6.1
need to understand that 33% is 13 and 60% is 35, then multiply the proportions together: 13 × 35
= 15 or 20% is the answer. If a town’s population is now 120% of what it was 10 years ago, when it was 50,000, the population is now 1.2 × 50,000, or 60,000. Once again we had to move from percentages to ratios to do the calculation.
In many cases where problems involve percentages the best way to proceed is to use real numbers rather than percentages. In the first example above, if 100 people were eligible to vote, 60 actually voted. Of these 33% or 33 out of 100 voted for the candidate, so 60 × 10033 , or 20 voted for them. This may seem
unnecessary in this simple case, but the value of this approach becomes clearer in the example below.
A blood test is carried out to screen suspects of a crime. 2% of the population of Bolandia possess ‘Factor AX’ which is identified by the test. However, the test is not perfect and 5%
of those not having Factor AX are found positive by the test (these are called false positives). Furthermore, in 10% of those with Factor AX, the test fails to identify them as having it (false negatives).
A suspect for a crime was tested and found positive for Factor AX. A lawyer for the defence asked what the chances were that somebody testing positive in the test actually had Factor AX.
Activity
Commentary
Although a fictitious situation, this is similar to many real problems which medical and legal professionals have to deal with on a regular basis, for example in cancer diagnosis.
The answer is much less obvious than it seems and many people will glance at the results and give an answer of 95%, which is 100 minus the percentage of false positives.
Let us now take the approach of putting in real numbers. In this case we will start with a very large number (as some of the percentages are quite small). Say the population of
Bolandia is 10,000. Then 2%, or 200, of these have Factor AX. Of these 200, 180 are found positive by the test (i.e. found to have Factor AX) and 20 are found negative. Of the 9800 without Factor AX, 5%, or 490, are found positive and 9310 are found negative. The table below shows the results.
Found positive
Found negative
Total
With Factor AX 180 20 200
Without Factor AX 490 9310 9800
Total 670 9330 10,000
We can now answer the question: 670 people are diagnosed positive. Of these, 180 have Factor AX. 180670 is 0.27 or 27%. This is the required answer, the percentage chance that a person found positive in the test has Factor AX. Working this out directly from the percentages would be very difficult.
Algebra
Consider the problem below. This is similar to one we encountered earlier. It can be solved using intuition or trial and error, but the algebraic method illustrated is quicker. Use of such techniques can be a particular help when working on thinking skills questions under time pressure.
A ferry travels at 20 km/hour downstream but only 15 km/hour upstream. Its journey between two towns takes 5 hours longer going up than coming down. How far apart are the two towns?
Before looking at the algebraic solution below, you may like to consider alternative ways of solving the question.
Activity
Commentary
If the distance between the two towns is x km, we have:
Time upstream = 15x hours Time downstream = 20x hours
Thus, since the difference between these times is 5 hours:
x
15 − 20x = 5
Multiplying both sides by 60:
4x − 3x = 300
So x, the distance between the towns, is 300 km. Put this answer back into the question to check that it is right.
This was a very simple example and hardly needed the formality of a mathematical solution. However, similar methods can be used for more complex questions to reduce them to equations that can be solved quite easily. Try the problem below.
Kara has just left the house of her friend Betsy after visiting, to walk home. 7 minutes after Kara leaves, Betsy realises that Kara has left her phone behind. She chases Kara on her bicycle. Kara is walking at 1.5 m/s;
Betsy rides her bike at 5 m/s.
How far has Kara walked when Betsy catches her?
Activity
6.1 Using other mathematical methods 233 answer is the LCM of 6 and 8. The prime
factors of 6 are 2 and 3; the prime factors of 8 are 2, 2 and 2. One of the 2s is common to both so the LCM is 2 × 2 × 2 × 3 = 24, the same answer as before.
In this case there is little to choose between the two methods, but if the counting method gave no coincidence for 30 or 40 values, the LCM method would be much faster. There is another lighthouse example in the end-of- chapter assignments, but with a twist. Problem- solving question-setters often use such twists to take problems out of the straightforwardly mathematical so that candidates must use their ingenuity rather than just knowledge. Even so, using the mathematics you do know can often reduce the time necessary for a question.
