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 $$ x \, ext{kg} $$. Instead of $$ D $$, if person $$ E $$ joins the room, the average weight of the persons in the room increases by $$ 2x \, ext{kg} $$. If the weight of $$ E $$ is 12 kg more than that of $$ D $$, then the value of $$ x $$ is:

Question

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 $$ x \, 	ext{kg} $$. Instead of $$ D $$, if person $$ E $$ joins the room, the average weight of the persons in the room increases by $$ 2x \, 	ext{kg} $$. If the weight of $$ E $$ is 12 kg more than that of $$ D $$, then the value of $$ x $$ is:

Accepted Answer

Let the average weight of persons $$ A, B, C $$ be denoted as $$ y $$. The total weight of these three persons is:

$$
A + B + C = 3y
$$

#### Step 1: When person $$ D $$ joins
The average weight decreases by $$ x $$, so:
$$
\frac{A + B + C + D}{4} = y - x
$$
Simplify:
$$
A + B + C + D = 4(y - x)
$$
Substituting $$ A + B + C = 3y $$:
$$
3y + D = 4y - 4x
$$
Thus:
$$
D = y - 4x
$$

#### Step 2: When person $$ E $$ joins
The average weight increases by $$ 2x $$, so:
$$
\frac{A + B + C + E}{4} = y + 2x
$$
Simplify:
$$
A + B + C + E = 4(y + 2x)
$$
Substituting $$ A + B + C = 3y $$:
$$
3y + E = 4y + 8x
$$
Thus:
$$
E = y + 8x
$$

#### Step 3: Relation between $$ E $$ and $$ D $$
It is given that $$ E $$ is 12 kg heavier than $$ D $$:
$$
E - D = 12
$$
Substitute values for $$ E $$ and $$ D $$:
$$
(y + 8x) - (y - 4x) = 12
$$
Simplify:
$$
12x = 12
$$

Thus:
$$
x = 1
$$

#### Final Answer:
The value of $$ x $$ is **1**, corresponding to option **D**.

Answer

Answer

Answer