Contents 1
Introduction
4
Percentages
6
1.1 1.2 1.3 1.4
Simple interest Percentage increases and decreases Calculating the original value Repeated percentage changes Ready to progress? Review questions Challenge Exponential growth
2
22
Equations and formulae
24
Multiplying out brackets Factorising algebraic expressions Expressions with several variables Equations with fractions Ready to progress? Review questions Investigation Body mass index
25 26 28 31 34 34
2.1 2.2 2.3 2.4
3
Polygons
3.1 3.2
Properties of polygons Interior and exterior angles of regular polygons 3.3 Tessellations and regular polygons Ready to progress? Review questions Mathematical reasoning Semi-regular tessellations
4
Using data
4.1 4.2 4.3
Scatter graphs and correlation Two-way tables Estimation of a mean from grouped data 4.4 Cumulative frequency diagrams 4.5 Statistical investigations Ready to progress? Review questions Challenge Census
2
7 10 14 17 20 20
36 38 39 44 46 48 48 50 52 53 59 62 65 70 72 72 74
5
Applications of graphs
5.1 5.2 5.3
Step graphs Time graphs Exponential growth graphs Ready to progress? Review questions Problem solving Mobile phone tariffs
6
Pythagoras’ theorem
6.1 6.2
Introducing Pythagoras’ theorem Using Pythagoras’ theorem to solve problems 6.3 The converse of Pythagoras’ theorem Ready to progress? Review questions Activity Practical Pythagoras
7
Fractions
7.1 7.2
Adding and subtracting fractions Multiplying fractions and mixed numbers 7.3 Dividing fractions and mixed numbers 7.4 Algebraic fractions Ready to progress? Review questions Investigation Fractions from one to six
8
Algebra
76 77 82 87 92 92 94 96 97 101 104 106 106 108 110 111 113 115 118 120 120 122 124
8.1
Expanding the product of two brackets 8.2 Expanding expressions with more than two brackets 8.3 Factorising quadratic expressions with positive coefficients 8.4 Factorising quadratic expressions with negative coefficients 8.5 The difference of two squares Ready to progress? Review questions Challenge Graphs from expressions
125 128 130 132 134 136 136 138
Contents
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9
Decimal numbers
140
Powers of 10 Standard form Multiplying with numbers in standard form 9.4 Dividing with numbers in standard form 9.5 Upper and lower bounds Ready to progress? Review questions Mathematical reasoning To the stars and back
141 143
9.1 9.2 9.3
10 Surface area and volume of cylinders 10.1 10.2 10.3
Volume of a cylinder Surface area of a cylinder Composite shapes Ready to progress? Review questions Problem solving Packaging soup
11 Solving equations graphically
147 149 151 154 154 156 158 159 162 166 170 170 172 174
11.1
Graphs from equations in the form ay Âą bx = c 11.2 Solving simultaneous equations by drawing graphs 11.3 Solving quadratic equations by drawing graphs 11.4 Solving cubic equations by drawing graphs Ready to progress? Review questions Challenge Maximum packages
175
12 Compound units 12.1 12.2 12.3
Speed More compound units Unit costs Ready to progress? Review questions Challenge Population density
181 184 184
189 193 196 200 200 202
13 Right-angled triangles
204
13.1
Introduction to trigonometric ratios 13.2 How to find trigonometric ratios of angles 13.3 Using trigonometric ratios to find angles 13.4 Using trigonometric ratios to find lengths Ready to progress? Review questions Investigation Barnes Wallis and the bouncing bomb 14.1 14.2
Practice Revision GCSE-type questions
207 210 215 218 218
220 223 232 239
Glossary
242
Index
245
186
Contents
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205
14 Revision and GCSE preparation 222
177 179
188
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How to use this book Learning objectives See what you are going to cover and what you should already know at the start of each chapter. About this chapter Find out the history of the maths you are going to learn and how it is used in real-life contexts. Key words The main terms used are listed at the start of each topic and highlighted in the text the first time they come up, helping you to master the terminology you need to express yourself fluently about maths. Definitions are provided in the glossary at the back of the book. Worked examples Understand the topic before you start the exercises, by reading the examples in blue boxes. These take you through how to answer a question step by step. Skills focus Practise your problem-solving, mathematical reasoning and financial skills.
