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(A) Let the average of prime numbers be $P$ and the average of composite numbers be $C$. Let the number of prime numbers be $x$,then the number of com

Vedclass · Accepted Answer

$\frac{7}{13}$

Vedclass · Answer

(A) Let the average of prime numbers be $P$ and the average of composite numbers be $C$. Let the number of prime numbers be $x$,then the number of composite numbers is $2x$.The average of all numbers is given by: $\frac{Px + C(2x)}{x + 2x} = 9 \Rightarrow \frac{Px + 2Cx}{3x} = 9 \Rightarrow P + 2C = 27$ (Equation $i$).If the counts are exchanged,the number of prime numbers becomes $2x$ and the number of composite numbers becomes $x$. The new average is $9 + 2 = 11$:$\frac{P(2x) + Cx}{2x + x} = 11 \Rightarrow \frac{2Px + Cx}{3x} = 11 \Rightarrow 2P + C = 33$ (Equation $ii$).Solving the system of equations:From $(i)$,$P = 27 - 2C$.Substitute into (ii): $2(27 - 2C) + C = 33 \Rightarrow 54 - 4C + C = 33 \Rightarrow 3C = 21 \Rightarrow C = 7$.Substitute $C = 7$ into $(i)$: $P + 2(7) = 27 \Rightarrow P + 14 = 27 \Rightarrow P = 13$.The ratio of the average of composite numbers to the average of prime numbers is $\frac{C}{P} = \frac{7}{13}$.

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