Out of the five intergral number C is the average of A and D. B is greater than C and less than D. Also, B is the average of A and E. The middle most number in the sequence is :

Options

A. A
B. B
C. C
D. D

Answer

To solve the problem step by step, we will analyze the relationships between the numbers A, B, C, D, and E based on the information given in the question.

### Step 1: Understand the relationships
We know that:
- C is the average of A and D.
- B is greater than C and less than D.
- B is also the average of A and E.

### Step 2: Express C in terms of A and D
Since C is the average of A and D, we can express it mathematically as:
\[ C = \frac{A + D}{2} \]
This means that C is located between A and D.

### Step 3: Establish the inequalities
From the information given:
1. Since C is the average of A and D, we have:
\[ A < C < D \]
2. Since B is greater than C and less than D, we can write:
\[ C < B < D \]
3. Since B is the average of A and E, we can express it as:
\[ B = \frac{A + E}{2} \]
This implies that:
\[ A < B \]

### Step 4: Combine the inequalities
From the inequalities established:
- We have \( A < C < B < D \)
- And since \( B = \frac{A + E}{2} \), we know that \( A < B \) implies \( A < \frac{A + E}{2} \), which leads to \( A < E \).

### Step 5: Order the numbers
Now we can summarize the order of the numbers:
- A is the smallest.
- C is greater than A but less than B.
- B is greater than C and less than D.
- D is the largest.

Thus, the order of the numbers is:
\[ A < C < B < D \]

### Step 6: Identify the middle number
Since we have five integral numbers and we know the order:
- The middle number in the sequence of A, C, B, D, and E will be B, as it is positioned between A and D.

### Conclusion
Therefore, the middle most number in the sequence is:
\[ \text{B} \]