PS
FS
Take it further Stretch your thinking by working through the Investigation, Problem solving, Challenge and Activity sections. By tackling these you are working at a higher level.
4
How to use this book
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Progress indicators Track your progress with indicators that show the difficulty level of each question.
Ready to progress? Check whether you have achieved the expected level of progress in each chapter. The statements show you what you need to know and how you can improve. Review questions The review questions bring together what you’ve learnt in this and earlier chapters, helping you to develop your mathematical fluency.
Activity pages Put maths into context with these colourful pages showing real-world situations involving maths. You are practising your problem-solving, reasoning and financial skills.
Interactive book, digital resources and videos A digital version of this Pupil Book is available, with interactive classroom and homework activities, assessments, worked examples and tools that have been specially developed to help you improve your maths skills. Also included are engaging video clips that explain essential concepts, and exciting real-life videos and images that bring to life the awe and wonder of maths. Find out more at www.collins.co.uk/connect
How to use this book
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1
Percentages
This chapter is going to show you: • how to calculate simple interest • how to use a multiplier to calculate percentage increases and decreases • how to calculate the original value after a percentage change • how to calculate the result of repeated percentage changes.
You should already know: • how to work out a percentage of a given number, with or without a calculator • how to write one number as a percentage of another number.
About this chapter Banks are in business to offer financial services, from which they can make profits. A lot of their income is generated through lending money to customers. When someone borrows money, they will have to pay it back, after an agreed time. Meanwhile, they have to pay the lender for the use of the money. They do this by paying interest, which is a percentage of the amount they have borrowed. The interest payments may be made at agreed time intervals, such as monthly, every three months or yearly. The most basic form is simple interest, which is one of the mathematical ideas you will learn about in this chapter. To understand it, though, you need to understand how percentages work.
6
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1.1 Simple interest Learning objectives
Key words
• To understand what is meant by simple interest
lender
• To solve problems involving simple interest
simple interest
If you take out a loan you usually have to pay interest to the lender. This is the person or organisation lending you the money. One type of interest is called simple interest. This is expressed as a percentage. As long as you still have the loan, you will pay the lender the percentage of the loan, at regular intervals.
Example 1 Tina takes a loan of £650. She agrees to pay simple interest of 5% every three months. How much interest will she pay in one year? The interest payment is 5% of £650. 5% of 650 = 0.05 × 650 = 32.5 She pays £32.50 four times in one year.
1 1 × 650 = 32.5 since 5% = 20 20 One year = 12 months or
She pays £32.50 × 4 = £130. Remember as well as paying the interest she will also have to pay back the value of the loan, which is £650 in this case.
Example 2 Wayne takes out a loan of £8290 to buy a car. He pays simple interest of 1.6% per month. Calculate amount of interest he pays in three years. He makes 36 payments of 1.6%. 8290 × 0.016 × 36 = 4775.04
3 years = 36 months 1.6% = 0.016
He pays £4775.04 interest over the three years.
Example 3 Lucy has a loan of £450. She pays interest of £10.80 per month. What is the rate of simple interest? 10.8 Divide the interest by the amount of the loan. = 0.024 450 0.024 × 100 = 2.4% Multiply the decimal by 100 to change it to a percentage.
If the numbers are straightforward you do not need a calculator. You should know how to use a calculator to work out percentages if you need to. 1.1 Simple interest
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Exercise 1A FS
FS
FS
1
2
3
Work out these percentages. Do not use a calculator. a 25% of £300 b 10% of £45 c 40% of £1000
d 75% of £600
e
h 150% of £80
30% of £2000
f
13% of £300
g
82% of £200
Use a calculator to work out these percentages. a 13% of £85 b 7% of £425 c 23% of £6500
d 3.5% of £230
e
h 0.5% of £325
1.4% of £620
f
38% of £560
g
3.8% of £1320
Sam takes a loan of £800. He pays simple interest of 3% per month for six months. Work out the total amount of interest Sam pays.
