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 Click on this button below to see the correct answer 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 Click this button to see if you are correct 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 Click on this button to see the correct answer 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.

Question 1
In a family of six, everybody plays football or hockey.
4 members of the family play both sports and 1 member of the family plays only hockey.

How many play only football?

Question 2
John's mum buys 5 portions of chips. All the portions have salt or vinegar on them.
Some have salt and vinegar. There are 2 portions with salt and vinegar and one portion with only vinegar.

How many portions have only salt on them?

Question 3
The diagrams below represent a class of children.
G is the set of girls and F is the set of children who like football.

Decide which diagram has the shading which represents:
(a) girls who like football
(b) girls who dislike football
(c) boys who like football
(d) boys who dislike football

Question 4
In a class of 32 pupils, 20 say that they like pancakes and 14 say that they like maple syrup.
There are 6 pupils who do not like either.

How many of them like both pancakes and maple syrup?

Question 5
On a garage forecourt there are 6 new cars, 12 red cars and no others.

(a) What is the maximum possible number of cars on the forecourt?
(b) What is the smallest possible number of cars on the forecourt?
(c) If 2 of the new cars are red, how many cars are on the forecourt?

Question 6
There are 20 people in a room.
Of these, 15 are holding newspapers and 8 are wearing glasses.
Everyone wears glasses or holds a newspaper.

How many people are wearing glasses and holding a newspaper?

Question 7
A pencil case contains 20 pens that are red or blue.
Of these, 8 are blue and 8 do not work.

How many of the blue pens do not work if there are 6 red pens that do work?

Question 8
In a school canteen there are 45 children.
There are 16 who have finished eating.
The others are eating either fish or chips, or both fish and chips.
There are 26 eating chips and 17 eating fish.

(a) How many are eating fish and chips?
(b) How many are eating fish without chips?
(c) How many are eating only chips?

Question 9
The diagrams below represent the activities chosen by youth club members.
They can choose to play tennis (T), badminton (B) or squash (S).

Decide which diagram has the shading which represents:
(a) those who play all three sports
(b) those who play tennis and badminton, but not squash
(c) those who play only tennis

Question 10
All the members of a group of 30 teenagers belong to at least one club.
There are 3 clubs, chess, drama and art.
  • 6 of the teenagers belong to only the art club.
  • 5 of the teenagers belong to all 3 clubs.
  • 2 of the teenagers belong to the chess and art clubs but not to the drama club.
  • 15 of the teenagers belong to the art club.
  • 2 of the teenagers belong only to the chess club.
  • 3 of the teenagers belong only to the drama club.
(a) How many of the group do chess and drama but not art?
(b) How many of the group belong to the chess club?
Question 11
In a class of 32 pupils:
  • 5 pupils live in New Town, travel to school by bus and eat school dinners
  • 3 pupils live in New Town, travel to school by bus but do not eat school dinners
  • 9 pupils do not live in New Town, do not travel to school by bus and do not eat school dinners
  • 11 pupils live in New Town and have school dinners
  • 16 pupils live in New Town
  • 9 pupils travel by bus and eat school dinners
  • 13 pupils travel by bus

How many pupils eat school dinners?


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Produced by A.J.Reynolds March 2011