3.5 Finding methods of solution 95
In this diagram, the outer box represents all
the students (the universal set); in this case
they all take physics. The left-hand circle
represents those taking chemistry (and
physics) and the right-hand circle represents
those taking biology (and physics).
We know that the number taking only
physics is 12; this is represented by the area
outside both circles. Those taking all three
sciences are represented by the intersection
of the two circles and shown as 9. The
number taking both physics and biology is
24; of these 9 take all three, so 15 take only
physics and biology. This is shown by the
outer section of the right-hand circle. We can
now calculate the number in the area marked
by the question mark, as this must be all the
students in the class minus the numbers in
the other three areas, i.e. 45 − 12 − 9 − 15 = 9.
This is the required answer: the number
studying chemistry and physics but not
biology is 9.
Interestingly, the number studying all three
was not used in the original calculation. It is,
in fact, not needed to solve the problem. We
used it in the Venn diagram solution so we
could calculate the numbers in all the areas on
the diagram.
Graphs, pictures and diagrams can often be
useful in solving problems as they help to
clarify the situation and represent the
numbers used in a more digestible manner.
This is covered in more depth in Chapter 6.2.
The activity above shows that problems can
often be solved in more than one way. It is
important to keep the mind open to
alternatives and not always to pursue a
method which is not apparently leading to a
solution.Another way of approaching problems is to
lay out the information in a different way. This
is especially so when the information is given
verbally – and therefore the connection
between the different pieces may not be
immediately obvious. Consider, for example,
the following problem:
In a group of 45 students at a school, all
students must study at least one science.
Physics is compulsory, but students may also
opt to study chemistry or biology or both. 9
students take all three sciences. 24 take
both physics and biology (with or without
chemistry). 12 students take only physics.
How many are studying chemistry and
physics but not biology?Activity
Commentary
With a bit of clear thinking, this may be solved
in a direct fashion by making an intermediate
calculation (those not studying biology). Since
all students take physics, the situation is
simplified. 24 study biology and there are 45
in total, so 21 do not study biology. These 21
comprise those studying physics alone and
those studying both physics and chemistry.
However, we know that 12 take only physics,
so 9 must take physics and chemistry but not
biology.
Although this appeared to be an easy
calculation, the method of approach was not
obvious. The situation can be made a lot
easier by using a Venn diagram:
Physics(^12) Chemistry Biology
? 915