Cambridge Additional Mathematics

30 Sets and Venn diagrams (Chapter 1)

Example 12 Self Tutor

Consider theOpening Problemon page 12 :

A city has three football teams in the national league: A,B, andC.

In the last season,20%of the city’s population saw teamAplay,24%saw teamB, and28%saw

teamC. Of these,4%saw bothAandB,5%saw bothAandC, and6%saw bothBandC.1%

saw all three teams play.

Using a Venn diagram, find the percentage of the city’s population which:

a saw only teamAplay b saw teamAor teamBplay but not teamC

c did not see any of the teams play.

We construct the Venn diagram in terms of percentages.

Using the given information,

a=1 f1%saw all three teams playg

a+d=4 f4%sawAandBg

a+b=6 f6%sawBandCg

a+c=5 f5%sawAandCg

) d=3, b=5, and c=4

In total,20%saw teamAplay,

so g+1+4+3=20 ) g=12

24%saw teamBplay,

so e+1+5+3=24 ) e=15

28%saw teamCplay,

so f+1+5+4=28 ) f=18

In total we cover100%of the population, so h=42.

a n(sawAonly) = 12% fshadedg

b n(AorB, but notC)

= 12% + 3% + 15%

= 30%

c n(saw none of the teams) = 42%

5 In a year group of 63 students, 22 study Biology, 26 study Chemistry, and 25 study Physics.

18 study both Physics and Chemistry, 4 study both Biology and Chemistry, and 3 study both Physics

and Biology. 1 studies all three subjects.

a Display this information on a Venn diagram.

b How many students study:

i Biology only ii Physics or Chemistry

iii none of Biology, Physics, or Chemistry iv Physics but not Chemistry?

A

C

B

U

(a)

(c) (b)

(d) (e)

(f)

(g)

(h)

A

C

B

U

(1)

(4) (5)

(3) (e)

(f)

(g)

(h)

A

C

B

U

(1)

(4) (5)

(3)

(42)

(15)

(18)

(12)

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_01\030CamAdd_01.cdr Thursday, 19 December 2013 1:47:52 PM GR8GREG