it would not be white.
6. Red gold and blue gold each have 75% gold and 25% of
another metal. In the case of red gold, the contributing metal
is copper; blue gold contains iron.
7. 19.2 carat
8. Sample responses can be found in the worked solutions in the
online resources.

170 Jacaranda Maths Quest 8