1a_ch3(1). 1a_ch3(2) 3.1simple problems involving percentages a using percentage to find a number b...

1A_Ch3(1)

1A_Ch3(2)

3.1 Simple Problems Involving Percentages

A Using Percentage to Find a

Number

B Finding the Percentage

C Finding the Original Number

from a Given Percentage

Index

1A_Ch3(3)

3.2 Percentage Change

A Percentage Increase

B Percentage Decrease

C Percentage Change

Index

1A_Ch3(4)

3.3 Profit and Loss

A Profit

B Loss

Index

Using Percentage to Find a Number

y% of a number A = A × y%

Index

A)

1A_Ch3(5)3.1 Simple Problems Involving Percentages

Example

Index 3.1

Find the value of each of the following.

(a) 850 × 25%

Index


(a) 25% of 850 (b) 10% of 12.8

= 850 × 0.25

= 212.5

(b) 12.8 × 10% = 12.8 × 0.1

= 1.28

Index


There are 1 400 staff in a company. It is

known that 88% of the staff in the company

are university graduates. How many staff in

that company are university graduates?

Number of staff who are university graduates

= 1 400 × 88%

10088

= 1 400 ×

= 1 232

Fulfill Exercise Objective

Use percentage to find a number.

Index


In 2002, the population of Hong Kong was 6.8 million and

16% of them were aged under 15. How many people were

15 or above?

Number of people aged under 15 = 6.8 × 16% million

= 6.8 × 0.16 million

= 1.088 million

∴ Number of people aged 15 or above

= (6.8 – 1.088) million

= 5.712 million


Use percentage to find a number.

Key Concept 3.1.1

Finding the Percentage

Index

B)


1. To find out what percentage of a is b,

we write .ba× 100%

2. To find out what percentage of b is a,

we write .ab× 100%

Example

Index 3.1

(a) What percentage of 45 is 36?

(b) What percentage of 36 is 45?

Index

1A_Ch3(10)


(a) The required percentage = %1004536

= 80%

(b) The required percentage = %1003645

= 125%

Index

1A_Ch3(11)


Among the 800 people in the election

team, 256 of the team members vote

for candidate A and the rest vote for

candidate B.

(a) What percentage of the votes has gone to

candidate A?

(b) What percentage of the number

of votes for A is that for B?

Index

1A_Ch3(12)


(a) The required percentage = %100800256

= 32%

(b) The number of votes that candidate B gets = 800 – 256

= 544

The required percentage = %100256544

= 212.5%


Find the required percentage.

Back to Question

Key Concept 3.1.2

Finding the Original Number from a Given Percentage

Index

C)

1A_Ch3(13)


Given : x% of the original number A = y

Theny

x%A =

Example

Index 3.1

Find the unknown in each of the following.

Index

1A_Ch3(14)


(a) $24 is 60% of $m. (b) 180 kg is 125% of n kg.

m × 0.6= 24

m = 24 ÷ 0.6

= 40

(a) $m × 60% = $24

n × 1.25 = 180

n = 180 ÷ 1.25

= 144

(b) n kg × 125%= 180 kg

Index

1A_Ch3(15)


A football team won 85% of its games

last year. If they won 34 games

altogether, how many games did the team

play?Let n be the total number of games the team played.

Then 85% of n is 34.

i.e. 85% × n = 34

n × 0.85= 34

n = 34 ÷ 0.85

= 40 ∴ The total number of games the team played was 40.


Find the original number.

Index

1A_Ch3(16)


In the academic year 1999 – 2000, 55%

of the students in the University of Hong

Kong were male. If the number of female

students in that year was 6 300, then what

was the total number of students?

Percentage of female students

= 100% – 55%

= 45%

Index

1A_Ch3(17)


Let n be the total number of students.

Then 45% of n is 6 300.

i.e. 45% × n= 6 300

n × 0.45= 6 300

n = 6 300 ÷ 0.45

= 14 000

∴ The total number of students was 14 000.


Find the original number.

Back to Question

Key Concept 3.1.3

Percentage Increase

Index

A)

1A_Ch3(18)


Example

Index 3.2

1. Increase = New value – Original value

2. Percentage increase = Increase

Original value× 100%

3. Increase = Original value × Percentage increase

4. New value = Original value × (1 + Percentage increase)

Find the percentage increase in each of the following.

Index

1A_Ch3(19)


(a) Increase = 156 – 120

= 36

∴ Percentage increase

= %10012036

= 30%

(a) An increase from 120 to 156.

(b) An increase of 10.5 mL from 25 mL.

