Cambridge International Mathematics

70 Sets (Chapter 2)

Method 2:

Example 13 Self Tutor

In a senior class all 29 students take one or more of biology, chemistry and physics. The headmaster

is informed that 15 take biology, 15 take chemistry, and 18 take physics.

10 take biology and chemistry, 5 take chemistry and physics, and 7 take biology and physics.

How many students take all three subjects?

Letxbe the number taking all three subjects.

) (10¡x) are inBandCbut notP

(5¡x) are inCandPbut notB

(7¡x) are inBandP but notC.

But 15 go into setB

) in the rest ofBwe have

15 ¡x¡(10¡x)¡(7¡x)

=15¡x¡10 +x¡7+x

=x¡ 2

Also, 15 go in setC

) in the rest ofCwe have

15 ¡x¡(10¡x)¡(5¡x)

=15¡x¡10 +x¡5+x

=x

and 18 go in setP

) in the rest ofPwe have

18 ¡x¡(7¡x)¡(5¡x)

=18¡x¡7+x¡5+x

=x+6

But x¡2+10¡x+x+7¡x+x+5¡x+x+6=29

) x+26=29

) x=3

So, 3 study all of the subjects.

U

FB

() 2

U

FB

() 2

() 6

() 6

() 5

2 boys play neither game, so 24 ¡2=22must be in

the union F[B.

However, 16 are in theFcircle, so 6 must go in the rest

ofB.

ButBcontains 11 in total, so 5 go in the intersection

F\B:

So, 5 play both games.

U

IGCSE01

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Y:\HAESE\IGCSE01\IG01_02\070IGCSE01_02.CDR Friday, 24 October 2008 4:37:51 PM PETER