There are three persons, A, B and C in a room. If a person D joins the room, the average weight of the persons in the room reduces by $x$ kg. Instead of D, if person E joins the room, the average weight of the persons in the room increases by $2x$ kg. If the weight of E is 12 kg more than that of D, then the value of $x$ is

The correct answer is 1.

Let's analyze the problem step-by-step.

Let the weights of persons A, B, and C be $W_A$, $W_B$, and $W_C$ respectively.
Let their average weight be $A$. Therefore, the sum of their weights is:
$W_A+W_B+W_C=3A$

Let the weight of person D be $W_D$ and the weight of person E be $W_E$.

Case 1: Person D joins the room.
There are now 4 persons in the room (A, B, C, and D). The new average weight reduces by $x$ kg, so the new average is $A-x$.
The sum of their weights is:
$3A+W_D=4(A-x)$
Simplifying this equation:
$3A+W_D=4A-4x$
$W_D=A-4x$ (Equation 1)

Case 2: Person E joins the room instead of D.
There are 4 persons in the room (A, B, C, and E). The new average weight increases by $2x$ kg, so the new average is $A+2x$.
The sum of their weights is:
$3A+W_E=4(A+2x)$
Simplifying this equation:
$3A+W_E=4A+8x$
$W_E=A+8x$ (Equation 2)

Finding the value of $x$:
We are given that the weight of E is 12 kg more than that of D:
$W_E-W_D=12$

Substitute Equation 1 and Equation 2 into the relation above:
$(A+8x)-(A-4x)=12$
$A+8x-A+4x=12$
$12x=12$
$x=1$

Thus, the value of $x$ is 1.