(b) Increase = 10.5 mL

∴ Percentage increase

= %100mL 25mL 5.10

= 42%

Increase each of the following quantities by the given percentage:

Index

1A_Ch3(20)


(a) New value = 82 × (1 + 15%)

= 82 × 1.15

= 94.3

(a) 82 by 15% (b) $144 by 35%

(b) New value = $144 × (1 + 35%)

= $144 × 1.35

= $194.4

Index

1A_Ch3(21)


The price increase of a movie recorded on DVD and

on video CD is both $5. If the original prices of a

DVD and a video CD are $125 and $40 respectively,

find the percentage increase in the price of

(a) a DVD,

(b) a video CD.

Index

1A_Ch3(22)


(a) Percentage increase in the price of a DVD

= %100125$5$

= 4%

(b) Percentage increase in the price of a video CD

= %10040$5$

= 12.5%


Find the percentage increase.

Back to Question

Index

1A_Ch3(23)


To celebrate the New Year, a certain

brand of chocolate beans added 10% to

the weight of each pack for free.

If the original weight of each pack of the chocolate beans

was 50 g, what was the weight of each pack after the

increase?

The new weight = 50 × (1 + 10%) g

= 50 × 1.1 g

= 55 gFulfill Exercise Objective

Find the new value. Key Concept 3.2.1

Percentage Decrease

Index

B)

1A_Ch3(24)


Example

Index 3.2

1. Decrease = Original value – New value

2. Percentage decrease = Decrease


3. Decrease = Original value × Percentage decrease

4. New value = Original value × (1 – Percentage

decrease)

Find the percentage decrease in each of the following.

Index

1A_Ch3(25)


(a) Decrease = 300 – 48

= 252

∴ Percentage decrease

= %100300252

= 84%

(a) An decrease from 300 to 48.

(b) An decrease of 3 g from 12 g.

(b) Decrease = 3 g

∴ Percentage decrease

= %100g 12g 3

= 25%

Decrease each of the following quantities by the given percentage:

Index

1A_Ch3(26)


(a) New value = 150 × (1 – 27%)

= 150 × 0.73

= 109.5

(a) 150 by 27% (b) 855 cm by 60%

(b) New value = 855 × (1 – 60%) cm

= 855 × 0.4 cm

= 342 cm

Index

1A_Ch3(27)


The worldwide population of the black r

hinoceroes has dropped from 65 000 in t

he early 1970s to 2 405 in the late 1990s.

What is the percentage decrease in the n

umber of black rhinoceroes?

The decrease = 65 000 – 2 405

= 62 595

The percentage decrease = %100000 65595 62

= 96.3%


Find the percentage decrease.

Index

1A_Ch3(28)


Kitty scored 80 marks in Mathematics

last term. If her score is decreased by

5% this term, what is her score in this

term?

The score of this term = 80 × (1 – 5%)

= 80 × 0.95

= 76Fulfill Exercise Objective

Find the new value. Key Concept 3.2.2

Percentage Change

Index

C)

1A_Ch3(29)


Example

Index 3.2

‧ Percentage change = New value – Original value


New value > Original value

New value < Original value

Change Sign of percentage change

Increase

Decrease

+

–

Index

1A_Ch3(30)


(a) If 100 is changed to 110, then

percentage change = %100100

100110

= +10%

(b) If 50 is changed to 40, then

percentage change = %10050

5040

= –20%

Index

1A_Ch3(31)


The price of a digital camera was

$4 000 last year. This year the price

becomes $3 400. Find the percentage

change in price.

Percentage change


Find the percentage change.

= %100000 4

000 4400 3

= %100000 4600

= –15% Key Concept 3.2.3

Profit

Index

A)

1A_Ch3(32)

3.3 Profit and Loss

1. When a merchant pays to buy goods, the amount he pays is called the cost price.

2. When the merchant sells goods at a price, the amount he receives is called the selling price.

3. If selling price > cost price, the merchant will make a profit.

4. If we compare the profit with the cost price of the goods and express the result as a percentage, the percentage is called the profit per cent (written as profit %).

Profit

Index

A)

1A_Ch3(33)

3.3 Profit and Loss

Example

Index 3.3

5. If selling price > cost price,

ii. Profit % = Profit

Cost price× 100%

iii. Profit = Cost price × Profit %

iv. Selling price = Cost price × (1 + Profit %)

i. Profit = Selling price – Cost price

The selling price of a toy car is $225, and the

cost price is $180.

Index

1A_Ch3(34)

3.3 Profit and Loss

(a) Find the profit of the toy car.

(b) Find the profit per cent of the toy car.

(a) Profit = $(225 – 180) = $45

(b) Profit % = %100180$45$

= 25%

Index

1A_Ch3(35)

Mr Wong bought a pair of jeans for $80

and sold them for $130. Mr Cheung bou

ght a T-shirt for $40 and sold it for $90.

