Statistics

16 Everyday statisticsOBJECTIVES Draw a a pictogram, b a bar chart, c a pie chart to show this information.By the end of this chapter you should beable to: Table 16.1 is a frequency table. The frequency is the number of students using each method of1 Present data in frequency tables, transport. pictograms, bar charts and pie charts Fig. 16.1 shows the required graphs.2 Interpret statistical graphs a3 Find the mean, median and mode of given data sets4 Apply measures of central tendency (mean, median, mode) to everyday issues5 Find the range of a given set of data.Teaching and learning materials Each symbol represents 5 students. bTeacher: Charts on mean, median andmode; frequency tables; bar charts; pie 15charts; real statistics from newspapers andofficial publications; chalk-board instruments(compasses, protractor, ruler)Student: Mathematical set; numerical datacollected from newspapers 10Frequency Bicyclel16-1  Presentation of data (revision) 5 Walk BusExample 1 revises four important ways of Carpresenting data: 0 Methods of transport a frequency table, a pictogram, c a bar chart, and a pie chart. Car Bicycle BusExample 1Table 16.1 shows how class 3A travel to school. Walk method bicycle walk bus car frequency 9 15 7 5Table 16.1 Fig. 16.1152Unit16.indd 152 28/3/08 13:25:42

Notes on Example 1: c Approximately how many bicycles werea Pictogram: Each method of transport is counted? represented by a suitable picture. In this case d Approximately how many lorries were each complete picture represents five students. counted?b Bar chart: Each bar represents a method of transport. The height of each bar is 2 Fig. 16.3 is a bar chart showing the heights proportional to the number of students using and names of the highest mountains of the that method of transport. continents of Asia, South America, Northc Pie chart: Each sector represents a method America, Africa and Europe respectively. of transport. The angle of each sector is proportional to the number of students who 10 000 use the method of transport shown in that sector. See Table 16.2. Height (m) Everest 5 000 Aconcagua McKinle ymethod frequency angle of sector Kilimanjarobicycle 9 Blancwalk 15 _​3 9_6  ​ 3 360° 5 90° 0bus 7 ​_31  _56  ​3 360° 5 150° Fig. 16.3 Mountaincar 5 ​_3 7_6   ​3 360° 5 70°total 36 ​_3 5_6  ​ 3 360° 5 50° a What is the height of Mount Kilimanjaro? b What is the height of Mount McKinley? 360° c Which mountain is 4 800 m high? d Find the difference in height betweenTable 16.2 Mount Everest and Mount Aconcagua.Exercise 16a  [Revision] 1 Fig. 16.2 is a pictogram of a traffic survey. 3 Fig. 16.4 is a pie chart showing how someone spent the first hour of a day after getting out of bed.bicycles cars Washing motorbikes Eating lorries Travelling to work buses Fig. 16.4Fig. 16.2  Each symbol represents 10 vehicles a What fraction of the hour was spent a Which kind of vehicle is the most eating? common? b What percentage of the hour was spent b Which kind of vehicle is the least washing? common? c How many minutes were spent travelling to work? 153Unit16.indd 153 28/3/08 13:25:43

4 Fig. 16.5 is a bar chart showing the estimated Draw a bar chart to show how the price population of a city by age group. increased.Age (years) 0·5 1·0 9 To support a ‘Clean Our Village’ campaign,80 and over Population (million) the local UBE school collected and burned used plastic bags that were lying around. The 60 – 79 Primary classes collected 60 kg of bags; the 40 – 59 JS classes collected 48 kg of bags. 20 – 39 a Draw a pie chart that compares the Under 20 performance of the two sets of classes. b Discuss the performance with your teacher. 0 10 During a 24-hour period in Kaduna, the sunFig. 16.5 shone for 10 hours, it was cloudy for 3 hours and it was dark for the remainder of the a Approximately how many people are under time. Draw a pie chart to show the periods of 20? sunshine, cloudiness and darkness during the day. b Estimate the total population of the city. c What fraction of the total population is 80 11 180 people attend a voluntary substance- abuse rehabilitation clinic. 