116 Unit 3 Problem solving: basic skills
3.10 Necessity and sufficiency
Another type of problem involves identifying
whether there is enough data to solve the
problem and, if not, which data is missing.
This is a useful building block in problem-
solving. It highlights one of the key elements
of problem solving, which is to find a way to
solve a problem without, in this case, having
to do any arithmetic.
The words ‘necessity’ and ‘sufficiency’ are
used in mathematics but have exactly the
same meaning as they do in normal language.
An individual piece of data is necessary to
solve a problem if we cannot solve the
problem without it. A set of data is sufficient
to solve a problem if it contains all the
information we need.
Identifying which data is needed to solve a
problem can save effort in finding unnecessary
data or in making unnecessary calculations.
Such questions are approached in a manner
similar to those described in earlier chapters.
To illustrate the type of question described
here, we start with a very simple example.
Suppose someone is taking a car journey. We
know their leaving time and we know the
average speed they will do. We want to know
their arrival time. Which other piece of
information is necessary for us to calculate
this?
The solution is very straightforward: we
need the distance of the journey. We can then
calculate the journey time (distance divided by
speed) and thus the arrival time. All of the
three pieces of data we now have are necessary
to do this calculation. The three pieces taken
together are sufficient.
Here is a slightly more complex example.I use the trip meter on my car to measure
the distance driven since I last had the car
serviced, so that I know when the next
service is due. The trip meter can be set to
zero by the press of a button and records
the kilometres driven since it was last reset.
I set the trip meter to zero after my
last service. The next service is due after
20,000 km have been driven. Some time
later, I lent the car to my brother. I forgot
to tell him about the trip meter; he
pressed the button to zero it and drove
575 km. I then started driving again
without realising what he had done.
What should the trip meter read
when the next service is due?
The above problem cannot be solved with the
information given. What additional piece of
information is needed to solve it?Activity
Commentary
This question is actually rather easier than it
may at first seem. The distance driven from
the last service when my brother returned the
car was the distance I had driven plus the
distance he had driven. I know how far he had
driven, so what I need to know was the
distance on the trip meter when I gave the car
to my brother.
In this case, like the previous example, we
were not asked to solve the problem, merely to
identify what pieces of information were
needed to solve it. In real-life problem solving,
the data is not generally given; it has to be