Who made a greater profit %?

Profit made by Mr Wong

= %10080$50$

3.3 Profit and Loss

= $(130 – 80)

= $50

∴ Profit %

= 62.5%

Index

1A_Ch3(36)

Profit made by Mr Cheung

= %10040$50$

3.3 Profit and Loss

= $(90 – 40)

= $50

∴ Profit %

= 125%

∴Mr Cheung made a greater profit %.


Find the profit %.

Back to Question

Index

1A_Ch3(37)

A car was bought for $350 000 and

was sold at a profit of 20%. How

much was it sold for?

Selling price

Key Concept 3.3.1

3.3 Profit and Loss

= $350 000 × (1 + 20%)

= $420 000

= $350 000 × 1.2


Find the selling price.

Loss

Index

B)

1A_Ch3(38)

3.3 Profit and Loss

1. If selling price < cost price, there will be a loss.

2. We can express loss as a percentage of the cost pric

e. The result is called the loss per cent (written as l

oss %).

Loss

Index

B)

1A_Ch3(39)

3.3 Profit and Loss

Example

3. If selling price < cost price,

ii. Loss % = Loss

Cost price× 100%

iii. Loss = Cost price × Loss %

iv. Selling price = Cost price × (1 – Loss %)

i. Loss = Cost price – Selling price

Index 3.3

Index

1A_Ch3(40)

If a bicycle is bought at $1 000 and sold for $800, then

loss = $(1 000 – 800)

= $200

and loss % = %100000 1$

200$

= 20%

3.3 Profit and Loss

Index

1A_Ch3(41)

A box of 100 coloured pencils costs $250. In

a sale, each pencil is sold at $2 a piece.What i

s the loss per cent after all of the pencils have

been sold?

Total cost price

Key Concept 3.3.2

3.3 Profit and Loss

= $250

Total selling price= $2 × 100 = $200

Total loss= $(250 – 200) = $50

∴ Loss % = %100250$50$

= 20%


Find the loss %.

Discount

Index

1A_Ch3(42)

3.4 Discount

1. The difference between the marked price and the

selling price is called the discount.

2. We can express the discount as a percentage of the

marked price, which is called the discount per cent

(written as discount %).

Discount

Index

1A_Ch3(43)

3.4 Discount

Example

3. i. Discount = Marked price – Selling price

ii. Discount % = Discount

Marked price× 100%

iii. Discount = Marked price × Discount %

iv. Selling price = Marked price × (1 – Discount %)

A sofa is marked at $2 800 and sold at $2 100 in

a sale. What is the discount and discount %?

Index

1A_Ch3(44)

Discount= $(2 800 – 2 100)

= $700

= 25%

3.4 Discount

Discount % %100800 2$

700$ =

Index

1A_Ch3(45)

A pack of E-Power Battery marked at $24 is now sold

for $21 only. A pack of D-cell Battery, on the other

hand, marked at $20 is now sold for $17.9.

Which pack of Battery is sold at a larger discount per

cent?

3.4 Discount

Index

1A_Ch3(46)

For E-Power Battery,

3.4 Discount

= $(24 – 21)

discount % = %10024$3$ = 12.5%

For D-cell Battery,

= $(20 – 17.9)

discount % = %10020$

1.2$ = 10.5%

= $3

= $2.1

∴ E-Power Battery is sold at a larger discount per cent.


Find the discount %.

discount

discount

Back to Question

Index

1A_Ch3(47)

After a 44% discount, a radio is now

sold for $112. What was the original

marked price of the radio?

3.4 Discount

44% off

Now $112 only

Let $P be the original marked price of the radio.

Then112 = P × (1 – 44%)

112 = P × 0.56

P = 112 ÷ 0.56

= 200

∴ The original marked price was $200.


Find the marked price.

Index

1A_Ch3(48)

A dictionary marked at $260 is sold at

a discount of 25%.

3.4 Discount

(a) Find the selling price.

(b) If the cost price of the dictionary is

$200, find the loss %.

Dictionary

25% off

(a) The selling price = $260 × (1 – 25%)

= $260 × 75%

= 10075

260$

= $195

Index

1A_Ch3(49)

Key Concept 3.4.1

3.4 Discount

(b) Loss = cost price – selling price

= $(200 – 195)

= $5

= %100200$

5$

= 2.5%

Loss %


Miscellaneous problems.

Back to Question

Simple Interest

Index

1A_Ch3(50)

3.5 Simple Interest

1. If the interest in each period is calculated on the

same principal, then the interest obtained is called

simple interest.