100 of them are or over? registered for smoking-related problems d Find the percentage of the total population (tobacco, cannabis), 50 for drink-related problems (alcohol dependency) and 30 for that is 20–39 years old. drug injection (heroin, cocaine, others). Draw a a pictogram, b a pie chart, to represent this5 A farmer has 100 cows, 40 sheep and 65 data. goats. Draw a pictogram to show the animals that he has. (Let one symbol represent 10 12 After 1 year, half of the people in question animals.) 11 are rehabilitated. The remaining 90 are registered as follows: smokers (30 in total),6 117 people travel from Kano to Lagos. 55 go drinkers (35) and injectors (25). by car, 39 go by air, and 23 go by train. Show a Draw a pie chart of this new data. this information on a pictogram. b By comparing with the pie chart in question 11, which area of substance abuse7 Table 16.3 shows the market prices of 1 kg of i was a success story for the clinic, plantain in four towns. ii appears to be most addictive?town Benin Auchi Agbor Sapele + Optional: Do Exercise 37 of your Students’price 5N 48 5N 42 5N 62 5N 80 Practice Book.Table 16.3 l16-2  Measures of central tendency Draw a bar chart to show this information.     (mean, median, mode) 8 Table 16.4 gives the price/tonne of An average is a single number that represents groundnuts in 2-year intervals from 2000 to a large set of data. The common averages are 2008. the mean, median and mode. These averages are often called measures of central tendency,year 2000 2002 2004 2006 2008 since an average usually has a central valueprice 55N32 400 55N66 000 55N100 000 55N22 400 5N27 200Table 16.4154Unit16.indd 154 28/3/08 13:25:43

within the data set. Example 2 revises the b Adult clinic:common averages and also emphasises the 28, 33, 45, 32, 37, 35, 27, 33, 36, 37importance of placing data in rank order. Find the mean, median and modal age for eachExample 2 clinic. a mean ageNine boys take a test. Their marks are: Ali 4, 5 ​1 _5__1__1_8_1___1_8_1__ _17_5_ 1___1 _2_1__​1_5_  _1__ 1_9_Ben 8, Dan 7, Isa 9, Joe 4, Kam 3, Mbu 6, 5 ​_1  _17_2 _​  5 16 yearsOlu 4, Sam 7. Place the marks in rank order. Arrange the ages in rank order:Hence find the mean, median and mode for the 19, 18, 18, 15, 15, 15, 12test. There are seven ages. Median 5 middle age (i.e. the 4th age)Table 16.5 gives the names, marks and 5 15 yearspositions of the boys in rank order. Mode 5 age appearing most often 5 15 yearsname mark position b mean ageIsa 9 1 5 ​2  _8_1___3_3_1__4_5__1_ _3_2__1__3 _7__11_0_3 _5_ _1__2_7_1_ _ _3_3_1_ __3 _6_1 __3_7_​Ben 8 2 5 _​3 1_4_03_  ​5 34?3 yearsDan 7 35 Arrange the ages in rank order:Sam 7 35 45, 37, 37, 36, 35, 33, 33, 32, 28, 27Mbu 6 5 There are ten ages.Ali 4 65 Median 5 ​5_  t_h__12_ _6_t​h_  5 ​3_  _5_12__ 3_​3_  Joe 4 65 5 34 yearsOlu 4 65 Mode 5 age appearing most oftenKam 3 9 Ages 37 and 33 both appear twice. The data are bimodal (37 years andTable 16.5 33 years).mean 5 ​9  _1___8_1__7__1__7__1_ _69__ 1__4_ 1___4_​1__  _4_1_ _3_ 5 ​_5 9_2_  ​5 5_​79   ​  Notice in Example 3b, when there is no single middle value in the data set, the median is The median is the middle mark when the calculated as the average of the middle twomarks are arranged in rank order. In this case, pieces of data. Notice also that data may beMbu is in the fifth (or middle) position; his bimodal.mark is 6. 155 median 5 6The mode is the mark which appears mostoften. Three of the boys score 4 marks. mode 5 4Example 3The ages of people attending two drug-abuseclinics are given below:a Young persons’ clinic: 15, 18, 18, 15, 12, 15, 19Unit16.indd 155 28/3/08 13:25:43