FS
4
Haritha pays 2.7% monthly interest on a loan of £5600. He pays simple interest for one year. Work out the total interest he has paid after one year.
FS
5
Jenny takes a loan of £6400. She pays 4.2% simple interest monthly, for two years, and then she pays back the loan. Calculate how much interest she pays altogether.
FS
6
Aaron has a loan of £1250. He pays simple interest of 0.5% per month for eight months. a
Calculate the total amount of interest Aaron pays.
b Show that the total interest is 4% of the original loan.
FS
7
Farouk pays 9% per year on a loan of £25 000. He pays interest for six years. a
Work out how much interest he pays altogether.
b What percentage of the loan is the total interest?
FS
8
Sania pays 0.5% per week interest on a loan of £300. a
How much interest has she paid after five weeks?
b How much interest has she paid after eight weeks? c
After a certain number of weeks she has paid £78 interest. How many weeks is this?
8
1 Percentages
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PS
9
Carly takes a loan of £5250 and pays 1.6% simple interest each month for ten months.
FS
James takes a loan of £5600 and pays 1.25% simple interest each month for a year. Who pays more interest, Carly or James? Justify your answer.
FS
10
Kwami has a loan of £600. He pays £102 per year simple interest. Work out the yearly rate of interest.
FS
11
Amy pays interest each month on a loan of £4000. After 3 months she has paid a total of £27 interest. Work out the monthly rate of interest.
FS
12
Huan pays simple interest each week on a loan of £350. After four weeks she has paid a total of £8.40 interest. Work out the weekly rate of interest.
PS
13
Jack pays monthly interest of £28 on a loan of £1600. a
FS
Work out the monthly interest rate.
b What percentage of the original loan will Jack have paid after: i three months
ii six months
iii one year
iv two years?
Challenge: Using a formula This is a formula you can use to work the total repayment (including the original loan)
(
A = P 1 + RT 100
)
where P is the initial loan, R is the percentage rate of interest, T is the number of payments and A is the total amount paid back (initial loan + interest). A Aaron takes out a loan of £500 at 2% per month for 6 months. In this case, P = 500, R = 2 and T = 6. Use the formula to work out the total amount he pays back. B Work out the total amount paid back on a loan of £250 at 1.7% monthly interest for a year. C Work out the total amount paid back on a loan of £6800 at 14.5% annual interest for four years. D Max took out a loan. He made interest repayments of 2.4% each month. After 42 months he repaid the original loan. Show that his total repayment was just over twice the original loan.
1.1 Simple interest
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1.2 Percentage increases and decreases Learning objectives
Key words
• To use the multiplier method to calculate the result of a percentage increase or decrease
decrease
increase
multiplier
original value
• To calculate the percentage change in a value A percentage change may be: • an increase if the new value is larger than the original value • a decrease if the new value is smaller than the original value.
There are different ways that you can calculate the new value but the multiplier method is often the most efficient. You just multiply the original value by an appropriate number (the multiplier) to calculate the result of the percentage change.
Example 4 Alicia buys a second-hand car for £4800. Calculate what she can sell it for, two years later, if: a the value has fallen by 20%
b the value has increased by 8%.
a The original value was 100% and it has decreased by 20%. 100% - 20% = 80% = 0.8
This is the multiplier for a 20% decrease.
The value is 0.8 × £4800 = £3840.
You could also work out answer.
4 5
× 3840 to get the
b The original value was 100% and it has increased by 8%. 100% + 8% = 108% = 1.08
This is the multiplier for an 8% increase.
The value is 1.08 × £4800 = £5184.
A multiplier is the most efficient method to increase or decrease by a percentage. For example: • to increase by 15% the multiplier is 100% + 15% = 115% = 1.15 • to decrease by 15% the multiplier is 100% - 15% = 85% = 0.85.
