7 Ratio, Proportion and Rate
C H A P T ER
Topics
Level
1 Ratio
Key words ratio, cancel, simplest form, map scale
CORE
2 Increases and decreases using ratios
increase, decrease
EXTENDED
3 Speed
CORE
average, speed, distance, time
4 Rates
CORE
rate
5 Direct proportion
CORE
unitary method, direct proportion, single unit value
6 Inverse proportion
CORE
inverse proportion
In this chapter you will learn how to: CORE ●
Demonstrate an understanding of: ●
ratio
●
direct and inverse proportion. (C1.11 and E1.11)
●
Use common measures of rate. (C1.11 and E1.11)
●
Divide a quantity in a given ratio. (C1.11 and E1.11)
●
Use ratio and scales in practical situations. (C1.11 and E1.11)
●
Calculate average speed. (C1.11 and E1.11)
EXTENDED ●
Increase and decrease a quantity by a given ratio. (E1.11)
Why this chapter matters We use ratio, proportion and speed in our everyday lives to help us compare two or more pieces of information. Ratio and proportions are often used to compare sizes; speed is used to compare distances with the time taken to travel them.
Speed
A 100-m sprinter
When is a speed fast? On 16 August 2009 Usain Bolt set a new world record for the 100-m sprint of 9.58 seconds. This is an average speed of 37.6 km/h. The sailfish is the fastest fish and can swim at 110 km/h. The cheetah is the fastest land animal and can travel at 121 km/h.
Sailfish
The fastest bird is the swift which can travel at 170 km/h.
Ratio and proportion facts •
Russia is the largest country.
•
Vatican City is the smallest country.
•
The area of Russia is nearly 39 million times the area of Vatican City.
•
Monaco has the most people per square mile.
•
Mongolia has the least people per square mile.
•
The number of people per square mile in Monaco to the number of people in Mongolia is in the ratio 10 800 : 1.
•
Japan has the highest life expectancy.
•
Sierra Leone has the lowest life expectancy.
•
On average people in Japan live over twice as long as people in Sierra Leone.
•
Taiwan has the most mobile phones per 100 people (106.5).
•
This is approximately four times that of Thailand (26.04).
Cheetah
Swift
Russia England
Monaco
Mongolia
Japan
Vatican City Taiwan Thailand Sierra Leone
Chapter 7: Ratio, Proportion and Rate
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Chapter 7 . Topic 1
7.1 Ratio A ratio is a way of comparing the sizes of two or more quantities. A ratio can be expressed in a number of ways. For example, if Tasnim is five years old and Ziad is 20 years old, the ratio of their ages is:
Tasnim’s age : Ziad’s age
which is:
5 : 20
which simplifies to:
1 : 4 (dividing both sides by 5)
A ratio is usually given in one of these three ways.
Tasnim’s age : Ziad’s age
Tasnim’s age to Ziad’s age or 5 to 20 or 1 to 4 Tasnim’s age 5 or 1 or Ziad’s age 20 4
or
5 : 20
or
1 : 4
Common units When working with a ratio involving different units, always convert them to the same units. A ratio can be simplified only when the units of each quantity are the same, because the ratio itself has no units. Once the units are the same, the ratio can be simplified or cancelled like a fraction. For example, you must convert the ratio of 125 g to 2 kg to the ratio of 125 g to 2000 g, so that you can simplify it.
125 : 2000
Divide both sides by 25: 5 : 80
Divide both sides by 5:
1 : 16 The ratio 125 : 2000 can be simplified to 1 : 16.
Example 1 Express 25 minutes : 1 hour as a ratio in its simplest form. The units must be the same, so change 1 hour into 60 minutes. 25 minutes : 1 hour = 25 minutes : 60 minutes
=
25 : 60
=
5 : 12
So, 25 minutes : 1 hour simplifies to 5 : 12
4
7.1 Ratio
Cancel the units (minutes) Divide both sides by 5
7.1 Ratios as fractions You can express ratio in its simplest form as portions of a quantity by expressing the whole numbers in the ratio as fractions with the same denominator (bottom number).
Example 2 A garden is divided into lawn and shrubs in the ratio 3 : 2. What fraction of the garden is covered by: a lawn b shrubs? You find the denominator (bottom number) of the fraction by adding the numbers in the ratio (that is, 2 + 3 = 5). a The lawn covers b The shrubs cover
3 5
2 5
of the garden. of the garden.