Example 4 a meanA woman buys three bottles of palm oil at 5 ​2  _3__1_1__1__1__3__1_2 _1_2_1_3_3_1_1 _1_33_1_1_6_ 6_3 _1_1_54__1 1__35 __3_ _1_5_1_ __3 _3__1_6_5N 180 per bottle. Next week she buys two  5 ​2 _2__1​__1_2__1__3 _9_ 12__0_ 8_4_ 1___7_5​_  1__ 4_8_bottles of palm oil at 5N 200 per bottle. Find the 5 _​2 2_8_00_  ​average cost per bottle for the two weeks. 5 143 bottles at 5N 180 cost 3 3 5N 180 5 5N 540 b There are 20 students. The median score is2 bottles at 5N 200 cost 2 3 5N 200 5 5N 400 the mean of the scores of the 10th and 11th students. The 10th and 11th students bothtotal cost of 5 bottles 5 5N 940 scored 14.average cost of 1 bottle 5 5N _​9  _54_0 _​   median score 5 14 5 5N 188 c The mode is the score that occurs most Example 5 often. Six students scored 14. A motorist travelled 96 km at an average speed of 60 km/h. She returned at an average speed mode 5 14 of 48 km/h. What was her average speed for the whole journey? Notice, in Example 6, that it is not necessary to 96 km at 60 km/h takes ​_69  _06_  ​hours 5 1?6 h make an ordered list of all the scores. Also note 96 km at 48 km/h takes _​49  _86_  ​hours 5 2?0 h that it is possible for some or all of the averages Altogether she travelled 192 km in 3?6 h. to be equal to each other. average speed 5 _​1 3_9?_62_  ​k  m/h 5 ​_1  _ 39_62_ 0_​   km/h Exercise 16b  [Revision] 5 ​_1  _36_ 0_​   km/h 5 53_​31   ​ k  m/h 1 A motorist drives 80 km at an average speedAlways remember that of 63 km/h. Which average is this: mean, median or mode? average speed 5 ​t_  o_t_a_tlo_dt_ai_sl_tta_inm_c_ee__tta_r ka_ ve_en_ l_  le_d​_ 2 Twenty people apply for jobs as policeExample 6 officers. There are only 10 jobs available. They are given to the 10 people who areTable 16.6 gives the scores of 20 students in a above the average height of those whomathematics test. applied. Which average is this: mean, median or mode?score 11 12 13 14 15 16 3 A trader sells shoes. He wants to know thefrequency   2   1   3   6   5   3 average size of shoe that people buy. Which average is most useful?Table 16.6 4 Arrange the following sets of numbers inFind: order of size. Find the mean, median anda the mean score, mode of each set.b the median score, a 7, 10, 7, 9, 7c the modal score. b 5, 3, 0, 7, 3, 6 c 1, 9, 5, 6, 1, 4 d 8, 3, 1, 7, 3, 4, 8, 3, 4, 9, 5 e 0·1, 0, 1?5, 0, 0?6, 1?1 f 159?5, 155?8, 153?7, 157?2, 155?8156Unit16.indd 156 28/3/08 13:25:43

5 The wages of five local government trainees 17 A lorry travelled 80 km at 40 km/h, 64 km are 5N 4 166, 5N 4 618, 5N 3 742, 5N 5 838, at 48 km/h and 40 km at 60 km/h. Find its 5N 5 366. average speed for the whole distance. Find a the mean, b the median wage. 18 In a test the average marks for three classes6 Six people together have a mass of ​_21   ​ tonne. were 74, 58 and 51. If the classes contained What is the average mass of the people in kg? 25, 22 and 23 students respectively, what was the average mark for the three classes7 A man caught four fish of mass 1?9 kg, together? 1?1 kg, 0?9 kg, 0?7 kg. Find a the mean, b the median mass of the fish. 19 Some students were asked how many brothers and sisters they had. The bar chart in8 The mean age of four women is 19 yr 11 mo. Fig. 16.6 shows the number of students who When a fifth woman joins them, the average had 0, 1, 2, 3, 4, 5, 6, 7, or 8 brothers and age of all five is 20 yr 7 mo. How old is the sisters. fifth woman?9 There are eight men and a woman in a boat. Frequency 5 The average mass of the nine people is 79 kg. Without the woman, the average is 81?5 kg. What is the mass of the woman?10 A trader mixes 3 kg of sugar at 5N 172/kg with 2 kg of sugar at 5N 152/kg. What is the cost/kg of the mixture?11 The mean daily rainfall for a week was 0 01 23 456 78 5?5 mm. For the first six days the mean Number of brothers and sisters rainfall was 1 mm. How much rain fell on the seventh day? Fig. 16.612 A bridge 1​_14    ​k  m long cost 5N 750 million. Find: What was the average cost per metre? a the number of students in the survey, b the modal number of brothers and sisters,13 The ages of a family of six children are 16?4, c the median, 14?8, 13?6, 11?10 and the twins are 9?1. d the mean number of brothers and sisters. Find the mean, median and modal ages of the family. (Take 16?4 to mean 16 years 4 20 Five