Example 5 The price of a car is decreased from £12 450 to £11 990. Calculate the percentage decrease. Original price × multiplier = new price 11990 new price So the multiplier = = original price 12 450 = 0.9630 . . . = 96.30 . . . % = 96.3%
Round to one decimal place.
The decrease is 100% - 96.3% = 3.7%.
10
1 Percentages
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Example 6 Over 50 years, the population of a city increased from 540 000 to 1 287 000. Calculate the percentage increase. Original population × multiplier = new population 1287 000 new population So the multiplier = = original population 540 000
Hint
Notice that an increase can be more than 100%.
= 2.3833 . . . = 238.33 . . . % The increase is 238% - 100% = 138%.
Round to the nearest whole number.
Exercise 1B 1
Peter has £4800 in a savings account. How much will he have if his savings: a increase by 7% b decrease by 7% c
2
increase by 47%
d decrease by 70%?
Work out the new prices. a Jacket was £225, price reduced by 40%. b Handbag was £132, price reduced by 30%. c
Shoes were £79, price reduced by 10%.
d Jeans were £85, price reduced by 85%.
3
4
Change each bill by the amount shown. a Electricity £325.40, increase by 8%
b Gas £216.53, decrease by 12%
c
d Council tax £781.20, decrease by 1.5%
Rent £475.50, increase by 14%
The price of a cooker was £585. a
Work out the new price after a decrease of 3.5%.
b The price is decreased from £585 to £499. Work out the percentage decrease. c
The price is increased from £585 to £605. Work out the percentage increase.
1.2 Percentage increases and decreases
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5
The multiplier for a price increase is 1.075. a
What is the percentage increase?
b What is the multiplier for a decrease of the same percentage?
FS
6
An antique painting was valued at £24 000. Work out the new value, if it increases by: a
FS
7
80%
b
180%
c
d
280%
330%.
Greg and Lynne bought a flat for £185 000. Six years later, they sold it for £429 000. Work out the percentage increase in the price.
FS
8
Ludmilla bought an antique necklace for £560. a
She wants to sell it for a profit of at least 20%. What is the minimum price she must sell it for?
b She sells it for £725. Work out her percentage profit. 9
Read these two statements. The price of a vintage car has increased by 150%. The price of a second-hand car has been reduced by 150%.
Explain why one statement is sensible and the other is not. 10
Work out the percentage change in each case. a An increase from £285 to £365 b A decrease from £365 to £285 c
11
An increase from 42.3 kg to 91.7 kg
d A decrease from 91.7 kg to 42.3 kg
Read what Beth is saying.
A change from 870 to 1240 is an increase of 42.5% and a change from 1240 to 870 is a decrease of 42.5%.
She is incorrect. Write a correct version.
12
1 Percentages
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12
This table shows the selling price of a house at different times. Year Selling price
2000
2006
2009
£195 000
£270 000
£235 000
Work out the percentage change in price: a 13
b
from 2000 to 2006
from 2006 to 2009
c
from 2000 to 2009.
Ayaan has a rectangular piece of card with sides of length 16 cm and 20 cm. 16 cm 12 cm
16 cm
20 cm
He cuts off a 2 cm border all around the edge. a
Work out the percentage change in the width of the card.
b Work out the percentage change in the perimeter. c
PS
14
Work out the percentage change in the area.
Read this newspaper headline.
V nezuela grapples Ve with 56% inflation
This means prices are increasing by 56% a year. The price of a shopping basket now is 400 Bolivars (the currency in Venezuela). a
What will the price be in a year’s time if it increases by 56%? Hint
The increase is not 56% of 400.
b What will the price be a year after that if it increases by another 56%? c
What will the price be a year later if, once again, it increases by 56%?
