rate and ratio 1 1.1rate 1.2ratio 1.3applications of ratios case study chapter summary

Rate and Ratio1

1.1 Rate

1.2 Ratio

1.3 Applications of Ratios

Case Study

Chapter Summary

P. 2

Mandy is planning to study abroad next year. She wants to compare the school fees among 3 different countries.

The exchange rate is the amount of Hong Kong dollars needed to exchange for one unit of a foreign currency. We can use the exchange rate to convert the foreign school fees into Hong Kong dollars first, and then compare them.

Case Study

Country Annual School Fee Exchange Rate

Australia AUD 17 000 AUD 1 HKD 5.85

Great Britain GBP 8400 GBP 1 HKD 14.80

U.S.A. USD 20 000 USD 1 HKD 7.78

The table below shows the school fees in different currencies and their corresponding exchange rates.

P. 3

In the figure, the price of milk per carton is found by comparing 2 different kinds of quantities: ‘the total price’ and ‘the number of cartons’.

Rate is the comparison of 2 quantities of different kinds.

When expressing the relationship in rate, we express the amount of one quantity as per unit of the other quantity, by the symbol ‘/’.

For example, the price of each brand of milk can be expressed as ‘$8/carton’ and ‘$7/carton’ respectively.

1.1 Rate

‘/’ means per.

P. 4

Example 1.1T

1.1 Rate

Solution:

Andy works as a programmer in a computer company. He earns a total of $51 000 in half a year. Find his income in the following units:(a) $/month (b) $/year

incomeMonthly (a)

incomeYearly (b)

/month8500$months 6

000 51$

000/year $102/year)128500$(

P. 5

Betty runs at a speed of 1.5 m/s.(a) Express her speed in the unit km/h.(b) How long does she take to run 1350 m? (Give the answer in minutes.)(c) How far can she run in 8 minutes? (Give the answer in m.)

(a) 1.5 m (1.5 1000) km 0.0015 km

hours 3600

1

minutes60

1s 1

h 3600

1km 0015.0Speed

Example 1.2T

1.1 Rate

Solution:

5.4 km/h

(b) Time required (1350 1.5) s 900 s (900 60)

minutes 15 minutes

(c) Distance run 1.5 (8 60) m 720 m

P. 6

Mrs. Wong gets 240 Euros (EU) for HKD 2386.(a) Find the exchange rate in the unit HKD/EU.(b) How much Hong Kong dollars can she get with 850 Euros?(Give the answers correct to 2 decimal places.)

(cor. to 2 d. p.)

Example 1.3T

1.1 Rate

Solution:(a) Exchange rate HKD 2386 EU 240

9.9417 HKD/EU 9.94 HKD/EU

(b) Amount of Hong Kong dollars she can get $(850 9.941

7) (cor. to 2 d. p.) $8450.45 For higher accuracy, we use 9.9417 as the exchange rate in the calculation in part (b).

P. 7

1.2 Ratio

In the figure, a fruit punch is mixed by adding a cup of soft drink into 2 cups of orange juice.

That means, the volume of orange juice in the fruit punch is always twice that of the soft drink.

We compare 2 quantities by division: ‘the volume of orange juice’ and ‘the volume of soft drink’ and these quantities are of the same kind.

Ratio is the comparison of quantities of the same kind. The ratio of

a to b is usually expressed as a : b or (where a 0 and b 0). b

a

A. Basic Concepts of Ratio

We say that the ratio of the volume of orange juice to that of the

soft drink is 2 : 1. This can be also written in the form .1

2

P. 8

1.2 Ratio

A ratio is usually expressed in its simplest form, e.g. 75 : 40 15 : 8.

A ratio can be written as a fraction and we know that the value of the fraction remains unchanged when we multiply (or divide) both the numerator and the denominator by the same non-zero number.

For example, 0.75 m : 40 cm 75 cm : 40 cm

40

75


15 : 88

15

P. 9

1.2 Ratio

If m : 4 (m 6) : 12, find the value of m.


Example 1.4T

Solution:m : 4 (m 6) : 12

12

6

4

mm

12m 4(m 6) 3m m 6 2m 6 m 3

P. 10

1.2 Ratio

Peter has 18 coins. Nancy has 6 more coins than Peter, and she has twice as many as Stella. Find the ratio of(a) Peter’s coins to Nancy’s coins,(b) Stella’s coins to Peter’s coins.


