Thinking Skills: Critical Thinking and Problem Solving

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
10.5% 24.5%
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