Reasoning: Population change A website gives these populations for three English cities. 1901
1951
1991
Derby
114 000
141 000
214 000
Liverpool
702 000
789 000
452 000
Plymouth
264 000
209 000
236 000
A Which city had the largest percentage change between 1901 and 1951? Justify your answer. B Which city had the largest percentage change between 1951 and 1991? Justify your answer. 1.2 Percentage increases and decreases
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1.3 Calculating the original value Learning objective • Given the result of a percentage change, to calculate the original value The number of swans in a wildlife reserve has increased by 20% since last year. There are now 450. How many were there last year? It would be incorrect to reduce 450 by 20% and get 450 × 0.8 = 360. This is because the 20% relates to the number last year, not the number now. The correct answer is that last year there were 375 swans. This is because if you increase 375 by 20% you get 375 × 1.2 = 450. In this section you will learn how to answer questions like this.
Example 7 Last year, the number of daily visitors to Hastings Museum increased by 30% to 1066. What was the number of visitors before the increase? The multiplier for the increase is 1.3.
100% + 30% = 130% = 1.3
Original number × 1.3 = 1066 1066 Original number = 1.3 = 820
The result of the increase is 1066. Find the original number by dividing by 1.3. 1066 ÷ 1.3 = 820
Example 8 In an annual survey, the number of birds seen in a wood was 150. This was a 40% reduction on the previous year. How many were there the previous year? The multiplier for a 40% reduction is 0.6. The previous year’s number × 0.6 = 150. 150 The previous year’s number = = 250. 0.6
100% - 40% = 60% = 0.6 Divide 150 by 0.6.
VAT Prices often include a tax called value added tax (VAT). This is set as a percentage of the original cost. You can use the multiplier method to find the cost before VAT is added.
14
1 Percentages
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Example 9 The price of a car repair, including 20% VAT, is £283.68. Work out the price excluding VAT. The multiplier for a 20% increase is 1.2.
100% + 20% = 120% = 1.2
Price excluding VAT × 1.2 = 283.68 Price excluding VAT = 283.68 1.2 = £236.40
Exercise 1C 1
a
After a 10% increase, a price is now £495. Work out the original price.
b After a 20% increase, the mass of a calf is 80 kg. Work out the mass before the increase. c
After a 40% increase, the height of a tree is now 4.9 m. Work out the original height.
d After adding 2.5% interest, a loan is now worth £6929. Work out the original loan. 2
a
After a 20% decrease in value, a car is now worth £2560. Work out the original value.
b After a 2.5% decrease, a price is now £468. Work out the original price. c
After a 12.5% decrease, a mass is now 154 kg. Work out the original mass.
d After an 80% decrease in membership, a club now has 36 members. Work out the original number of members. 3
The number of trees in a wood is 552. This is a 15% increase from five years ago. a
Work out the number of trees that were in the wood five years ago.
b If the numbers increase by 15% in the next ten years, how many will there be?
4
The amount of cereal in a packet has increased by 10%. The mass of the contents is now 495 g. a
Work out the mass of the contents before the increase.
b If the mass of the contents is increased by another 10%, what will it be?
1.3 Calculating the original value
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5
The number of people employed by Acme Electronics increased by 60% over the last year. The company now employs 448 people. a
How many people did Acme Electronics employ a year ago?
b If they employ 60% more in the coming year, how many will they employ a year from now?
FS
6
The price of concert tickets offered on a website is reduced by 15%. The price now is £57.80. Harry buys six tickets. How much will he save, because of the reduction?
FS
7
These are some prices including VAT at 20%. Work out the price excluding VAT in each case. a
FS
8
9
b Garage bill, £462.24
c TV, £899.50
The rate of VAT on energy bills is 8%. These are some energy bills, including VAT. Work out each bill before VAT is added. a
FS
Restaurant bill, £101.52
£128.15
b £255.40
c £324.16
The cost of a new boiler, including 20% VAT, is £849.00. How much is the VAT?
FS
10
The standard rate of VAT in Ireland is different than in England. a
A camera costs €265 excluding VAT and €325.95 including VAT. Work out the rate of VAT in Ireland.
b A sofa in Ireland costs €2099 including VAT. How much is the VAT? 11
Between 1901 and 2001 the population of England rose by 63.4% to 49 139 000. a
What was the population in 1901?
Between 1801 and 1901 the population of England rose by 287.8%. b What was the population in 1801?