Example 1.5T

Solution:

Note that a : b b : a.

Number of coins that Nancy has 18 6

(a) Required ratio 18 : 24

(b) Required ratio 12 : 18

3 : 4 2 : 3

24Number of coins that Stella has 24 2

12

P. 11

1.2 Ratio

mL 13

6650

mL 13

7650

Since the volumes of juice are in the ratio 6 : 7, we can imagine that the bottle of apple juice is divided into (6 7) 13 equal parts.


Example 1.6TEmily bought a bottle of apple juice of volume 650 mL. She pours the juice into 2 cups such that the volumes of juice in these cups are in the ratio 6 : 7. Find the volume of juice in these 2 cups.

Solution:

Volume of the cup with less juice

mL 300

Volume of the cup with more juice

mL 350

P. 12

1.2 Ratio

Alternative Solution:

mL 13

6650

mL )300650(

)76(

6

bottle of Volume

juice less with cup theof Volume

If we divide the juice into 13 parts, then 6 parts belong to the cup with less juice and the other 7 parts belong to the cup with more juice.


Example 1.6T

13

6

Volume of the cup with less juice mL 300

Volume of the cup with more juice mL 350

We can compare the ratios directly, without finding the exact value of each small part.

Emily bought a bottle of apple juice of volume 650 mL. She pours the juice into 2 cups such that the volumes of juice in these cups are in the ratio 6 : 7. Find the volume of juice in these 2 cups.

P. 13

1.2 Ratio

(a) Number of male teachers : Number of female teachers 27 : (57 27)

Educational Secondary School has a total of 57 teachers, of which 27 of them are male teachers.(a) Find the ratio of the number of male teachers to the number of

female teachers.


Example 1.7T

10:9 27 :

30

Solution:

P. 14

1.2 Ratio

7

6

30

35

x

x

∴ 5 female teachers has been hired.


Example 1.7T

(b) Let x be the number of female teachers hired.Number of male teachers hired 8 x.

Educational Secondary School has a total of 57 teachers, of which 27 of them are male teachers.(b) The principal has just hired 8 new teachers. The ratio of male

teachers to female teachers now becomes 6 : 7. How manyfemale teachers has the principal hired?

[27 (8 x)] : (30 x) 6 : 7

6(30 x) 7(35 x)

6(30 x) 7(35 x) 180 6x 245 7x 13x 65 x 5

Solution:

P. 15

1.2 Ratio

We can also use ratio to compare 3 or more quantities of the same kind.

For example, the expression

a : b : c 4 : 5 : 9

compares the 3 quantities a, b and c, with

a : b 4 : 5, b : c 5 : 9 and a : c 4 : 9.

Such an expression is called a continued ratio.

For 3 quantities given, if we only know the ratio between individual quantities, we can rewrite the ratios into a continued ratio.

Continued ratios can only be expressed in the form a : b : c, but not in a fraction.

B. Continued Ratio

P. 16

1.2 Ratio

If 3a 5b 4c, find the ratio a : b : c.

B. Continued Ratio

Example 1.8T

Solution:Since 3a 5b 4c, we have 3a 5b and 5b 4c.

3

5b

a

5

4c

b∴ and

∴ a : b 5 : 3 and b : c 4 : 5

1. First, find the ratios a : b and b : c.

2. Then, make the common terms equal in both ratios.

15:12:20:: cba

a : b 5 :3b : c 4 : 5

20 :12

12 : 15

5 4 : 3 4 4 3 : 5 3

P. 17

1.2 Ratio

There are 540 seats in a plane. The number of economy class seats and business class seats are in the ratio 12 : 1. The number of business class seats and first class seats are in the ratio 2 : 1.(a) Find the ratio of the number of economy class seats : the number

of business class seats : the number of first class seats.(b) Find the number of first class seats.

