6.1 6.1 Using other mathematical methodsUsing other mathematical methods 231231
Some types of question may be answered in a
more straightforward manner by using
mathematical techniques of a slightly higher
level than those required so far. In particular,
simple algebra can be used to give a clear
statement of the problem, which can then be
solved by standard mathematical methods.
Other areas where some mathematical
knowledge can help are those such as
probability, permutations and combinations,
and the use of highest common factors and
lowest common multiples. Although these
techniques are beyond the elementary
methods we have used so far, they are dealt
with in the early stages of secondary
education, and most candidates for thinking
skills examinations will have some knowledge
and skill in these areas. Probability is covered
in Chapter 6.3.
Percentages
Most people have a grasp of simple
percentages: if a candidate gets 33% of the
vote in an election it is quite easy to
understand that this means about^13 of voters
voted for them. Things become a little more
complicated when we try to multiply or divide
percentages or deal with percentages over 100.
There are, however, very easy ways to tackle
these to make them easier to understand. In
the example above, suppose only 60% of those
eligible to vote actually voted in the election.
What percentage of the total number eligible
did the candidate get? Once again, most
people will be able to handle this, but it is
easier to move away from percentages to do it.
Multiplying 33 by 60 does not help a lot; we
Using other mathematical
methods
Unit 6 Problem solving: further techniques
6.1
need to understand that 33% is^13 and 60% is^35 ,
then multiply the proportions together:^13 × 35
= 15 or 20% is the answer. If a town’s
population is now 120% of what it was 10
years ago, when it was 50,000, the population
is now 1.2 × 50,000, or 60,000. Once again we
had to move from percentages to ratios to do
the calculation.
In many cases where problems involve
percentages the best way to proceed is to use
real numbers rather than percentages. In the
first example above, if 100 people were eligible
to vote, 60 actually voted. Of these 33% or 33
out of 100 voted for the candidate, so 60 × 10033 ,
or 20 voted for them. This may seem
unnecessary in this simple case, but the value
of this approach becomes clearer in the
example below.A blood test is carried out to screen suspects
of a crime. 2% of the population of Bolandia
possess ‘Factor AX’ which is identified by the
test. However, the test is not perfect and 5%
of those not having Factor AX are found
positive by the test (these are called false
positives). Furthermore, in 10% of those with
Factor AX, the test fails to identify them as
having it (false negatives).
A suspect for a crime was tested and
found positive for Factor AX. A lawyer for the
defence asked what the chances were that
somebody testing positive in the test actually
had Factor AX.Activity