Thinking Skills: Critical Thinking and Problem Solving

206 Unit 5 Advanced problem solving

Commentary

This is mainly a data-extraction type question.

Such questions are normally quite

straightforward but this one includes a large

amount of information to digest, and a

method of solving it also needs to be found.

There are three important things (the first

skill is to identify these):

1 The scoring system, which means

that with two villages left to vote, the

maximum extra votes that any one

village can score is 16.

2 The fact that a village cannot vote for

itself, which means that Riverton and

Runcastle can only receive a maximum

of 8 more votes.

3 Some villages might score no more, so

any village that can pass the mark of 24

can still win.

Given these three things, the method becomes

much clearer. The appropriate maximum

available must be added to each team and the

result compared with 24. The allocation of the

lesser votes is unimportant, as they could go to

villages who have no hope anyway.

Adding 16 votes to each of the first eight

villages, we see that four of them can exceed

24: Longwood, Gigglesford, White Stones and

Martinsville. Adding 8 votes to each of the last

two, we see that Riverton cannot reach 24 but

Runcastle can reach 25. So five teams can still

win. Runcastle would be best advised not to

vote for Longwood or Martinsville!

You may see that this question required

no new skills, and the mathematics was

limited to simple addition and counting. The

difficulty in this question was in using the

information correctly and seeing how best

to proceed.

The next activity gives an example in which

the main problem is in identifying a method

of proceeding. The information in this case is

much simpler.

A survey of Bolandian petrol prices showed

the average to be 82.5¢ per litre. Filling

stations in the province of Dorland made up

5% of the survey and the Dorland average

was 86¢ per litre.

On average, how much more expensive is

petrol in Dorland than in the rest of the

country?

Activity

Commentary

This problem is not, in principle, any harder

than those we have encountered earlier. It is

mathematically slightly more complex and a

clear idea of the meaning of an average must

be retained.

We can quickly note that 5% is^120 of the

total. One easy way to proceed is to assume

that there were 20 filling stations in the

survey, one of which was in Dorland.

The sum of the prices at all Bolandian filling

stations must have been 20 × 82.5¢ = 1650¢.

The price in the Dorland filling station was

86¢. Therefore the sum of the prices in the

remaining 19 was 1650¢ – 86¢ = 1564¢. The

average in the rest of the country was

1564

19

82^6

19

= 

or about 82.3¢. So Dorland prices are, on

average, 3.7¢ more expensive than in the rest

of the country.

Since all the numbers are just over 80¢, we

could make life easier by subtracting 80¢ from

everything, leaving smaller numbers to work

with. As long as we remember to add the 80¢

back on at the end, this will still give the right

answer. For example, if we wanted the average

of 82¢ and 86¢, we could say this was

(82¢ + 86¢) ÷ 2 = 84¢. It would be much easier

to note that the average of 2¢ and 6¢ is 4¢, then