Thinking Skills: Critical Thinking and Problem Solving

6.2 Graphical methods of solution 235

6.2 Graphical methods of solution

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

N

8 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.