Thinking Skills: Critical Thinking and Problem Solving

232 Unit 6 Problem solving: further techniques

Commentary

Although a fictitious situation, this is similar

to many real problems which medical and

legal professionals have to deal with on a

regular basis, for example in cancer diagnosis.

The answer is much less obvious than it seems

and many people will glance at the results and

give an answer of 95%, which is 100 minus the

percentage of false positives.

Let us now take the approach of putting in

real numbers. In this case we will start with a

very large number (as some of the percentages

are quite small). Say the population of

Bolandia is 10,000. Then 2%, or 200, of these

have Factor AX. Of these 200, 180 are found

positive by the test (i.e. found to have Factor

AX) and 20 are found negative. Of the 9800

without Factor AX, 5%, or 490, are found

positive and 9310 are found negative. The

table below shows the results.

Found

positive

Found

negative

Total

With Factor AX 180 20 200

Without Factor AX 490 9310 9800

Total 670 9330 10,000

We can now answer the question: 670 people

are diagnosed positive. Of these, 180 have

Factor AX.^180670 is 0.27 or 27%. This is the

required answer, the percentage chance that a

person found positive in the test has Factor

AX. Working this out directly from the

percentages would be very difficult.

Algebra

Consider the problem below. This is similar to

one we encountered earlier. It can be solved

using intuition or trial and error, but the

algebraic method illustrated is quicker. Use of

such techniques can be a particular help when

working on thinking skills questions under

time pressure.

A ferry travels at 20 km/hour downstream

but only 15 km/hour upstream. Its journey

between two towns takes 5 hours longer

going up than coming down. How far apart

are the two towns?

Before looking at the algebraic solution

below, you may like to consider alternative

ways of solving the question.

Activity

Commentary

If the distance between the two towns is x km,

we have:

Time upstream = 15 x hours

Time downstream = 20 x hours

Thus, since the difference between these times

is 5 hours:

x

15 −

x

20 = 5

Multiplying both sides by 60:

4 x − 3x = 300

So x, the distance between the towns, is

300 km. Put this answer back into the question

to check that it is right.

This was a very simple example and hardly

needed the formality of a mathematical

solution. However, similar methods can be

used for more complex questions to reduce

them to equations that can be solved quite

easily. Try the problem below.

Kara has just left the house of her friend

Betsy after visiting, to walk home. 7 minutes

after Kara leaves, Betsy realises that Kara

has left her phone behind. She chases Kara

on her bicycle. Kara is walking at 1.5 m/s;

Betsy rides her bike at 5 m/s.

How far has Kara walked when Betsy

catches her?

Activity