Thinking Skills: Critical Thinking and Problem Solving

3.11 Choosing and using models 121

Commentary
If we just look at the data as it stands the
pattern is not clear, other than that the price
per passenger drops with the number of
passengers. Since we are looking at the charge
made by the taxi company, it is preferable to
look at the total cost of the taxi in each case.
This may be carried out by multiplying the
cost per passenger by the number of
passengers, as shown in the table below.
The pattern now becomes much clearer.
Allowing for some small variations (it was stated
that there was a small variation in journey
time), the first two values are the same and they
then increase by $4 per passenger. We can,
therefore, conclude that the taxi company hire
fee includes one or two passengers, then there is
an extra charge of $4 per additional passenger.
The $40 ‘basic’ fee covers the hire charge, the
distance charge and an average time charge. We
have no information which will enable us to
separate these three items. A model can only be
as good as the data on which it is based.

Number of passengers 1 2 3 4 5

Charge per journey per passenger $40.00 $19.98 $14.68 $12.03 $10.38

Total charge per journey $40.00 $39.96 $44.04 $48.12 $51.90

A graph can be a very useful tool for

analysing data such as in the table below, and

can also help in developing models. Try

graphing the data, for both the cost per

passenger and the total cost per journey. Does

this help in clarifying the charging structure?

The activity above introduced the idea that

models usually are approximations to the real

world. The model used did not allow for

variations in the time of the journey. This is

why the word ‘model’ is used. Almost all

models are approximate – the model car does

not usually have an internal combustion

engine. Economic models cannot take into

account factors such as the weather.

Many people use models in their everyday

lives without even realising it. An efficient

shopkeeper will, for example, have a set of

rules that tells her how much ice cream to

order so she has plenty in the summer months

and less stock in the winter.

•   We have learned how a mathematical

or graphical model may be used to

approximate real-life processes.

•   We have seen how models can be used

to simulate changes in cost structure and

their effects.

•   We have used real data to calculate the

constants used in a mathematical model,

for example the starting rate and charge

per mile for a taxi fare.

•   A graph of any sort is a model from which

it is possible to get a picture of how

variations can occur.

Summary