students took a test in four subjects. months.) Their results are given in Table 16.7.14 A man travelled 30 km in a car at an average speed of 40 km/h. He returned at an average English History Maths Science 52 69 54 57 speed of 60 km/h. Find his average speed for Rose 68 60 67 73 Efosa 80 73 49 42 the whole journey. Mfon 26 14 37 35 Emem 34 44 38 4815 A woman travelled for ​_12    ​hour in a car at an Vera average speed of 40 km/h. For the next ​_21    ​hour her average speed was 60 km/h. What was her average speed for the whole time?16 Of a journey of 216 km, 56 km were on Table 16.7 untarred road. For this part of the journey, a motorist could only average 28 km/h. If Find: his average speed for the whole journey was a the mean mark of each student, 48 km/h, what was his average speed for the b the mean mark in each subject, part of the journey on tarred road? c the median mark in each subject. 157Unit16.indd 157 28/3/08 13:25:43

If you had difficulty with Exercise 16b, revise Example 9Chapter 22 in Book 1. The midday temperatures in °C for a week in16-3  Interpretation of averages l towns A and B are given below: Town A: 23, 24, 22, 23, 22, 24, 23Remember that an average, such as a mean, Town B: 29, 29, 26, 23, 19, 18, 17median or mode, is a number which represents awhole set of numbers. Sometimes averages can a Calculate the mean midday temperature forbe misleading. each town.Example 7 b Compare the results with the weather for each town.A construction company has 20 labourers, twosupervisors and two directors. Labourers get a Town A: mean temperature5N 6 000 per week, supervisors get 5N 20 000per week and directors get 5N 40 000 per week. 5 ​2 _3__1__2_4_1___2_2_1__ _27_3_ 1___2 _2_​1__2_4_  _ 1__ 2_3_ °CCalculate the company’s average weekly wage. 5 _​1  _76_ 1_​   °C 5 23 °C Town B: mean temperature mean weekly wage 5​_  _to_t_a_l_m__o_n_e_y_ e_ a_r_n _ e_d_ ​ 5 ​2 _9__1__2_9_1___2_6_1__ _27_3_ 1___1 _9_​1__1_8_  _ 1__ 1_7_ °C 5 _​1  _76_ 1_​   °C 5 23 °C number of employees b The mean temperature is the same for both 5 ​2 _0__3__5N__ 6_ 0_0_0__1_ _2_23_0_1_5N_ 2_ 20_1 _0_02_0 _1 _ _2__3_ _5_N _ 4_0 _ 0_0_0_​ towns. However, the weather is different. In town A the temperature remained steady 5 ​5_ N_ 1_2_0_ _0_0_0_1__5_N_  42_0_4 0_ 0_0_ _1__5_N​_ 8 _ 0_ 0_ 0_0_ at around 23 °C. In town B the week began 5 _​5 N__ 2_42_04_  0_0_​0_  5 5N 10 000 with hot weather, but the temperature fell daily so that the weather was cool by the end of the week. A labourer joining the company in Example In Example 9, the temperatures for town A7 is told that the average wage is 5N 10 000. He range from 22 °C to 24 °C and those for townwill be disappointed when he gets only 5N 6 000. B range from 17 °C to 29 °C. The range ofIn this case it may be more helpful to give the temperatures is different in each case. However,average wage in terms of the mode, 5N 6 000. the results of Example 9 show that it is possible to get the same mean from different ranges ofRange l numbers.The range of a set of numbers is the difference Example 10between the largest and the smallest numbers inthe set. The same test was given to two groups of 20 students. The test was marked out of 5. TheExample 8 results of the groups are given in Table 16.8.Find the range of the following percentages: 1st group84%, 56%, 72%, 64%, 33%, 49%. mark 012345Arranging the set in rank order:84%, 72%, 64%, 56%, 49%, 33% frequency 2 5 6 4 2 1The range is 84% 2 33% 5 51%.158Unit16.indd 158 28/3/08 13:25:43

