The class X of a Vidhyalaya has four sections: A, B, C and D. The average weight of the students of A, B, C together and A, C, D together are 45 kg and 55 kg respectively, while the average weight of the students of A, B, D together and B, C, D together are 50kg and 60kg respectively. Which of the following could be the average weight of the students of all the four sections together?

Options

A. 47.6 kg
B. 52.5 kg
C. 53.7 kg
D. 56.5 kg

Answer

To solve the problem step by step, we will use the information given about the average weights of different combinations of sections A, B, C, and D.

### Step 1: Calculate the total weight of sections A, B, and C
The average weight of students in sections A, B, and C is given as 45 kg. Since there are 3 sections, we can find the total weight of these sections.

\[ \text{Total weight of A, B, C} = \text{Average} \times \text{Number of sections} = 45 \text{ kg} \times 3 = 135 \text{ kg} \]

**Hint:** Remember that the total weight can be calculated by multiplying the average weight by the number of sections.

### Step 2: Calculate the total weight of sections A, C, and D
The average weight of students in sections A, C, and D is given as 55 kg. Again, we can find the total weight of these sections.

\[ \text{Total weight of A, C, D} = \text{Average} \times \text{Number of sections} = 55 \text{ kg} \times 3 = 165 \text{ kg} \]

**Hint:** Use the same formula as in Step 1 to find the total weight for different combinations.

### Step 3: Calculate the total weight of sections A, B, and D
The average weight of students in sections A, B, and D is given as 50 kg. We can find the total weight of these sections.

\[ \text{Total weight of A, B, D} = \text{Average} \times \text{Number of sections} = 50 \text{ kg} \times 3 = 150 \text{ kg} \]

**Hint:** Keep applying the average weight formula to different combinations of sections.

### Step 4: Calculate the total weight of sections B, C, and D
The average weight of students in sections B, C, and D is given as 60 kg. We can find the total weight of these sections.

\[ \text{Total weight of B, C, D} = \text{Average} \times \text{Number of sections} = 60 \text{ kg} \times 3 = 180 \text{ kg} \]

**Hint:** This step follows the same pattern as the previous steps. Always multiply the average by the number of sections.

### Step 5: Set up equations to find the total weights of each section
Now, we have the following equations based on the total weights calculated:

1. \( A + B + C = 135 \) (from Step 1)
2. \( A + C + D = 165 \) (from Step 2)
3. \( A + B + D = 150 \) (from Step 3)
4. \( B + C + D = 180 \) (from Step 4)

### Step 6: Solve the equations
We can solve these equations to find the individual weights of A, B, C, and D.

1. From equations (1) and (2):
\[ (A + C + D) - (A + B + C) = 165 - 135 \implies D - B = 30 \implies D = B + 30 \]

2. From equations (3) and (4):
\[ (B + C + D) - (A + B + D) = 180 - 150 \implies C - A = 30 \implies C = A + 30 \]

3. Substitute \( C = A + 30 \) into equation (1):
\[ A + B + (A + 30) = 135 \implies 2A + B + 30 = 135 \implies 2A + B = 105 \implies B = 105 - 2A \]

4. Substitute \( B = 105 - 2A \) into \( D = B + 30 \):
\[ D = (105 - 2A) + 30 = 135 - 2A \]

### Step 7: Find the total weight of all sections
Now we can find the total weight of all sections A, B, C, and D:

\[ A + B + C + D = A + (105 - 2A) + (A + 30) + (135 - 2A) \]
\[ = A + 105 - 2A + A + 30 + 135 - 2A = 270 - 2A \]

### Step 8: Calculate the average weight of all sections
To find the average weight of all four sections, we divide the total weight by the number of sections (4):

\[ \text{Average weight} = \frac{A + B + C + D}{4} = \frac{270 - 2A}{4} \]

### Conclusion
Now we can find possible values for the average weight based on the values of A. The average weight must be one of the options given: 47.6 kg, 52.5 kg, 53.7 kg, or 56.5 kg.

By substituting values for A, we can find that the average weight of 52.5 kg is achievable.

**Final Answer:** The average weight of the students of all four sections together could be **52.5 kg** (Option 2).