The mean of the distribution is 50.

Find the value of a and hence the frequencies of 30 and 70.

Explanation

We know that,
Mean of the distribution `(barx)= (underset(i=1)overset(5)Sigmaf_(i)x_(i))/(underset(i=1)overset(5)Sigmaf_(i))=50` (Given)
`rArr (f_(1)x_(1) + f_(2)x_(2)+f_(3)x_(3)+f_(4)x_(4)+f_(5)x_(5))/(f_(1)+f_(2)+f_(3)+f_(4)+f_(5)) = 50`
`rArr ((17)(10) + (5a+a)(30)+(32)(50)+(7a-11)(70)+(19)(90))/(12a+60)=50`
`rArr (3570-770+640a)/(12a+60)=50`
`rArr 2800 + 640a = 600a +300)`
`rArr 40a=200 rArr a=5)`
`therefore` Frequency of 30 = 5a+3=5(5)+3 (put a=5)
=25+3=28
and frequency of 70 = 7a-11=7(5)-11 (put a=5)
=35-11=24
Hence, the frequencies of 30 and 70 are respectively, 28 and 24.