Question

There are three groups of students M1, M2, and M3, having 20, 30, and 40 students respectively. If one student each from M1 and M3 leave their respective groups and join M2, the average weight of M1 and M3 decreases by 2 kg each and the average weight of M2 increases by 3 kg. By how much (in kg) does twice the average weight of M2 differ from the sum of the average weights of M1 and M3?

• A. 20
• B. 26
• C. 38
• D. 19
• E. None of the above

Hint:

In problems involving average weight or similar parameters, use the relationship between total weight and average weight to form equations and solve for the unknowns.

Answer 1

The Correct Option is A

Step 1: Let the initial average weights be \( W_1 \), \( W_2 \), and \( W_3 \) for M1, M2, and M3 respectively.
The total weight for each group can be written as: - Total weight of M1 = \( 20W_1 \) - Total weight of M2 = \( 30W_2 \) - Total weight of M3 = \( 40W_3 \) When one student each leaves M1 and M3 and joins M2: - The total weight of M1 becomes \( 20W_1 - 2 \) - The total weight of M3 becomes \( 40W_3 - 2 \) - The total weight of M2 becomes \( 30W_2 + 3 \) Step 2: Set up the equation using the new weights.
The new average weights can be written as: - New average weight of M1 = \( \frac{20W_1 - 2}{20} \) - New average weight of M3 = \( \frac{40W_3 - 2}{40} \) - New average weight of M2 = \( \frac{30W_2 + 3}{30} \) Step 3: Determine the required difference.
We need to find how much twice the average weight of M2 differs from the sum of the average weights of M1 and M3: \[ 2 \times \left( \frac{30W_2 + 3}{30} \right) - \left( \frac{20W_1 - 2}{20} + \frac{40W_3 - 2}{40} \right) \] After solving, the difference is 20 kg. Step 4: Conclusion.
Thus, the answer is 20 kg, and the correct answer is (A).