2nd group Notice in Example 10 that the range of marks is the same for each group: from 0 to 5.mark 012345 However, the mode and mean are different for both groups. This shows that it is possible tofrequency 1 1 2 4 7 5 get different averages from the same range of numbers. In Fig. 16.7, notice the position of theTable 16.8 mode in each group: in general, the higher the mode, the higher the mean.a Draw bar charts for each group.b Find the mode and mean for each group. Exercise 16cc Explain the difference between the two 1 Find the range and mean of: results. a 1, 3, 5, 7, 9; b 1, 5, 7, 8, 9.a Fig. 16.7 gives the required bar charts. Compare the two results. 1st group 2nd group10 10 2 Find the range and mean of: a 35, 37, 39, 41, 48;Frequency b 21, 27, 28, 29, 95. Frequency Compare the two results. 00 1 234 5 00 1 234 5 Marks Marks 3 A trader buys 200 ball-point pens at 5N 28 each and 10 gold pens at 5N 4 900 each.Fig. 16.7 a Find the mean cost per pen. b Is this a sensible value to give for theb The mode is the mark with the highest average cost of a pen? frequency. 4 A factory employs 100 workmen, 161st group: mode 5 2 marks supervisors and four directors. The workmen2nd group: mode 5 4 marks get 5N 4 800 per week, the supervisors get 5N 16 000 per week and the directors get1st group: mean 5N 36 000 per week. a Calculate the mean weekly wage at the 5 ​2  _3__0_1__5_3__1_ 1__6_3__ 2_1_2_04 _3_ _3__1_2_ _3_4__ 1 __1_ 3__​5 factory. 5 ​0 __1__5__1__1_2_ 12__0_ 1_2 _1___8​ _ 1__ 5_ b Is this average representative of the wages 5 _​24  _02_  ​5 2?1 marks that the people get?2nd group: mean 5 The daily rainfall in mm for two weeks is 5 ​1  _3__0_1__1_3__1_ 1__2_3__ 2_1_2_04 _3_ _3__1_7_ _3_4__ 1 __5_ 3__​5 given below: 5 ​0 __1__1__1__4_1__ 2_10_2 _1_ _2__8_​1  __2 _5_ 5 _​27  _00_  ​5 3?5 marks Week 1: 13, 12, 10, 14, 17, 12, 13c The results show that the second group Week 2: 28, 32, 21, 10, 0, 0, 0 did better on the test than the first group. a Find the mean daily rainfall for each week. There are a number of possible reasons for b Compare your results with the weather for this: they may be older and had covered the each week. subject matter of the test many times before; the first group may not have been taught the 6 In a five-a-side basketball match between subject matter of the test. teams A and B, the points scored by the players are as follows: Team A: 8, 8, 10, 26, 30 Team B: 14, 15, 17, 18, 18 a Find the mean score/player for each team. 159Unit16.indd 159 28/3/08 13:25:44

b Compare your results with the actual 10 In a survey, two groups of 20 women were scores of the players in the two teams. asked how many children they had. The results are given in Table 16.12.7 The numbers of days absent during a term for a class of 30 students is given in Table 1st group 16.9. no. of children 0 1 2 3 4 5 days absent   0 1 2 3 5 35 frequency 853211 frequency 14 5 4 3 3   1 2nd groupTable 16.9 no. of children 0 1 2 3 4 5 Find a the mode, b the median, c the mean number of days absent. frequency 224543 d Which average is most representative of the data? Table 16.12 8 A club collected money from its members to a Draw a bar chart for each group of buy a DVD player. 30 members of the club women. made the following contributions. b Find the mode and mean for each group.amount in 5N 100 200 500 1 000 5 000 c Give a possible explanation for thefrequency    4   10    8 53 difference between the two sets of averages.Table 16.10 11 In an election the percentage of the registered a Find the mean, median and modal amounts voters who vote is called the ‘average of money contributed. turn-out’. Table 16.13 shows the numbers registered and those who actually voted in an b Which average is most representative of LGA election. the data? Give reasons. Number Men Women Total9 The label on a box of Strike matches states, registered 1 808 1 377 3 185 ‘Average Contents 40 Matches’. The contents Number of 30 boxes of Strike matches were counted who voted 1 419 1 165 2 584 and the results are given in Table 16.11. Table 16.13no. of matches 37 38 39 40 41 42 43 a What is the total average turn-out in the election (nearest whole percent)?frequency  4  5  6  7  2  4  2 b Who has the higher average turn-out: theTable 16.11 men or the women? a Find the mean, median and modal number c Give possible explanations why the of matches per box. number of women registered is so much lower than the number of men. b Is the statement on the box true or not? 160 28/3/08 13:25:44Unit16.indd 160

