Thinking Skills: Critical Thinking and Problem Solving

236 Unit 6 Problem solving: further techniques

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

RMRWBW

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