Question

In a set of prime and composite numbers, the composite numbers are twice the number of prime numbers and the average of all the numbers of the set is 9. If the number of prime numbers and the composite numbers are exchanged then the average of the set of numbers is increased by 2. If during the exchange of the numbers the average of the prime numbers and composite numbers individually remained constant, then the ratio of the average of composite numbers to the average of prime numbers (initially) was?

Answer 1

Initially let there be n prime numbers.  So there are 2n number of composite numbers.  Totally there are 3n numbers. Let the average of prime numbers be p and the average of composite numbers be c.

     Average of 3n numbers = 9
     (p * n + c * 2n) / (3n)  = 9        =>   p + 2 c = 27    --- (1)

After the exchange of the number of prime  & composite numbers:
    (averages p & c remain the same.)

    avg of 3n numbers = 11
    (p * 2n + c * n) / (3n) = 11      => 2 p + c = 33      ---- (2)

Solving (1) and (2) gives:    p = 13,   c = 7

Ratio = c : p = 7 : 13