Form 4 & 5 Unit 6 Lesson 5: Story problems with three sets



At the end of the lesson, students should be able to:

  • Represent relationships between sets.
  • Solve story problems involving three sets.

 So far we have dealt with the intersection of two sets. It is, however, quite usual for there to be intersections between three or more sets.

When representing three sets on a Venn diagram, draw them intersecting like this:





4) 37 students sat an examination in physics, 48 sat chemistry and 45 sat biology. 15 students sat physics and chemistry, 13 sat chemistry and biology, 7 sat physics and biology and 5 students sat all three subjects.

a. Draw a venin diagram to represent this information.

b. Calculate n(P U C U B).

5) Out of 136 students in a school, 60 take French, 100 take chemistry and 48 take physics. If 28 take French and chemistry, 44 take chemistry and physics and 20 take French and physics, how many students take all three subjects?

6) Out of 68 pupils in a school, 30 take English, 50 take mathematics and 24 take physics. If 10 take mathematics and physics, 14 take English and physics and 22 take English and mathematics, how many pupils take all three subjects?

7) In an examination, 180 candidates offered French, 240 offered English and 150 offered German. If 60 offered French and English, 45 English and German and 75 French and German and 30 offered all three, how many candidates sat the examination?

8) Out of three sports, rugby, association football and hockey, 240 young people were asked to state their preference. 120 preferred association football, 172 preferred rugby and 128 preferred hockey. 64 liked rugby and association football, 76 liked rugby and hockey whilst 68 liked association football and hockey. How many liked all three sports?

9) In a class of 60 students, 47 study mathematics, 33 study mathematics and physics, 31 study mathematics and chemistry, 29 study physics and chemistry and 20 study all the three subjects. If the number of students who study only physics is equal to that of those who study only chemistry, illustrate the given information on a Venn diagram and find the number of students who study:

a. Only Physics

b. Chemistry

c. Only one subject

10) In a class of 60 students, some study at least one of the following subjects: mathematics, Economics and Accounting. 8 students study none of them. The following table gives further details of the subjects studied.


Subjects Number Subjects Number
Mathematics only 6 All three subjects 7
Economics only 1 Mathematics & Accounting 18
Accounting only 5 Economics and Accounting 17

a. Illustrate the above information on a Venn diagram.

b. Find the number of students who study:

i) Mathematics or Accounting or both but not Economics.

ii) Economics.

NOTE: Please check for the Answers in the (Additional) Material section at the top


a) 35
b) 27
c) 30
d) 11
e) 10
f) 9
g) 4
h) 662.
a) {Claire, Kath, harry, Eric, Alan}
b) {Eric, Beth, Freda, Alan, John}
c) {Harry, Alan, John, David, George, Lynne}
d) {Eric, Alan}
e) {Alan, Harry}
f) {John}
g) {Alan}
h) {Martin}3. a) 5
b) 53
c) 52
d) 54
e) 15
f) 10
g) 12
h) 5
b) 1005. 206. 10

7. 420

8. 28

9. a) 2     b) 42     c) 7

b) i) 22    ii) 30


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