Unit 1 Section 5 : Logic Problems and Venn Diagrams
Venn diagrams can be helpful in solving logic problems.
Example Questions
Question 1
In a class there are:
 8 students who play football and hockey
 7 students who do not play football or hockey
 13 students who play hockey
 19 students who play football
How many students are there in the class?

We can draw a venn diagram to show the numbers playing football (F) or hockey (H).
We can put in the numbers for the first two facts straight away, as seen on the left.
The 8 students who play both hockey and football go in the intersection because they need to be in both circles.
The 7 students who don't play either sport go on the outside because they shouldn't be in either circle.


We have to be careful with the other two facts. There are 13 students who play hockey, so the numbers in the hockey circle should add up to 13.
We already have 8 in the intersection, so there must be 5 who play hockey but not football. In the same way there are 19 students who play
football so the number who play football but not hockey must be 19 – 8 = 11.
To find out the number of students in the class we add up all the sections: 11 + 8 + 5 + 7 = 31
There are 31 students in the class.

Question 2
In a class there are 30 students.
 21 students like Maths
 16 students like English
 6 students don't like Maths or English
How many students like both Maths and English?

We can draw a venn diagram to show the numbers who like Maths (M) or English (E),
but this time we can only put in a number for one of the facts straight away.
The 6 students who don't like either subject go on the outside because they shouldn't be in either circle.
We know the total in the Maths circle needs to be 21 but we can't put this in because we don't know how many should go in the intersection
(if they like both subjects) and how many should go on the left (if they only like Maths).


We know there are 30 students in the class, and if there are 6 students outside the circles then the other three sections must add up to 24.
We know there are 21 students who like Maths, so the middle and left section must add up to 21. This leaves 3 on the right because 24 – 21 = 3.


There are 16 students who like English so the two parts of the English circle should add up to 16,
so we can find the number in the intersection by doing 16 – 3 = 13.
There are 21 students who like Maths, and 21 – 13 = 8, so the number who like Maths but not English must be 8.
If we check all four facts we were given, we can now see they are all true.
There are 13 students who like both Maths and English.

Practice Question
Work out the answers to each question part below then click
to see whether you are correct.

There are 18 cars on a garage forecourt.
 12 cars are diesels.
 5 cars are automatics.
 3 cars are automatic diesels.
To find out how many cars are not automatic
and not a diesel, work through the stages below.

(a) Where do the 3 automatic diesels get marked on the diagram?
(b) If there are 12 diesels, how many diesels are not automatic?
(c) How many of the automatic cars are not diesels?
(d) How many cars are not automatic and not a diesel?
Exercises
Work out the answers to the questions below and fill in the boxes. Click on the
button to find out whether you have answered correctly. If you are right
then will appear and you should move on to the next
question. If appears then your answer is wrong. Click
on to clear your original answer and have another go.
If you can't work out the right answer then click on
to see
the answer.
It may help to have a pencil and paper handy so you can sketch venn diagrams to help you answer the questions.
You have now completed Unit 1 Section 5
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Produced by A.J.Reynolds March 2011