Exercise 7A CORE
1 Express each of these ratios in its simplest form. a 6 : 18 b 5 : 20 c 16 : 24 d 24 : 12
e 20 : 50 f 12 : 30 g 25 : 40 h 150 : 30
2 Write each of these ratios of quantities in its simplest form. (Remember that you must always express both parts in a common unit before you simplify.) a 40 minutes : 5 minutes b 3 kg : 250 g c 50 minutes to 1 hour d 1 hour to 1 day
e 12 cm to 2.5 mm f 1.25 kg : 500 g g 75 cents : $2 h 400 m: 2 km
3 A length of wood is cut into two pieces in the ratio 3 : 7. What fraction of the original length is the longer piece?
4 Tareq and Hassan find a bag of marbles that they share between them in the ratio of their ages. Tareq is 10 years old and Hassan is 15 years old. What fraction of the marbles did Tareq get? 5 Mona and Petra share a pizza in the ratio 2 : 3. They eat it all. a What fraction of the pizza did Mona eat? b What fraction of the pizza did Petra eat?
Chapter 7: Ratio, proportion and rate
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CORE
6 A camp site allocates space to caravans and tents in the ratio 7 : 3. What fraction of the total space is given to:
a the caravans b the tents?
7 In a safari park at feeding time, the elephants, the lions and the chimpanzees are given food in the ratio 10 to 7 to 3. What fraction of the total food is given to: a the elephants
b the lions c the chimpanzees?
8 Paula wins three-quarters of her tennis matches. She loses the rest.
What is the ratio of wins to losses?
9 Three brothers share some cash.
The ratio of Marco’s and Dani’s share is 1 : 2.
The ratio of Dani’s and Paulo’s share is 1 : 2.
What is the ratio of Marco’s share to Paulo’s share?
Dividing amounts in a given ratio To divide an amount in a given ratio, you first look at the ratio to see how many parts there are altogether. For example, the ratio 4 : 3 has 4 parts and 3 parts giving 7 parts altogether.
7 parts is the whole amount.
1 part can then be found by dividing the whole amount by 7.
3 parts and 4 parts can then be calculated from 1 part.
Example 3 Divide $28 in the ratio 4 : 3 4 + 3 = 7 parts altogether. So 7 parts = $28. Dividing by 7:
1 part = $4
4 parts = 4 × $4 = $16 and 3 parts = 3 × $4 = $12
So $28 divided in the ratio 4 : 3 is $16 : $12
6
7.1 Ratio
7.1 Map scales Map scales are often given as ratios in the form 1 : n.
Example 4 A map of New Zealand has a scale of 1 : 900 000. The distance on the map from Auckland to Hamilton is 11.5 centimetres. What is the actual distance? 1 cm on the map = 900 000 centimetres on the ground.
= 9000 metres (100 centimetres = 1 metre)
= 9 kilometres (1000 metres = 1 kilometre).
The distance is 11.5 Ă— 9 kilometres = 103.5 kilometres.
Exercise 7B a 400 g in the ratio 2 : 3 b 280 kg in the ratio 2 : 5
CORE
1 Divide each according to the given ratio. c 500 in the ratio 3 : 7 d 1 km in the ratio 19 : 1 e 5 hours in the ratio 7 : 5 f $100 in the ratio 2 : 3 : 5
g $240 in the ratio 3 : 5 : 12 h 600 g in the ratio 1 : 5 : 6 i $5 in the ratio 7 : 10 : 8 j 200 kg in the ratio 15 : 9 : 1
2 The ratio of female to male members of a sports club is 7 : 3. The total number of members of the group is 250.
a How many members are female? b What percentage of members are male?
3 A store sells small and large TV sets.
The ratio of small : large is 2 : 3.
The total stock is 70 sets.
a How many small sets are in stock? b How many large sets are in stock?
4 When a supermarket checked a total of 357 confectionery products for sugar content, they found the ratio of products without sugar to those with sugar was 1 : 16.
How many of those products contained no sugar?
Chapter 7: Ratio, proportion and rate
7
CORE
5 Joshua, Aicha and Mariam invest $10 000 in a company.
The ratio of the amounts they invest is:
Joshua : Aicha : Mariam = 5 : 7 : 8
How much does each of them invest?
6 Rewrite each of these scales as a ratio in the form 1 : n. a 1 cm to 4 km b 4 cm to 5 km c 2 cm to 5 km
d 4 cm to 1 km e 5 cm to 1 km f 2.5 cm to 1 km g 8 cm to 5 km h 10 cm to 1 km i 5 cm to 3 km
7 A map has a scale of 1 cm to 10 km. a Rewrite the scale as a ratio in its simplest form.
b What is the actual length of a lake that is 4.7 cm long on the map?
Advice and Tips 1 km
= 1000 m
= 100 000 cm
c How long will a road be on the map if its actual length is 8 km?
8 A map has a scale of 2 cm to 5 km. a Rewrite the scale as a ratio in its simplest form.
b How long is a path that measures 0.8 cm on the map? c How long should a 12 km road be on the map?