Permutations and combinations
Another area where a little mathematics can help is in problems involving permutations and combinations. Here is another simple example.
Three married couples and three single people meet for a dinner. Everybody shakes hands with everybody else, except that nobody shakes hands with the person to whom they are married.
How many handshakes are there?
Activity
Commentary
Without the twist of the married couples, this would be very straightforward – the answer is
9 8×
2 = 36. You have to divide by 2 because the
‘9 × 8’ calculation counts A shaking hands with B and B shaking hands with A. The married couples can be taken care of easily, because they would represent three of the handshakes, so the total is 33.
The alternative way to do this is to count:
AB, AC, AD . . . AI, BC, BD, etc. This is very time-consuming.
Commentary
Once again, there is more than one way of answering this question, but algebra can make it much more straightforward. If Kara has walked x metres when Betsy catches her, the time taken in seconds from Kara leaving Betsy’s house is 1 5x. . The time for Betsy to cycle this distance is x5. We know that Kara takes 7 minutes (420 seconds) longer than Betsy, so:
x
1.5 − 15x = 420
Multiplying both sides by 15:
10x − 3x = 420 × 15 = 6300, so x = 900 metres
900 metres takes Kara 600 seconds and takes Betsy 180 seconds – a difference of 420 seconds or 7 minutes as required. We could also calculate that it takes Betsy 3 minutes to catch Kara.
Lowest common factors and multiples
Another example follows of a problem that can be solved using a simple mathematical technique.
From a boat at sea, I can see two lighthouses. The Sandy Head lighthouse flashes every 6 seconds. The Dogwin lighthouse flashes every 8 seconds. They have just flashed together. When will they flash together again?
Activity
Commentary
There is a straightforward way of solving this with little mathematics; just list when the flashes happen:
Sandy Head: 6, 12, 18, 24, 30 seconds later Dogwin: 8, 16, 24 seconds later
So they coincide at 24 seconds. Those with a little more mathematical knowledge will spot that this is an example of a lowest common multiple (LCM) problem. The
• This chapter has shown how knowledge of a few relatively simple mathematical techniques can make the solution of some problem-solving questions quicker and more reliable.
• Percentage calculations can be simplified by replacing the percentages with real numbers.
• The use of algebra, lowest common factors and multiples, and permutations and combinations can aid the finding of methods of solution and shorten the work required for some problems.
Summary
3 From my boat at sea I can see three lighthouses, which flash with different patterns:
• Lighthouse A flashes 1 second on, 2 seconds off, 1 second on, 1 second off, then repeats.
• Lighthouse B flashes 1 second on, 3 seconds off, 1 second on, 2 seconds off, 1 second on, 3 seconds off, then repeats.
• Lighthouse C flashes 2 seconds on, 1 second off, 1 second on, 2 seconds off, then repeats.
They have all just started their cycles at the same time. When do they next all go on at the same time?
4 Four friends have a photograph taken with them all throwing their graduation hats in the air. Afterwards they pick up the hats and find they all have the wrong hat. How many different combinations of picking up the hats are there? In how many of these combinations do they all have the wrong hat?
Answers and comments are on pages 333–34.
1 Rita has a small shop. 40% of the money she receives from selling cornflakes is profit. Next week she is having a sale and is selling cornflakes at three packets for the price of two. What percentage profit will she make on cornflakes sold under this offer?
2 At my local baker’s, the price of bread rolls is 25¢ and I went with exactly the right money to buy the number I needed. When I got there, I found they had an offer giving 5¢ off all rolls if you bought eight or more.
Consequently, I found I could buy three more for exactly the same money. How many was I originally going to buy?
End-of-chapter assignments
6.2 Graphical methods of solution 235
Graphical methods of solution
6.2
It can often be useful to draw a simple picture when trying to analyse a problem. This can take the form of a map, a diagram or a sketched graph. Some examples where such pictures can help are given below.
Map
The town of Perros is connected to Queenston then to Ramwich and finally Sandsend and back to Perros by a circular bus service. Ramwich has a bus service to Upperhouse via Tempsfield. Queenstown has a bus service to Ventham via Tempsfield.