summary PUZZLE CORNER 1 Statistical information is called data. 2 Data can be represented pictorially or Think of a Number graphically by pictograms, bar charts and 123456789 pie charts. 3 An average is a single number that is Choose any number from 1 to 9. representative of a larger set of numbers. Multiply it by 10 and add 1. 4 Mean, median and mode are three kinds of Then multiply the answer by 10 and add 7. average. Make a note of your answer. • The mean is the value obtained by Do it again with a different number. What do you notice? dividing the sum of the numbers in a set [Now you can pretend to be a mind reader of data by the total number of numbers in with your friends.] the data. • The median is middle value when the PUZZLE CORNER numbers are arranged in order of size. • The mode is the value with the highest Remainders frequency. 5 The range of a set of numbers is the What is the smallest number that has a difference between the largest and smallest remainder of number in the set. 1 when divided by 5, PUZZLE CORNER 5 when divided by 6, and 3 when divided by 7? How Many Triangles? (2) Can you find a pattern for all the higher How many equilateral triangles have their numbers? corners at the dots of this 3 x 3 triangular lattice? 161 What about a 4 x 4 lattice, … etc? 28/3/08 13:25:45Unit16.indd 161

Chapters Revision exercises and tests 9–16Revision exercise 4  (Chapters 9, 10, 14) 2 The exact value of 2852 2 2152 is:1 Expand the following. A 70 B  140 C  500 D  4 900 a (x 2 3)(x 2 4) b (a 1 2b)(a 2 3b) E 35 000 c (3p 2 2q)2 d (4x 1 1)(4x 2 1) 3 If ​_56 _  ​5 _​d3 _  ​then d 52 Expand the brackets in (x 1 5)(x 2 3) 5 A ​_ 15_8  ​  B ​_ 25  ​ C  2_​12   ​  D  3_​53   ​  E  90 x(x 2 2) and hence solve the equation.3 Factorise the following. 4 A girl gets x marks in a test. Her friend gets a x2 1 10x 1 24 b x2 2 8x 1 15 c x2 2 11x 2 26 d 2ax2 2 24ax 1 22a 4 marks less. The ratio of the two marks is 4 : 3. Which one of the following equations4 Factorise the following. can be solved for x? a 1 2 25x2 b x2 2 18x 1 81 A ​_ x_2_x _4_  ​5 _​34 _  ​ B ​ x__2_x _4_​  5 ​_34 _  ​ C ​ _x_1_x _4_  ​5 _​34 _  ​ c 9 1 6a 1 a2 d (5a 2 2)2 2 9a2 D ​x_  _1_x _4_​  5 ​_34 _  ​ E ​ x_x_12__44_  ​5 _​34 _  ​5 a Expand (a 2 b)2. 5 If x 1 3y 5 7 and x 2 3y 5 7, then y 5 b Using the fact that 9952 5 (1 000 2 5)2, find the exact value of 9952 without long A 24​_23   ​  B  22​_13    ​ C  0 D  2_​13    ​ E  4​_32   ​  multiplication.6 Solve the following. 6 Simplify a ​_ 2_x1_28__ 1_  ​5 3 b ​_ a_27__ 4_  ​5 _​a  _25__ 2_  ​ (a 1 5b)(a 2 2b) 1 (a 2 3b)(a 2 4b).7 A girl walks 6 km at a speed of v km/h. 7 Factorise the following. a x2 1 10x 1 16 b r2 2 16r 1 28 a Write the time taken in hours in terms of v. c m2 2 7m 2 18 d y2 2 8y 2 20 e a2 1 8a 1 16 f πa2 2 πb2 b Find the value of v if her journey takes 1 h 20 min. 8 Solve the following. a ​_ x_2_2_ 1_0_  ​1 _​31 _  ​5 0 8 Solve graphically the simultaneous equations b ​ 2__3x_12__13_ x_​  5 _​34 _  ​ y 5 15 2 4x and y 5 2x. (Hint: Make tables of values of y corresponding to 9 Solve the following pairs of simultaneous x 5 0, 2, 4.) equations. a 8y 1 4z 5 7, 6y 2 8z 5 419 Solve the following pairs of simultaneous b 2y 5 3x 1 2, 9x 1 8y 5 1 equations. a 3x 1 2y 5 12, 5x 2 3y 5 1 10 A nut and a bolt together have a mass b 2a 2 3b 5 7, 4a 1 5b 5 3 of 98 g. The mass of four bolts and two nuts is 336 g. Find the mass of a nut and10 Two pencils and one rubber cost 5N 66; five of a bolt. pencils and three rubbers cost 5N 174. How much does each cost?Revision test 4  (Chapters 9, 10, 14) Revision exercise 5  (Chapters 11, 13, 15, 16) 1 The coefficient of x in the expansion of 1 5N 40 000 is saved for 2 years at 9% (x 2 2)(x 1 9) is: compound interest per annum. Find: A 218 B 22 C 11 D 17 E 19 a the amount, b the interest.162Unit16.indd 162 28/3/08 13:25:45