B. Continued Ratio

Example 1.9T

Solution:(a) Economy : Business 12 :1

Business : First 2 : 1 24 :

2 2 : 1

Required ratio 24 : 2 : 1

P. 18

1.2 Ratio

Number of first class seats 27

1540

B. Continued Ratio

Example 1.9TThere are 540 seats in a plane. The number of economy class seats and business class seats are in the ratio 12 : 1. The number of business class seats and first class seats are in the ratio 2 : 1.(a) Find the ratio of the number of economy class seats : the number

of business class seats : the number of first class seats.(b) Find the number of first class seats.

Solution:(b) We can imagine that the total number of seats can be divided

into (24 2 1) 27 equal parts.

20

P. 19

A. Similar Figures


If we compare the lengths of the corresponding sides in the 2 photos, we will have:

In general, similar figures have the following property:

I photo ofLength

II photo ofLength

I photo ofHeight

II photo ofHeight

If 2 figures have the same shape but their sizes are not the same, then the 2 figures are said to be similar.

For 2 similar figures, the ratios of the corresponding sides are always the same.

P. 20

In the figure, the 2 parallelograms are similar to each other. Find x and y.

x

5

3

6

6x 15

83

6 y


A. Similar Figures

Example 1.10T

3y 48

x 2.5 (m)

y 16 (m)

Solution:

When finding the side lengths of similar figures, we should identify which of them are the corresponding sides.

P. 21

In the figure, a boy with a height of 1.8 m stands in front of the tree. Assume that ABC and DEF are similar triangles. What is the length of his shadow?

Let y m be the length of his shadow, i.e., EF y m.

y

3.1

8.1

2.5

∴ The length of his shadow is 0.45 m.


A. Similar Figures

Example 1.11T

Solution:

45.0y

34.22.5 y

P. 22

If we want to draw something which is very large or small in size, such as a country or an insect, we need to reduce or enlarge it according to a specified ratio in a diagram.

This kind of drawing is called scale drawing.

When using scale drawing, we need to specify the ratio in which the object is enlarged or reduced in the picture.

This ratio is called the scale of the drawing, and is usually represented in the form 1 : n or n : 1.


B. Scaling

Note that 1 : n n : 1.

P. 23

For example, the map of Hong Kong Island shown has a scale of 1 : 1 500 000.

This means that a length of 1 cm on the map represents an actual length of 1 500 000 cm.

In the figure, a length of 1 cm on the figure represents an actual length of 0.2 cm.

Thus the scale is 1 : 0.2, i.e., 5 : 1.

B. Scaling

We can also express the scale in the form1 cm : 15 km.


P. 24

The picture on the right shows the top view of a tennis court of actual length 36 m. If the length of the picture is 4.8 cm, find the scale of the picture.

Scale of the picture 4.8 cm : 36 m

750:1

B. Scaling

Example 1.12T

Solution:

4.8 cm : 3600 cm

3600

8.4

750

1


P. 25

Consider a map of a city with a scale of 1 : 20 000. If the distance between 2 buildings is 3.2 cm, find the actual distance between them. Give the answer in the unit of km.

B. Scaling

Example 1.13T

Solution:Actual distance (3.2 20 000) cm 64 000 cm 640 m

0.64 km


P. 26

According to the floor plan, find the ratio of the actual area of the master bedroom to the actual area of the kitchen.(Hint: Assume the scale of the floor plan to be 1 cm : n m.)


B. Scaling

Example 1.14T

Solution:Actual side length of the master bedroom (2.5 n) m

∴ The required ratio (2.5n 2.5n) m2 : (2n 1.5n) m2

2.5n mSimilarly, the actual length and the actual width of the kitchen are 2n m and 1.5n m respectively.

6.25 : 3 25 : 12

P. 27

Chapter Summary

1.1 Rate

Rate is the comparison of 2 quantities of different kinds.

P. 28

Chapter Summary

1.2 Ratio

2. If the ratios a : b and b : c are given, we can find the continued ratio a : b : c by finding the L.C.M. of the values corresponding to the common term b.

1. Ratio is the comparison of quantities of the same kind. The ratio of

a to b is usually expressed as a : b or (where a 0 and b 0).b

a

P. 29

Chapter Summary


1. Similar figures For 2 similar figures, the ratios of the corresponding sides are

always the same.

2. Scale drawing If we reduce or enlarge the drawing of the real object by a certain

scale, the drawing is similar to the original object.

rate and ratio 1 1.1rate 1.2ratio 1.3applications of ratios case study chapter summary

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