84 Unit 3 Problem solving: basic skills
having a ‘Buy two get a third free’ offer on
this item. The supermarket does not want to
lose money on this offer, so it expects the
manufacturers to reduce their prices so
SuperSave will make the same actual profit
on every three bottles sold.
By how much will the manufacturers have
to reduce their prices?A (^16) B 14 C 13 D 12 E (^23)
problems before and you think carefully about
the information given.
Finally, to be sure that you have found the
correct solution, check the answer. The profit
on one bottle was $1.20 − 80¢ = 40¢; the profit
on three bottles under the offer is $2.40 −
$1.20 = $1.20, or 40¢ per bottle. That’s correct!
You should have learned a little about finding
a method of solution from this example. The
guesswork method can only work by luck. This
may be called the ‘pirate’s gold’ approach – we
know the treasure is on the island somewhere so
we dig a hole. If it’s not there, we dig another
one somewhere else. Sometimes this method
may seem to work, but it is usually because a
little previous experience has been used, even
unknowingly. The trial and error method,
sometimes using a common sense strategy
which turns it into a partial search, can be
effective for solving some problems. Other
problems may need an exhaustive search to
solve; these are discussed in Chapter 3.6.
In the case above – and in many others – the
method of finding a clear strategy was the
most efficient. Strategies are not always found
by rigorous methods; the discovery of an
appropriate strategy usually depends on past
experience of similar problems.
• We have looked at some methods of
solving problems, investigating how
different methods may be used in different
circumstances.
• We have recognised the value of
experience in identifying problem types and
appropriate methods of solution.
• We have seen how important it is to read
and understand the information and the
question.
• We have looked at the relative merits
of guesswork, searching and strategic
methods of solution.
Summary
Commentary
This could be solved in a variety of ways. We
could just guess. As we are letting^13 of the
bottles go for free, option C,^13 , is tempting.
This is wrong.
It could be done by trial and error: for
example, start with the manufacturers
charging 60¢ (this would be option B) and see
what that leads to. For three bottles they will
charge $1.80 and the supermarket sells for
$2.40, so their mark-up is 60¢ for three bottles
or 20¢ each. This is not enough, so the
manufacturers’ price must be lower.
In fact there is a straightforward, systematic
way of solving this which is made clear by
writing down all the relevant values which can
be calculated:
Normally SuperSave sell at $1.20, so they
buy at 80¢ (selling at 50% more than they
buy), so each bottle is sold for 40¢ more than
the price at which it is bought.
Under the offer, they will sell three bottles
for the price of two, i.e. three for $2.40, or 80c
each. If they are still selling for 40¢ more than
the price at which they have to buy, they will
be buying from the manufacturer at 40¢. So,
the manufacturers will have to halve their
price. Option D is correct.
This method was quite quick, and certainly
quicker than the trial and error method. It is
the sort of solution that you are more likely to
come up with if you have seen a lot of similar