In a survey conduced on 800 students of a school , 250 students were found to like tea and 300 like coffee , 150 like both tea and coffee .Find how many students like neither tea nor coffee ?

Answer

400

Explanation

Let
U be the set of all students who took part in the survey
T be the set of students who like tea , and
C be the set of the students who like coffee .
According to the question ,
`n(u ) = 800 ,n( T ) = 250 , n( c) = 300 , n ( T cap C ) = 150 `
To find the number of students taking neither tea nor coffee.ie,
we have to find `n( T cap C ) `
`n( T cap C) = n ( T cup C) `
`n( U)- n( T cup C ) `
`=n( U) -n ( T cup C) `
`= n( U) - [n ( T)+ n(C)- n( T cap C )]`
=800-[250+300-150]=400
Hence , 400 student like neither tea nor coffee.
Alternative method : ltbr gt
We can use Venn diagram as shown in the figure.
In the above figure regions are
`r rarr` students liking tea and coffee both`=n(T cap C)`
`p rarr `students liking tea only `n(T)-n( T cap C)`
`q rarr ` students liking coffee only .`= bn( C) - n ( T cap C)`
`s rarr `students liking neither tea neither tea noe coffee `= n( T cap C)`
Here r= 150 ( given )
Now , p+ r student liking tea = 250 (given)
`rArr p= 250 - 150 = 100`
q+r = student liking coffee = 300 ( given )
`rArr q = 300 - 150 = 150 `
p+ q+r = student liking at least one of tea and coffee
= 100 + 150 + 150 = 400
s = student liking neither tea nor coffee = 800 - 400= 400