Correct Option (4)
Let 'x' represent the number of persons who read magazine X only.
Let 'y' represent the number of persons who read magazine Y only.
Let 'z' represent the number of persons who read both magazines X and Y.
From the problem statement:
- "The number of persons who read magazine X only is thrice the number of persons who read magazine Y." Interpreting "magazine Y" as "magazine Y only":
x = 3y (Equation 1) - "The number of persons who read magazine Y only is thrice the number of persons who read magazine X." Interpreting "magazine X" as "magazine X only":
y = 3x (Equation 2)
Substitute Equation 1 into Equation 2:
y = 3 * (3y)
y = 9y
8y = 0
y = 0
Substitute y = 0 back into Equation 1:
x = 3 * 0
x = 0
Thus, the number of persons who read magazine X only (x) is 0, and the number of persons who read magazine Y only (y) is 0. The number of persons who read both magazines (z) is not determined by these statements and can be any non-negative integer.
Now, let's evaluate the given conclusions:
Conclusion 1: "The number of persons who read both the magazines is twice the number of persons who read only magazine X."
In terms of our variables: z = 2x
Substitute x = 0: z = 2 * 0
z = 0
This conclusion implies that z must be 0. However, the initial problem statements (x=0, y=0) do not constrain z to be 0. For example, if z were 10, then 10 = 0 would be false. Therefore, Conclusion 1 does not necessarily follow.
Conclusion 2: "The total number of persons who read either one magazine or both the magazines is twice the number of persons who read both the magazines."
The total number of persons who read at least one magazine is x + y + z.
In terms of our variables: x + y + z = 2z
Substitute x = 0 and y = 0: 0 + 0 + z = 2z
z = 2z
This implies z = 0.
Similar to Conclusion 1, this conclusion implies that z must be 0. The initial problem statements do not constrain z to be 0. For example, if z were 10, then 10 = 2 * 10 (i.e., 10 = 20) would be false. Therefore, Conclusion 2 does not necessarily follow.
Since neither Conclusion 1 nor Conclusion 2 necessarily follows from the given statements, the correct answer is "Neither 1 nor 2".
Incorrect Options:
Options 1, 2, and 3 are incorrect because, as demonstrated in the explanation for the correct option, neither of the conclusions can be definitively drawn from the provided statements. The initial conditions lead to the determination that the number of persons reading only magazine X and only magazine Y is zero, but they do not impose any constraint on the number of persons reading both magazines (z). Both conclusions require z to be zero, which is not a necessary outcome of the problem's premises.