2. The calculation of interest is based on a percentage of

the principal and that percentage is called the interest

rate.

Simple Interest

Index

1A_Ch3(51)

3.5 Simple Interest

3. Let $I stand for the simple interest, $P stand for the

principal, R% stand for the interest rate per annum,

T years stand for the period of time, $A stand for

the amount, then

Examplei.100

TRPI

ii. )100

1(RT

PIPA Example

Index

1A_Ch3(52)

If $1 000 is deposited in a bank at an interest rate of 6%

p.a., find the interest received a year later.

3.5 Simple Interest

Interest received = $1 000 × 6%

= $60

Index

1A_Ch3(53)

Deposit $6 000 in a bank at an interest rate of 8% p.a.

for 2 years. What is the simple interest received?

3.5 Simple Interest

Simple interest received

=100

28000 6$

= $960

Index

1A_Ch3(54)

Simon wants to deposit $30 000 in a bank

for 2.5 years. Bank A offers an interest

rate of 8% p.a. and Bank B offers 7.5%

p.a. How much more simple interest will

Bank A give to Simon than Bank B?

Simple interest given by Bank A

3.5 Simple Interest

=100

5.28000 30$

= $6 000

Index

1A_Ch3(55)

Simple interest given by Bank B

3.5 Simple Interest

=100

5.25.7000 30$

= $5 625

The difference between the simple interests

∴ Bank A will give $375 more simple interest to Simon

than Bank B.


Find the simple interest.

= $(6 000 – 5 625)

= $375

Back to Question

Index

1A_Ch3(56)

Kimmy borrows some money from a bank to start her o

wn business. The interest rate is 5% p.a.

Find the amount of money she borrows

from the bank if she has to pay a total

of $30 000 as simple interest

after 2 years.

3.5 Simple Interest

Index

1A_Ch3(57)

Let $P be the amount of money Kimmy borrows, then

3.5 Simple Interest

30 000 = 100

25P

30 000 = 10P

P = 30 000 × 10

= 300 000

∴ The amount of money Kimmy borrows is $300 000.


Find the principal.

Back to Question

Key Concept 3.5.1

Index

1A_Ch3(58)

A bank pays simple interest for money deposited at an

interest rate of 10% p.a. If $5 000 is deposited in the bank,

find the amount received after 3 years.

3.5 Simple Interest

Amount received after 3 years

=

100310

1000 5$

= $6 500

Index

1A_Ch3(59)

A bank pays simple interest at an

interest rate of 8% p.a. If $25 000 is

deposited in the bank, find

3.5 Simple Interest

(a) the amount received after 10 months, correct to the nearest $100,

(b) the time required to get the amount $27 500.

(a) Amount received after 10 months

=

100

81000 25$

1012

= $26 700 , cor. to the nearest 100

Index

1A_Ch3(60)

3.5 Simple Interest

(b) Let T years be the time required, then

27 500 =

1008

125000T

1008

1T =

000 25500 27

1008 T

= 0.1

T =8

10

= 1.25

∴ The time required is 1.25 years, i.e. 1 year and 3 months.


Find the amount. Find the period of time.

Back to Question

Index

1A_Ch3(61)

John deposits $1 000 000 in a bank where simple interest

is calculated at a rate of R% p.a.

3.5 Simple Interest

(a) If the amount received by John doubles the principal

after 10 years, find the value of R.

(b) Suppose R = 8.

(i) Find the amount received by John after 20

years.

(ii) Peter borrows a sum of money from John

and the interest rate is 8% p.a. If the amount that

Peter has to pay back John after 2.5 years is

$120 000, how much does Peter borrow?

Soln

Soln

Index

1A_Ch3(62)

3.5 Simple Interest

(a) The principal is $1 000 000.

The amount received by John after 10 years

Hence 2 000 000 =

10010

1000 000 1R

2 = 1 + 0.1R

1 = 0.1R

∴ R = 10

= $1 000 000 × 2

= $2 000 000

Back to Question

Index

1A_Ch3(63)

3.5 Simple Interest

(b) (i) Amount received after 20 years=

100208

1000 000 1$

= $2 600 000

(ii) If $P is the sum of money that Peter borrows from John,

then120 000 =

1005.28

1P

120 000 = P(1 + 0.2)

P =2.1000 120

= 100 000

∴ Peter borrowed $100 000 from John.


Find the interest rate. Find the principal.

Key Concept 3.5.1

Back to Question

1a_ch3(1). 1a_ch3(2) 3.1simple problems involving percentages a using percentage to find a number b...

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simple problems

required percentage

original number

number of staff

number of people

percentage increaseindexa1a

awhat percentage

percentage changeindex1a