2 The population of a town increases by about 120 km. What is the average speed for the 4% each year. Two years ago the population whole journey? was 38 000. What is the population now? (Give your answer correct to 2 s.f.) 13 The midday temperatures in °C for a week in a certain town were 27, 29, 29, 33, 28, 24, 3 A year ago a new refrigerator cost 5N 36 000. 26. Today the same kind costs 5N 41 400. a What is the modal temperature? a Calculate the percentage rate of b State the range of temperatures. inflation. c What is the median temperature? b If inflation stays at the same rate, what d Calculate the mean temperature. will the same kind of refrigerator cost next e What is the greatest deviation from the year? mean i above, ii below? 4 A boy walks at a steady speed and takes 14 Table R2 gives the estimated number of 90 min for a certain journey. How many students (in thousands) between the ages of minutes will a girl take if she cycles at 2​_21   ​  15 and 18 in a certain country. times the speed of the boy? age   15   16   17   18 5 In 2007 £1 sterling was equivalent to about number (3 1 000) 228 206 152 134 5N 250. a Draw a graph connecting £ to 5N for Table R2 amounts up to 5N 4 000. b Read off the equivalent amounts in the Show this information a in a pie chart, b in a other currency of: bar chart. i £15, £11, £8 ii 5N 2 500, 5N 1 900, 5N 3 600. Revision test 5  (Chapters 11, 13, 15, 16) 6 Find the value of R if ​_R1 _ ​5 _​1 1_5_   ​1 _​2 1_3_   .​ Give 1 A new table costs 5N 10 000. Find the cost your answer correct to 2 s.f. of buying the same kind of table in 2 years’ time if the rates of inflation for those years 7 By drawing and measurement find are 20% and 10%. approximately: A 5N 10 000 B 5N 11 500 C 5N 12 000 a the value of cos 44°, D 5N 13 000 E 5N 13 200 b the size of the angle whose sine is ​_54    ​. 2 A bookshelf can hold 84 books each 3 cm 8 Use 4-figure tables to find the value of: wide. How many books of width 4 cm can it a sin 38?2° b cos 51?7° hold? b cos 63?24° d sin 82?63° A 56 B  63 C  84 D  105 E  112 9 In nABC, Aˆ 5 38°, Bˆ 5 90° and 3 The reciprocal of 0?02 is AC 5 9 cm. Calculate a Cˆ , b AB, c BC. A 500 B  50 C  5 D  0?5 E  0?0510 The diagonals of a rectangle are 10 cm Use Fig. R9 to answer questions 4 and 5. long and intersect at an angle of 120°. Make a sketch of the rectangle. Hence use Pq R trigonometry to calculate the length and breadth of the rectangle. rp11 A man’s average wage for four weeks was Q 5N 6 152. In the first three weeks his wages were 5N 5 560, 5N 6 672 and 5N 6 024. Calculate Fig. R9 his wage in the fourth week. 4 In Fig. R9, sin Pˆ 5 D ​_ pr_  ​ E ​_ pr_ ​12 A motorist averages 48 km/h for the first A ​_ qp_  ​ B  _​pq _  ​ C ​_ qr_ ​ 30 km of a journey and 64 km/h for the next 163Unit16.indd 163 28/3/08 13:25:45

5 Which of the following statements is (are) b Draw a graph connecting speed and true in Fig. R9? reciprocal of time. I sin Pˆ 5 cos Rˆ c Use your graph to find the time taken II cos Pˆ 5 sin Rˆ when the speed is 45 km/h. III tan Pˆ 5 tan Rˆ 10 a Use reciprocal tables to find the value of A none of them B I and II only ​ _v3a_?16l_u 9_e  ​scoorfrebct_​3  _t61_o 9_ 2 ​ans.df.cH​_0 e_?n31_c6_ e9_w  .​ rite down the C II and III only D I and III only 11 By drawing and measurement, find E all of them approximately a the value of sin 37°, b the 6 The average age of 21 students is 14 years size of the angle whose cosine is ​_1 7_0  . ​ 3 months. If the oldest student is not counted, the average drops to 14 years 2 months. The 12 A ladder 6 m long leans against a vertical age of the oldest student is: wall and makes an angle of 80° with the A 14 years 4 months B 14 years 5 months horizontal ground. Calculate, to 2 s.f., how C 15 years 11 months D 16 years far up the wall the ladder reaches. E 21 years 13 The ages of 14 students in years and months 7 Fig. R10 is a pie chart of the information in are: Table R3. 