246 Unit 6 Problem solving: further techniques
6.4 Have you solved it?
This chapter considers how you may check
and be sure that your answer to a problem-
solving question is correct. In real life, there
might be several possible answers, or even no
answer to a problem. (Can you fit a square peg
into a round hole?) However, in examinations,
especially those with multiple-choice answers,
there must be a correct answer. One of the
options in some cases might, of course, be that
the task cannot be done. This means that,
when you have an answer, you must have a
way of being sure that it is correct.
Different problems need to be checked in
different ways. Sometimes it is possible to put
the answer back into the question and see if it
‘fits’. This is probably the easiest way. For
example, look at the question in Chapter 3.5
(page 94) about Amy and David passing on the
road. We concluded that they would pass at
10.40 a.m. We can now go back and see where
they both were at 10.40 a.m. Amy left at 8 a.m.
so by 10.40, at 120 km/hr, she had travelled
320 km. Similarly, David, leaving at 10 a.m.,
had travelled 80 km. The total is 400 km,
which is the distance between their two
houses, so the answer is correct.
Similarly, the first problem in Chapter 3.3
(page 86), with the table showing
participation in exercise, could be checked by
putting the correct figure into the table and
seeing whether all the rows and columns
added up correctly.
You can go back and check the answers for a
lot of the activities and examples in Unit 3 of
the book by using the ‘put it back in’ method.
However, the second problem in Chapter
3.3, involving a graph showing temperatures,
cannot be checked in this way. We are simply
being asked to extract the right value from thegraph and there is no way of putting this back
in to see whether it is right. In cases like this,
the answer simply has to be checked carefully.
What exactly was the question asking? Is this
what we answered? Is the numerical value of
the answer about what we would expect?
The same applies to questions requiring a
search. ‘Putting the answer back’ will tell you
whether your answer fits the criteria asked for
in the question but will not tell you whether it
is the lowest (or largest) possible answer. If the
search is not too large, you can sometimes
check, if you are looking for the lowest answer,
that all smaller answers will not work. This can
be time-consuming and impractical if the
search is large. It is often better to check your
method and be sure that it will come up with
the correct answer.
Approximation, or a feel for the magnitude
of results, is a skill that can be refined through
practising this type of question. This is
particularly valuable when questions depend
on getting the decimal point in the right
place. A minimum temperature of 10°C might
be acceptable when 100°C would not.
The end-of-chapter assignment considers
several problems that may have a variety of
ways of checking. It is always preferable to use
a different method for checking the problem
from that which you originally used to solve it.
If you simply repeat your original calculation,
it is possible that any mistake you made in the
first instance you will make again.
Checking the answers of questions
involving searches (see Chapter 3.6) can be
more difficult. There can often be more than
one way of searching but, if you have done the
question efficiently, any other way may be
time-consuming. It is often more important to