Correct Option: B Step-by-step Solution: 1. **Understand the concept of Average:** The average of a set of values is defined as the sum of the values divided by the number of values. Consequently, the sum of values can be calculated as the average multiplied by the number of values. 2. **Formulate the total sums of weights for each group:** * For the initial group of $P$ students, with an average weight of $W_1$ kg, their total combined weight is $P \times W_1$ kg. * For the $Q$ new students, with an average weight of $W_3$ kg, their total combined weight is $Q \times W_3$ kg. * After the $Q$ new students join, the total number of students in the class becomes $P+Q$. The new overall average weight for all students is $W_2$ kg. Therefore, the total combined weight of all $(P+Q)$ students is $(P+Q) \times W_2$ kg. 3. **Establish the governing equation based on the conservation of total weight:** The total weight of all students in the class after the new students join must be equal to the sum of the total weight of the initial students and the total weight of the new students. This can be expressed as: \[ P W_1 + Q W_3 = (P+Q) W_2 \] 4. **Perform algebraic manipulation to simplify and identify the correct relationship:** First, expand the right-hand side of the equation: \[ P W_1 + Q W_3 = P W_2 + Q W_2 \] Next, rearrange the terms to group similar variables (i.e., terms involving $P$ and terms involving $Q$) on opposite sides of the equation: \[ P W_1 - P W_2 = Q W_2 - Q W_3 \] Factor out $P$ from the left-hand side and $Q$ from the right-hand side: \[ P(W_1 - W_2) = Q(W_2 - W_3) \] This is a core relationship derived from the problem statement. Now, we need to compare this derived relationship with the given options. 5. **Evaluate the given options:** * **Option A: $P(W_1 - W_2) = Q(W_3 - W_2)$** Comparing this with our derived equation $P(W_1 - W_2) = Q(W_2 - W_3)$, it would imply $Q(W_3 - W_2) = Q(W_2 - W_3)$. Assuming $Q \neq 0$, this leads to $W_3 - W_2 = W_2 - W_3$, which simplifies to $2W_3 = 2W_2$, or $W_3 = W_2$. This condition is not always true; $W_3$ can be different from $W_2$. * **Option B: $P(W_2 - W_1) = Q(W_3 - W_2)$** Let's take our derived equation $P(W_1 - W_2) = Q(W_2 - W_3)$ and multiply both sides by $-1$: \[ -P(W_1 - W_2) = -Q(W_2 - W_3) \] \[ P(-(W_1 - W_2)) = Q(-(W_2 - W_3)) \] \[ P(W_2 - W_1) = Q(W_3 - W_2) \] This exactly matches Option B. Therefore, Option B is the correct mathematical relationship. * **Option C: $P(W_1 - W_3) = Q(W_2 - W_1)$** This relationship involves different groupings of weights compared to our derived equation and is not generally true. * **Option D: $P(W_1 + W_2) = Q(W_2 - W_3)$** The left-hand side, $P(W_1 + W_2)$, differs from our derived left-hand side, $P(W_1 - W_2)$. This option is incorrect. **Alternative Approach (Principle of Sum of Deviations):** The fundamental property that the sum of deviations of all individual values from their overall arithmetic mean is zero can also be applied. * The average weight of the initial $P$ students is $W_1$. The deviation of their average from the new overall average $W_2$ is $(W_1 - W_2)$. The total sum of deviations for this group is $P(W_1 - W_2)$. * The average weight of the $Q$ new students is $W_3$. The deviation of their average from the new overall average $W_2$ is $(W_3 - W_2)$. The total sum of deviations for this group is $Q(W_3 - W_2)$. According to the principle of sum of deviations, the sum of these total deviations must be zero: \[ P(W_1 - W_2) + Q(W_3 - W_2) = 0 \] \[ P(W_1 - W_2) = -Q(W_3 - W_2) \] \[ P(W_1 - W_2) = Q(-(W_3 - W_2)) \] \[ P(W_1 - W_2) = Q(W_2 - W_3) \] To match Option B, multiply both sides by $-1$: \[ -P(W_1 - W_2) = -Q(W_2 - W_3) \] \[ P(W_2 - W_1) = Q(W_3 - W_2) \] This confirms that Option B is the correct relationship.