15,5; 15,0; 15,3; 14,5; 14,7; 15,4; 15,10; B C 14,9; 15,2; 13,11; 15,1; 16,1; 15,5; 14,11. A a State the range of these ages. b Calculate the average of the highest and ED lowest ages and use this as a working mean to find the average age of all theFig. R10 students. c What is the greatest deviation from the shoe size 6 7 8   9 10 mean, i above, ii below? number of people 5 6 9 10   6 14 After 12 games a basketball player’s points average was 18?5. How many points must he score in the next game to raise his average to 20? Table R3 Which sector in Fig. R10 represents the General revision test B  (Chapters 9–16) people who wear size 8 shoes? 1 If x2 1 14x 1 k is a perfect square, then k 5 8 5N 200 000 is saved at 10% per annum A 4 B  7 C  28 D  49 E  196 compound interest. Find a the amount, b the interest, after 3 years. 2 Which of the following is (are) factors of 9 Table R4 gives some average speeds and 3x2 2 6x 2 9? corresponding times for a certain journey. I 3 II (x 2 3) III (x 1 1) speed (km/h) 15 30 50 A I only B II only C III only time (h) 10 5 3 D I and II only E I, II and III Table R4 3 If ​_5 f_  ​5 4 2 _​7 f_   ​, then f 5 a Copy Table R4 and add an extra line for A 21 B ​_ 13  ​ C ​_ 21 ​  D  1 E  3 values of ​_t i_m1_ e_  .​ 4 A herdsman bought a cow for 5N 25 000. He sold it to a butcher at a profit of 20%. The164Unit16.indd 164 28/3/08 13:25:46

butcher then sold it at a profit of 20%. What c z2 1 11z 2 26 d y2 2 9y 2 36was the final selling price of the cow? e 16 1 8h 1 h2 f 9 2 (a 1 2)2A 5N 2  9 000 B 5N 29 200 C 5N 32 000D 5N 35 000 E 5N 36 000 13 Solve the following.5 When a sum of money is shared between a ​_ 1k_5_  ​5 _​3k _  ​1 3 b ​_ 1k_5_  ​5 _​k  _13__ 2_  ​ c ​_ x_11__ 5_  ​2 _​4  _x_12_ _7_  ​5 040 people they each get 5N 800. If the samemoney is shared equally between 50 people, 14 A motor bike costs 5N 750 000. It depreciates byhow much will each get? 20% in the first year and by 15% in each of theA 5N 600 B 5N 640 C 5N 810 following years. Find the value of the motor bikeD 5N 960 E 5N 1 000 to the nearest 5N 10 000 after 3 years.6 The reciprocal of 0?08 is 12?5. The reciprocal 15 A man saves 5N 200 000 at 6% per annum compound interest. At the end of each yearof 80 is he adds 5N 18 000 to his total savings. Find his total amount after 3 years.A 0?001 25 B 0?012 5 C  0?125D 1?25 E 125 0007 If x 5 2y 2 1 and 2x 5 3y 1 2, then x 5 16 A long straight road makes an angle of 7° A 29 B  24 C  4 D  7 E  9 with the horizontal. Two posts are 1 km apart on the road. Calculate in metres: 8 Two of the angles of a triangle are x° and a the horizontal distance between the posts, (90 2 x)°. Which of the following is (are) b the difference in height between the posts. correct? 17 First simplify, then solve, the following I sin x° 5 cos(90 2 x)° simultaneous equations: II cos x° 5 sin(90 2 x)° 3(2x 2 y) 5 x 1 y 1 5 III tan x° 5 ​_t a_n_(_9_01__ 2_ _x_ )_° ​ 5(3x 2 2y) 5 2(x 2 y) 1 1 A I only B I and II only C III only 18 A cyclist travels for x hours at 20 km/h and D all of them E none of them then for y hours at 12 km/h. Altogether she travels 60 km in 4 hours.Table R5 gives the approximate hand spans of 20 a Express the total time taken in terms of xstudents. and y. b Express the total distance travelled inhand span (cm) 19 20 21 22 23 terms of x and y. c Hence find x and y.frequency  3  2  5  8  2Table R5 19 The acute angles of a rhombus are 76°. If the shorter diagonal is of length 8 cm, calculateUse Table R5 to answer questions 9 and 10. the length of the sides of the rhombus to the nearest mm. (First make a good sketch.) 9 What is the median size of hand span? A 20?8 cm B 21 cm C 21?2 cm 20 A small factory employs 10 workers, two D 21?5 cm E 22 cm supervisors and one manager. The workers get 5N 7 000 per week, the supervisors get10 What is the mean hand span? 5N 11 000 per week and the manager gets A 20?8 cm B 21 cm C 21?2 cm 5N 17 200 per week D 21?5 cm E 22 cm a Calculate the mean weekly wage at the factory.11 Expand the brackets in (4a 1 3)(a 2 2) 2 b Compare this result with the modal wage. (2a 2 3)(2a 1 3) 5 0 and hence solve the Which average is most representative of equation. the weekly wages?12 Factorise the following. a n2 1 12n 1 32 b x2 2 14x 1 33 165Unit16.indd 165 28/3/08 13:25:46