Orla is visiting the area and wants to look at all these towns starting and finishing at Perros. What is the smallest number of stages (i.e. journeys from one town the next) she can do the journey in?
A 7 B 8 C 9 D 10
Activity
Commentary
It would be very difficult to answer this question without some sort of picture. Our sketch of the towns and bus services only has to be quite rough and is shown below.
Q
P
S R
V
T U
This now becomes a straightforward problem.
In order to achieve the minimum number of stages, the shortcut between Q and T must be taken either on the way out or on the way back (but not both as we need to visit R). It is
possible to go either way round, but both will result in the same number of stages. One minimum route is:
P-Q-T-U-T-V-T-R-S-P The answer is C, 9 stages.
Graph
Two buses run services between Southbay and Norhill. One is an express service which completes a one-way journey in one hour. The other is a stopping service which takes 1 hour 45 minutes. The express service starts at Southbay at 8 a.m. and the stopping service starts at Norhill at 8 a.m. When each bus reaches its destination, it waits for 15 minutes before setting off again. This continues throughout the day. The last journey of the day is the last to finish before 8 p.m., each bus stopping at the town where it started.
How many times do the drivers pass each other in opposite directions on the road during the day?
Activity
Commentary N8 a.m.
8 a.m.
8 p.m.
8 p.m.
12 p.m.
12 p.m.
S
The black line shows the express service bus which starts from Southbay at 8 a.m. This takes one hour to reach Norhill, where it stops for 15 minutes; the next line shows the return journey and so on through the day.
Commentary
A Venn diagram for this problem is shown below. The rectangle represents all those who voted. We do not need to consider the non- voters as the exit poll does not categorise whether non-voters can be defined as Blue, Red, Men or Women. We just need to remember that only 70% of the electorate voted.
BM
RM RW BW
The left circle represents the Red voters and the right circle represents Women voters. R represents Red, B represents Blue, W represents Women and M represents Men.
We know that the Red vote was 60% of those who voted, so the areas:
RM + RW = 0.6 × 0.7 = 0.42, i.e. 42% of the electorate, and
BM + BW = 0.4 × 0.7 = 0.28, i.e. 28% of the electorate
We know that 50% of the electorate were women; 70% of these voted; of these, 30%
voted Red and 70% voted Blue, so:
RW = 0.5 × 0.7 × 0.3
= 0.105, i.e. 10.5% of the electorate, and BW = 0.5 × 0.7 × 0.7
= 0.245, i.e. 24.5% of the electorate We can now calculate the proportion of the electorate in each area of the diagram:
RW = 10.5%, BW = 24.5%, RM = 31.5%
and BM = 3.5%
We can check that this is correct as these add up to 70% – the turnout, and both men and women add to 35% – equal numbers.
The proportion of women voting Red is 10.5/(10.5 + 24.5) = 30% and the proportion of Red voters is (10.5 + 31.5)/70 = 60%.
Similarly, the coloured line shows the stopping service, starting at Norhill at 8 a.m.
and taking 1 hour 45 minutes to reach
Southbay, where it waits for 15 minutes before starting the return journey.
The intersections, shown by circles, indicate where the buses pass in opposite directions:
five times in total. There is also one point where the fast bus overtakes the slower one and various positions when they are at either Southbay or Norhill at the same time.
This question would have taken a very long time to solve without the diagram as the crossing points would have had to be inferred from a timetable.
Venn and Carroll diagrams
Venn diagrams were introduced in Chapter 3.5.
The problems considered there were relatively simple and could have been solved without the diagrams, just by using a bit of clear thinking.
In this chapter we are going to look at problems that are more complicated and, although they could be solved without the use of diagrams, the diagram makes the solution much more straightforward.
Taking a problem of a similar nature to that which was used to introduce Venn diagrams, the extension to one more category makes analysis of the problem much more complex, as shown below.
Elections have just been held in the town of Bicton. There were two parties, the Reds and the Blues. Turnout to vote was 70%. The Reds got 60% of the vote and the Blues the remaining 40%. An exit poll showed that 30%
of women voting voted Red, whilst 70% voted Blue. (There are equal numbers of men and women registered to vote and the percentage turnout was the same for men and women.)
What proportion of men in the total electorate voted Blue?