9 The scale of a map is 5 cm to 1 km. a Rewrite the scale as a ratio in the form 1 : n.
b How long is a wall that is shown as 2.7 cm on the map? c The distance between two points is 8 km; how far will this be on the map?
10 You can simplify a ratio by changing it into the form 1 : n. For example, 5 : 7 can be rewritten as: 5 : 7 = 1 : 1.4 5 5 Rewrite each of these ratios in the form 1 : n. a 5 : 8 b 4 : 13 c 8:9 d 25 : 36 e 5 : 27 f 12 : 18 g 5 hours : 1 day h 4 hours : 1 week i £4 : £5
8
7.1 Ratio
7.1 Calculating with ratios when only part of the information is known Example 5 A fruit drink is made by mixing orange squash with water in the ratio 2 : 3. How much water needs to be added to 5 litres of orange squash to make the drink? 2 parts is 5 litres. Dividing by 2:
1 part is 2.5 litres
3 parts = 2.5 litres × 3 = 7.5 litres
So 7.5 litres of water is needed to make the drink.
Example 6 Two business partners, Lubna and Adama, divided their total profit in the ratio 3 : 5. Lubna received $2100. How much did Adama get? Lubna’s $2100 was 1 8
3 8
of the total profit. (Check that you know why.)
of the total profit = $2100 ÷ 3 = $700
So Adama’s share, which was 58 , amounted to $700 × 5 = $3500.
Exercise 7C CORE
1 Sean, aged 15, and Ricki, aged 10, shared some sweets in the same ratio as their ages. Sean had 48 sweets. a Simplify the ratio of their ages.
b How many sweets did Ricki have? c How many sweets did they share altogether?
2 A blend of tea is made by mixing Lapsang with Assam in the ratio 3 : 5. I have a lot of Assam tea but only 600 g of Lapsang. How much Assam do I need to make the blend using all the Lapsang?
3 The ratio of male to female spectators at a hockey game is 4 : 5. 4500 men watched the match. What was the total attendance at the game? 4 A teacher always arranged the content of each of his lessons as ‘teaching’ and ‘practising learnt skills’ in the ratio 2 : 3. a If a lesson lasted 35 minutes, how much teaching would he do? b If he decided to teach for 30 minutes, how long would the lesson need to be?
Chapter 7: Ratio, proportion and rate
9
CORE
Chapter 7 . Topic 2 5 A ‘good’ children’s book has pictures and text in the ratio 17 : 8. In a book I have just looked at, the pictures occupy 23 pages.
a Approximately how many pages of text should this book have to be a ‘good’ children’s book? b What percentage of a ‘good’ children’s book will be text?
6 Three business partners, Ren, Shota and Fatima, put money into a business in the ratio 3 : 4 : 5. They shared any profits in the same ratio. Last year, Fatima received $3400 from the profits. How much did Ren and Shota receive last year? 7 a Iqra is making a drink from lemonade, orange and ginger ale in the ratio 40 : 9 : 1. If Iqra has only 4.5 litres of orange, how much of the other two ingredients does she need to make the drink? b Another drink made from lemonade, orange and ginger ale uses the ratio 10 : 2 : 1.
Which drink has a larger proportion of ginger ale, Iqra’s or this one? Show how you work out your answer.
8 There is a group of boys and girls waiting for school buses. 25 girls get on the first bus. The ratio of boys to girls at the stop is now 3 : 2. 15 boys get on the second bus. There are now the same number of boys and girls at the bus stop. How many students altogether were originally at the bus stop? 9 A jar contains 100 cm3 of a mixture of oil and water in the ratio 1 : 4. Enough oil is added to make the ratio of oil to water 1 : 2. How much water must be added to make the ratio of oil to water 1 : 3?
7.2 Increases and decreases using ratios Sometimes increases and decreases can be expressed in terms of ratios. Suppose a recipe for six people requires 450 g of flour. How much is needed for 10 people? You need to increase 450 g in the ratio 10 : 6 = 5 : 3. Think of 450 as 3 parts and you need to find 5 parts. You need to find 450 ×
5 3
5 3
of 450 g.
= 750 g
If the recipe was to be changed to feed four people you would need to decrease the amount of flour in the ratio 4 : 6 = 2 : 3. You need to find 450 ×
10
2 3
2 3
of 450 g.
= 300 g
7.2 Increases and decreases using ratios
E
Chapter 7 . Topic 3
Exercise 7D
EXTENDED
1 Increase 200 in each ratio. a 3 : 1 b 3 : 2 c 10 : 1 d 7 : 4 e 6 : 5 f 11 : 10
2 Decrease 80 in each ratio a 1 : 4 b 3 : 4 c 1 : 10 d 7 : 10 e 1 : 5 f 4:5
3 A projector enlarges an image in the ratio 20 : 1.
a What will be the size of an enlargement of a picture that is 8 cm by 6 cm? b What will be the size of the image if the ratio is changed to 15 : 2?
