3.8
In the introductory chapters we saw that problems involving making inferences from data or suggesting reasons for the nature of variations in the data may appear in either the critical thinking or the problem-solving sections of thinking skills examinations.
Such examples are usually based on
quantitative (numerical or graphical) data and may arise from such areas as finance or
science. They require analysis of the data given in order to reach some conclusions that may be drawn from the data or to suggest reasons for the nature of the data.
The example below is based on a scientific scenario. While this requires a little
understanding of basic scientific concepts, most of the skills involved in coming to a solution depend on clear, logical thinking.
The graph shows the results of an experiment to determine the growth of a culture of yeast in a nutrient medium. The liquid containing the nutrient was made up and a small amount of yeast introduced. At regular intervals afterwards, the solution was stirred, a small sample taken and the concentration of yeast measured. The graph represents a smooth line drawn through the results.
Yeast concentration
Time
Activity
Which of the following explanations are consistent with the shape of the curve?
(Identify as many as apply.)
A Yeast cells divide when they have grown enough, so grow exponentially if they have enough nutrient.
B The rate of increase of yeast cells depends only on the amount of nutrient.
C Eventually, the growth of yeast cells is limited by lack of nutrient.
D Yeast cells die when there is insufficient nutrient.
E The shape of the curve is explained by a linear growth in yeast and a linear decrease in nutrient.
So A and C are the explanations for the shape of the curve.
The activity below is firmly in the
Cambridge Thinking Skills syllabus category of
‘suggesting hypotheses for variations’. You are given a scenario incorporating numerical data, and asked which, of a number of possible situations, could explain the nature of the data.
Nikul runs exercise classes at his local gym, and gets there each day by train and bus.
Classes start at different times each day, but always either on the hour or at half past the hour. He always gets to the railway station 45 minutes before he is due to start teaching and the train journey takes 20 minutes, after which he takes a bus to the gym, which takes 10 minutes. Trains leave every 20 minutes, starting on the hour. Some days Nikul finds that he gets to work 5 minutes early. On all the other days he finds that he gets there 5 minutes late.
Which one of the following could explain the times that Nikul arrives at the gym?
A The buses leave at 5 and 35 past each hour.
B The buses leave at 15 and 45 past each hour.
C The buses leave at 25 and 55 past each hour.
D The buses leave at 5, 25 and 45 past each hour.
E The buses leave at 15, 35 and 55 past each hour.
Activity
Commentary
Nikul arrives at the station at either 15 or 45 past the hour. Therefore, he takes the train either on the hour or at 20 past the hour. He gets to the bus stop at 20 or 40 past the hour.
Buses at 5, 25 and 45 past the hour would therefore fit the requirements:
• He would get the bus at 25 past the hour if he arrived at 20 past. This would mean that he arrived at work 5 minutes late.
• He would get the bus at 45 past the hour if he arrived at 40 past. This would mean that he arrived at work 5 minutes early.
• He would never use the bus at 5 past the hour, so it doesn’t matter that this one doesn’t fit the arrival times.
The correct answer is D. It is also illustrative to see why the wrong answers do not work:
A Buses at 5 and 35 past the hour would always get Nikul to work on a quarter hour which could not be 5 minutes early or late.
B Buses at 15 and 45 past the hour would mean that Nikul was always 5 minutes early.
C Buses at 25 and 55 past the hour would mean that Nikul was always 5 minutes late.
E Because Nikul arrives on the train at 20 or 40 past the hour, he would be getting the 35 or 55 past the hour bus. The bus at 35 past the hour would get Nikul to work at 15 or 45 past the hour which is neither 5 minutes early nor 5 minutes late.
Longer questions at A Level can involve analysing quite complex data and determining what conclusions may be drawn from it. The activity below is of this type. It looks at identifying reasons for variations in data.
B A rise in Danotian inflation would only cause the fall in the ratio shown if the rise in inflation in Wembling was smaller or negative: we know nothing about this.
C If food prices rose less in Wembling than in Danotia as a whole, this could explain why the ratio fell – even if inflation in Wembling was rising.
D Seasonal fluctuations would only manifest themselves within a year, not between years.
E Even if the inflation rate in Wembling is falling, we know nothing about the inflation rate in the whole of Danotia, so we cannot conclude that the ratio would fall.
Thus C is the only reasonable answer. The others depend either on reading the graph incorrectly or reading more into the graph than we can safely conclude. This illustrates the importance of reading and understanding the information given (both verbal and in other forms) and of reading the question correctly. Beyond that, the deductions that can be made follow from the application of correct logic.
As in the previous chapter, solving these two types of problem also depends on the skill of recognising an identity between data presented in two forms. As with the other skills described here, this comes with practice and it can be useful to look at data in
newspapers to see how they are presented and to consider whether they are always presented in the clearest way.
The next activity uses logic based more on manipulating numbers. In this regard, it has elements of the ‘finding methods of solution’
skill from Chapter 3.5. However, the nature of the question places it closer to the ‘suggesting hypotheses for variations’ category of questions.
Commentary
Again we will look at these five answers in turn. It is important to remember that the graph represents the inflation in Wembling as a percentage of the inflation in Danotia, not the actual inflation rate in Wembling.
A The graph represents only the ratio between Wembling and the whole of Danotia; high inflation in Danotia cannot explain the shape of a graph of the ratio.
The graph shows the inflation rate in the province of Wembling as a percentage of inflation in the country of Danotia as a whole over the period from 2006 to 2012.
2006 2007 2008 2009201020112012 100
90
70 80
Wembling ination as a percentage of Danotian ination
Which of the following is the most plausible explanation for the variations shown in the graph?