Challenge: At the gym A gym increases its membership fee by 25% to £160. The number of members in the gym decreases by 25% to 180. Did the gym gain or lose money by increasing the fees? Justify your answer.
16
1 Percentages
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1.4 Repeated percentage changes Learning objective • To calculate the result of repeated percentage changes Suppose Jim’s pay increases by 15% and then later decreases by 5%. You could calculate the result of the first increase and then do a second calculation to find his final pay. However, you can do all the calculations at once, as the next example shows.
Example 10 Jasmine has a part-time job and earns £8.40 per hour. She is given a 15% pay rise. Later she has to take a 5% pay cut. a Calculate her rate of pay after the two changes. b Work out the overall percentage change. a The multiplier for a 15% pay rise is 1.15.
100% + 15% = 115% = 1.15
The multiplier for a 5% pay cut is 0.95.
100% - 5% = 95% = 0.95
8.40 × 1.15 × 0.95 = 9.177
Multiply by each of the multipliers.
She is paid £9.18 per hour.
Round to the nearest penny.
b 1.15 × 0.95 = 1.0925
1.0925 is the multiplier for an increase of 9.25%. 1.0925 = 109.25%
Notice that the overall percentage is not 15% - 5% = 10%. This is because the 10% and the 5% are percentages of different amounts. You can illustrate the changes in a diagram, like this. 0.95
1.15 1.0925
Original
15% increase 5% decrease
1.4 Repeated percentage changes
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Exercise 1D 1
The mass of a young child is 7.2 kg. The mass increases by 20% and then by a further 10%. Work out the final mass of the child.
2
The population of a village in 1990 was 350. The population increased by 20% between 1990 and 2000. The population increased by 30% between 2000 and 2010. Calculate the population in 2010.
FS
3
William bought a new car for £18 500. After one year the value has fallen by 20%. In the next year the value falls by 12%. Work out the value of the car after two years.
FS
4
Gabrielle put £2500 in a savings account. At the end of one year she was paid interest of 2% and she left this in her account. At the end of the second year she was paid interest of 2.5%. How much did she have at the end of the second year?
5
Sam’s usual heart rate is 60 beats per minute. After jogging for five minutes, his heart rate has increased by 120%. After resting for one minute it has decreased by 35%. What is Sam’s heart rate now?
6
The number of fish in a lake in 2012 was 270. In 2013 the number increased by 20%. In 2014 the number decreases by 20%. a
How many fish were there in the lake in 2014?
b What is the overall percentage change from 2012 to 2014? 7
A tree is 1.5 m tall. After a year, the height of the tree has increased by 12%. After another year, the height has increased by a further 14%. a
Work out the height of the tree after two years.
b Work out the percentage increase in height over the two years. 8
The price of a sculpture increases by 16% and then increases by 25%. Show that the overall percentage increase in price is 45%.
18
1 Percentages
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PS
9
In 2013 the price of electricity increased by 10%. In 2014 the price increased by a further 10%. What was the overall percentage rise for the two years?
PS
10
The cost of gold increased by 20%. Then it increased by a further 30%. Work out the overall percentage increase.
PS
11
The cost of a mobile phone was reduced by 30% and then by a further 40%. What was the overall percentage reduction?
12
The mass of a young lion increased by 40% and then increased by a further 40%. Show that the lion’s mass has almost doubled.
13
a
What is the overall effect of an increase of 25%, followed by a decrease of 20%?
b What is the overall effect of a decrease of 20%, followed by an increase of 25%? c
What is the overall effect of an increase of 20%, followed by a decrease of 25%?
Investigation: Up and down A Show that an increase of 10%, followed by a decrease of 10%, is equivalent to a decrease of 1%. B Find the single percentage change that is equivalent to: a an increase of 20%, followed by a decrease of 20% b an increase of 30%, followed by a decrease of 30% c an increase of 40%, followed by a decrease of 40%. C Generalise the results of parts A and B, to describe the single change that is equivalent to an increase of 10N% followed by a decrease of 10N%. D Predict the result of an increase of 5%, followed by a decrease of 5%. Check your prediction.
