Read the information given below and answer the questions that follow :
There are 60 students in a class. These students are divided into three groups A, B and C of 15, 20 and 25 students each. The groups A and C are combined to form group D.
If all the students of the class have the same weight, then which of the following is false?

Options

A. The average weight of all the four groups is the same.
B. The total weight of A and C is twice the total weight of B.
C. The average weight of D is greater than the average weight of A.
D. The average weight of all the groups remains the same even if the number of students are shifted from one group to another.

Answer

To solve the problem, we need to analyze the information provided about the students in the class and the groups they are divided into. Let's break it down step by step.

### Step 1: Understand the composition of the groups
- There are a total of 60 students in the class.
- The students are divided into three groups:
- Group A: 15 students
- Group B: 20 students
- Group C: 25 students

### Step 2: Calculate the number of students in Group D
- Group D is formed by combining Group A and Group C.
- Therefore, the number of students in Group D = Number of students in Group A + Number of students in Group C
\[ \text{Number of students in Group D} = 15 + 25 = 40 \]

### Step 3: Analyze the average weight of the groups
- We assume that each student has the same weight, denoted as \( x \) kg.
- The total weight of each group can be calculated as follows:
- Total weight of Group A = \( 15x \)
- Total weight of Group B = \( 20x \)
- Total weight of Group C = \( 25x \)
- Total weight of Group D = \( 40x \)

### Step 4: Calculate the average weight of each group
- The average weight for each group is calculated by dividing the total weight by the number of students in that group:
- Average weight of Group A = \( \frac{15x}{15} = x \)
- Average weight of Group B = \( \frac{20x}{20} = x \)
- Average weight of Group C = \( \frac{25x}{25} = x \)
- Average weight of Group D = \( \frac{40x}{40} = x \)

### Step 5: Evaluate the statements
1. **The average weight of all four groups is the same.**
- True, as all averages calculated above are equal to \( x \).

2. **The total weight of A and C is twice the total weight of B.**
- Total weight of A and C = \( 15x + 25x = 40x \)
- Total weight of B = \( 20x \)
- \( 40x \) is indeed twice \( 20x \), so this statement is also true.

3. **The average weight of D is greater than the average weight of A.**
- This statement is false because the average weight of D is \( x \) and the average weight of A is also \( x \). Therefore, they are equal, not greater.

4. **The average weight of all the groups remains the same even if the number of students are shifted from one group to another.**
- True, because shifting students with the same average weight does not change the overall average.

### Conclusion
The false statement among the options provided is:
- **The average weight of D is greater than the average weight of A.**