Activity
6.2 Graphical methods of solution 237 easier to understand than the Venn diagram
and the various subdivisions and sums may be more easily seen and totalled.
A general household repairs business has 15 workers. Two are managers and do not have specialised skills. Five are plumbers and do not do other jobs. There are six electricians and a number of carpenters. Of these, three can work as either electricians or carpenters.
How many are carpenters but not electricians?
Activity
Commentary
The Venn diagram for this problem is shown here.
Managers 2
Electricians
3 Carpenters
3 ?
Plumbers 5
As none of the plumbers are either electricians or carpenters, their area does not intersect with the other two. The entire outer box represents the 15 workers. The ‘2’ shown on the diagram outside the circles represents the two managers who do not fit any of the other categories. The 5 plumbers are shown in their circle. The intersection between electricians and carpenters represents the 3 which fall into both categories. As there are 6 electricians, there must be 3 who are not also carpenters.
We now have 13 accounted for so the remainder, 2, must be carpenters but not electricians.
The area BM indicates that 3.5% of the electorate were men who voted Blue. Since half the electorate are men, we can now answer the original question: 7% of men voted Blue.
This question can also be solved using a Carroll diagram (originally devised by Lewis Carroll, author of Alice’s Adventures in Wonderland), which is really just a table representing the areas shown in the Venn diagram. Some people may find Carroll diagrams easier to understand. Venn and Carroll diagrams become more complicated when there are more categories of things involved, but a problem involving more than three categories is unlikely to appear in a thinking skills examination. A Carroll
diagram for two categories is just a 2 × 2 table (it has four areas, just like the Venn diagram).
You might like to revisit the Venn diagram activity in Chapter 3.5 using a Carroll diagram.
The Carroll diagram for three categories may be drawn with an inner rectangle
expressing one level of the third category (e.g.
non-voters) and, for the problem above, would appear as shown:
Red Blue
24.5%
10.5%
Non-voters 30%
31.5% 3.5%
Men Women
The inner rectangle is not subdivided as it represents the non-voters. In this case (and, in fact, in many cases) the Carroll diagram is
• In this unit we have seen how various diagrams may be used to represent and solve problems in categorisation, logic and searching.
• We have looked at using sketched maps and graphs to clarify and simplify quite complicated problems.
• More advanced Venn and Carroll diagrams have been introduced for problems involving three levels of categorisation.
Summary
2 Draw a Venn diagram for three categories to sort the numbers from 1 to 39 according to whether they are even, multiples of three or square numbers.
Write each number in the appropriate part of the diagram.
3 The island of Nonga has two ferry ports:
Waigura and Nooli. All ferries from Waigura go to Dulais on a neighbouring island.
Some ferries from Nooli also go to Dulais.
Some of the ferries that serve Dulais are fast hydrofoil services; those going elsewhere are slow steamboats.
Which of the following statements can safely be concluded from the information given above?
A No hydrofoils go to Dulais from Nooli.
B All hydrofoils going to Dulais leave from Waigura.
C Some hydrofoils from Nooli go to places other than Dulais.
D Some steamboats from Waigura go to Dulais.
E All hydrofoils from Waigura go to Dulais.
1 Winston is organising a dinner to raise money for his football team. The hall he has hired is a square room measuring 15 metres by 15 metres. The tables are rectangular. Each one measures 2 metres by 80 centimetres and can seat up to eight people, as indicated in this diagram:
To fit as many people as possible into the hall, Winston plans to put the tables together, end to end, to create parallel rows. He can use as many tables as he can fit in, but he has to make sure there is a gap of at least 1.5 metres between the edge of any table and the edge of the room, and also a gap of at least 1.5 metres between rows of tables.
What is the maximum number of people that could sit down to eat at Winston’s dinner?
A 190 B 192 C 228 D 240 E 288
End-of-chapter assignments
6.2 Graphical methods of solution 239 for 34 hour. Over a long period, what is the
percentage of times they will coincide at the gym?
Answers and comments are on pages 334–35.
4 (Harder task) Anna and Bella both go to the gym on the same three days each week. The gym is open from 8 a.m. to 10 p.m. and either may arrive, quite randomly, any time between 8 a.m. and 3 p.m. Anna stays for one hour and Bella