A Danotian inflation has been high over the period shown.
B Danotian inflation has risen over the period shown.
C Food prices in Wembling have risen less than in Danotia as a whole.
D Food prices in Wembling are subject to higher seasonal fluctuations than in Danotia as a whole.
E The inflation rate in Wembling is falling due to high unemployment.
Activity
they leave Whitesea at 9.00 a.m., 9.20 a.m., 9.40 a.m., etc. They then leave Greylake 1 hour 10 minutes later, at 10.10 a.m., 10.30 a.m., 10.50 a.m., etc. A driver leaving Whitesea, for example, at 11.20 a.m. will see the trams which left Greylake at 10.30 a.m., 10.50 a.m.,
11.10 a.m., 11.30 a.m., 11.50 a.m. and 12.10 p.m. – six in total.
Looking at statement A, the first tram leaving Whitesea at 9.00 a.m. reaches Greylake at 10.00, leaves at 10.10 and returns to
Whitesea at 11.10, in time to become the 11.20 service. In the meantime, other trams will have left at 9.20, 9.40, 10.00, 10.20, 10.40 and 11.00, i.e. six more, so there must be seven trams to run the service. A is incorrect.
Looking at statement C, if I sit near the midpoint from 11.15 a.m. to 12.15 p.m., I will see the 11.00, 11.20 and 11.40 trams going one way and the 10.50, 11.10 and 11.30 trams going the other way, so I will see six in total. C is correct.
Can you confirm these answers by constructing a timetable?
This kind of problem will be revisited in Chapter 6.2 where we look at graphical solutions to problems.
• We have encountered examples where we are required to suggest a hypothesis or a reason for the nature of variation in data.
• The terms hypothesis, reason, explanation and inference are used in exactly the same way in problem solving as in critical thinking, the only difference being that the information given is in the form of data rather than verbal description.
• We have seen that extended examples where more data is supplied can require analysis that may lead to a range of conclusions.
Summary
Commentary
We can test the statements by making a
timetable. However, to do this, we need to make an assumption about the departure intervals. It is, in fact, better to carry out a little analysis first.
When a tram driver leaves Whitesea, all those trams which left Greylake up to one hour earlier will already be on their journey.
During the hour it takes him to travel to Greylake, more trams will be leaving Greylake.
So, in total, he will see all those trams which left up to an hour before he left, and also all those which leave up to an hour after he left.
This means the six trams he sees must have left Greylake in a two-hour period. As they leave at regular intervals, one must leave every
20 minutes, so B is incorrect.
This may be illustrated by looking at some actual times. If trams leave every 20 minutes, (Harder task) A tram company runs a service along the seafront from Whitesea to Greylake.
Trams leave Whitesea at regular intervals, starting from 9 a.m., taking one hour to reach Greylake. They then turn around and start back 10 minutes after arriving. A driver in the middle of the day, in his journey from Whitesea to Greylake, always sees six trams travelling in the opposite direction. Some of these will have set off from Greylake before he left, and some will have set off after he left.
Which of the following must be true? There may be more than one. If any of the
statements are not true, can you correct them?
A It takes six trams to run the service.
B The trams run every ten minutes.
C If I sit on the seafront from 11.15 a.m.
to 12.15 p.m., I will see six trams going past.
Activity
at half the time shown in the graphs, patients have stopped taking it.
Assuming that the effects of the two drugs are independent, what would be the expected shape of the graph of effectiveness for a patient on the regime described?
2 At a local school, 70% of the students studied French and 45% studied German.
Which of the following can be confirmed from the information given?
A All students study either French or German.
B 23 of those studying German do not study French.
C 25% of the students study neither French nor German.
D At least 15% of students study both French and German.
3 I was shopping at a market in Northern Bolandia and asked a local how much an orange was. He said that an orange and a lemon together cost $2. He then further confused me by saying that a grapefruit and a lemon cost $3 and that all three were different prices.
Based on this rather unhelpful
information, which one of the following can be confirmed?
A An orange costs more than a lemon.
B A lemon costs more than a grapefruit.
C A grapefruit costs more than a dollar.
D An orange costs less than a dollar.
1 In order to treat a particular disease effectively, patients are initially given two drugs. Drug A alone has the effect shown on the graph below. (10 = total relief from symptoms. 1 = no relief.)
1 10
Time
Average eectiveness
Drug A
The effect of drug B showing how it varies with time is shown in the graph below (on the same effectiveness and time scales).
Average eectiveness
1 10
Time Drug B
The reason the patients are given two drugs is that drug A, whilst being very effective, has long-term harmful side effects. Drug B takes some time to
become effective, and has a lower eventual effect but can be taken indefinitely. The regime used by doctors is to give both drugs starting at the same time, then to withdraw A at a uniform steady rate until,
End-of-chapter assignments
Which of the following is possible?
A Liam works on Thursday.
B Nadila works on Sunday and Monday.
C Orla works on Monday and Tuesday.
D Three people work on a Monday.
Answers and comments are on pages 319–21.
4 The Fitland health centre swimming pool is open seven days a week. There are four lifeguards, Liam, Moses, Nadila and Orla, each working four days a week. None works four consecutive days and at least two lifeguards are on duty each day, with three on duty on Saturday and Sunday.
We know that:
• Liam doesn’t work Monday or Saturday.
• Moses doesn’t work Tuesday, Thursday or Friday.
• Nadila doesn’t work Wednesday or Friday.
• Orla always works on Thursday.
Approximately what proportion of the two tiles will be needed to cover the whole floor?
A three white to one black B two white to one black C equal quantities of both D two black to one white E three black to one white