1.4 Repeated percentage changes
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Ready to progress? I can calculate percentages, including simple interest. I can find the result of a percentage increase or decrease. I can use the multiplier method to calculate the result of a percentage increase or decrease. I can calculate the original value, given the result of a percentage increase or decrease. I can use the multiplier method to find the result of a repeated percentage change.
Review questions 1 The height of the Delta Tower is 55 m. a b c
FS
The Epsilon Tower is 23% taller than the Delta Tower. Work out the height of the Epsilon Tower. The Zeta Tower is 23% shorter than the Epsilon Tower. Work out the height of the Zeta Tower. Calculate the height of the Zeta Tower as a percentage of the height of the Delta Tower.
2 Noah has a loan of £3600. He pays simple interest of 1.3% per month for two years. a b
FS
Work out the total amount of interest he will pay. What percentage of his original loan is this?
3 Ava has a loan of £600. She pays 2.2% per month simple interest for six months and then she pays back the original loan. Calculate the total cost to her of the £600 loan.
FS
4 The charge for using a credit card to pay a bill is 2%. Work out the total cost when Oscar uses his credit card to buy four tickets, costing £64.50 each, to watch a football match.
20
1 Percentages
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FS
5 The cost of getting some gardening done is £282. This includes 20% VAT. How much was the VAT? 6 These are some sale prices. Work out the percentage reduction in each case.
PS
a
From £89 to £65
b From £445 to £385
c
From £1275 to £825
d From £25.95 to £8.49
7 Ella draws a square. Each side is 12 cm long. a
Oliver draws a square with a perimeter 150% longer. Work out the length of each side of Oliver’s square b Freya draws a square with an area that is 56.25% larger. Work out the length of each side of Freya’s square 8 The number of road accidents in a county is increasing by 10% per year. This year there were 252. a b
How many accidents were there last year? If this trend continues, how many accidents will there be in two years’ time?
9 The population of a town in 1990 was 14 000. The population increased by 18% from 1990 to 2000 and then decreased by 4% from 2000 to 2010. What was the population in 2010? 10 The number of badgers in a rural area has decreased by 20% since last year. There are now 96. a b
PS
How many were there a year ago? If this trend continues, how many will there be in two years’ time?
11 A restaurant bill includes 20% VAT and a 10% service charge. The total bill is £111.21. Work out the cost, excluding VAT and service.
Review questions
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Challenge Exponential growth
There are 1000 deer on an island. An ecologist is studying how the number of deer might change over the years.
1 Suppose the number of deer increases by 20% each year. a Show that in two years’ time there will be 1440 deer. b Copy and complete this table. Number of years 0 Number of deer 1000 c Plot your values on a graph. The axes should look like this.
2 1440
3
4
5
3000
2500
2000
Population
The points are not in a straight line. Join them with a smooth curve.
1
1500
1000
500
0
22
0
1
2
Years
3
4
5
1 Percentages
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2 Now suppose the number of deer increases by 30% each year. a Complete a new table to show how the numbers will increase over 5 years. b Plot these points on the same graph as for question 1. Join them with a smooth curve. 3 This time, suppose the number of deer increases by 40% each year. Calculate values and plot a third line on your graph to show this. The three lines you have drawn are examples of exponential growth. This means that the numbers increase by the same percentage each time. 4 Finally, suppose that the numbers decrease by 20% each year. a Show that after 2 years there will be 640 deer. b Copy and complete this table. Number of years Number of deer
0 1000
1
2 640
3
4
5
c Draw a new line on your graph to show these numbers.
5 a Draw a line on your graph to show the numbers, if they fall by 30% each year. b Draw a line on your graph to show the numbers, if they fall by 40% each year. These are examples of exponential decay. The numbers decrease by the same percentage each time. People sometimes talk about ‘exponential growth’ when they mean ‘rapid growth’. Exponential growth can be very slow at first but it always increases as time goes on – the graph keeps getting steeper.
Challenge
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