4 A photocopier enlarges a photograph in the ratio 5 : 4.
a What is the size of the enlargement of a photograph that is 10 cm by 12 cm? b When the setting of the photocopier is changed, the size of the enlargement is 15 cm by 18 cm. Write the setting as a ratio, as simply as possible.
5 A photograph measures 12 cm by 20 cm.
It is made smaller in the ratio 3 : 4.
a What are the dimensions of the new photograph? b The new photograph is again made smaller in the ratio 3 : 4. What are the dimensions now?
6 $5000 is invested and after a year the value has increased in the ratio 3 : 2. a What is the value of the investment now? b What is the percentage increase over one year? The value of the investment continues to grow and in the second year it increases in the ratio 5 : 4. c What is the value after two years? d What is the percentage increase in the second year?
e What is the overall percentage increase over two years? f Show that the overall increase could be written as a ratio as 15 : 8.
7 Prices are going up in the ratio 6 : 5. a Show that this is a 20% increase. b Write a 10% increase as a ratio. c Write a 10% decrease as a ratio.
7.3 Speed The relationship between speed, time and distance can be expressed in three ways: distance distance distance = speed × time time = speed = speed time Chapter 7: Ratio, proportion and rate
11
In problems relating to speed, you usually mean average speed, as it would be unusual to maintain one exact speed for the whole of a journey. This diagram will help you remember the relationships between distance (D), time (T) and speed (S). D S
D = S × T S = T
D D T = T S
Units for speed include km/h (kilometres per hour, or ‘the number of kilometres travelled in an hour’) and m/s (metres per second).
Example 7 Paula drove a distance of 270 kilometres in 5 hours. What was her average speed? Paula’s average speed = distance she drove = 270 = 54 kilometres per hour (km/h) 5 time she took
Example 8 Renata drove from her home to Frankfurt in 3 21 hours at an average speed of 60 km/h. How far is it from Renata’s home to Frankfurt? Since: distance = speed × time the distance from Renata’s home to Frankfurt is given by: 60 × 3.5 = 210 kilometres Note: You need to change the time to a decimal number and use 3.5 (not 3.30).
Example 9 Maria is going to drive to Rome, a distance of 190 kilometres. She estimates that she will drive at an average speed of 50 km/h. How long will it take her? distance she covers 190 = = 3.8 hours her average speed 50 Change the 0.8 hour to minutes by multiplying by 60, to give 48 minutes.
Maria’s time =
So, the time taken for Maria’s journey will be 3 hours 48 minutes.
Remember: When you calculate a time and get a decimal answer, as in Example 9, do not mistake the decimal part for minutes. You must either: • leave the time as a decimal number and give the unit as hours, or • change the decimal part to minutes by multiplying it by 60 (1 hour = 60 minutes) and give the answer in hours and minutes.
12
7.3 Speed
7.3
Exercise 7E
Advice and Tips
What was her average speed?
2 How far along a road would you travel if you drove at 110 km/h for 4 hours?
3 I can drive from my home to see my aunt in about 6 hours. The distance is 315 kilometres.
Remember to convert time to a decimal if you are using a calculator, for example, 8 hours 30 minutes is 8.5 hours.
What is my average speed?
4 The distance from Leeds to London is 350 kilometres.
CORE
1 A cyclist travels a distance of 90 kilometres in 5 hours.
Advice and Tips
The train travels at an average speed of 150 km/h.
km/h means kilometres per hour.
If I catch the 9.30 am train in London, at what time should I expect to arrive in Leeds?
5 How long will an athlete take to run 2000 metres at an average speed of 4 metres per second?
m/s means metres per second.
6 Copy and complete this table. Distance travelled
Time taken
a
150 km
2 hours
b
260 km
40 km/h
c
5 hours
35 km/h
d
3 hours
80 km/h
e
544 km
f
Average speed
g
8 hours 30 minutes 3 hours 15 minutes
215 km
100 km/h 50 km/h
7 Eliot drove a distance of 660 kilometres, in 7 hours 45 minutes.
a Change the time 7 hours 45 minutes to a decimal. b What was the average speed of the journey? Round your answer to 1 decimal place.
8 Johan drives home from his son’s house in 2 hours 15 minutes. He says that he drives at an average speed of 70 km/h.
a Change the 2 hours 15 minutes to a decimal. b How far is it from Johan ’s home to his son’s house?
9 The distance between Paris and Le Mans is 200 km. The express train between Paris and Le Mans travels at an average speed of 160 km/h. a Calculate the time taken for the journey from Paris to Le Mans, giving your answer as a decimal number of hours. b Change your answer to part a to hours and minutes.
Chapter 7: Ratio, proportion and rate
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