p,q,r are the decimal numbers (e.g. 5.8) and [x] means the greatest integer less than or equal to x and `A = [p + q + r] and B = [p] + [q] + [r]`, then the maximum value of `A - B` is:

Options

A. a.0
B. b.2
C. c.`2.99`
D. d. none of these

Answer

To solve the problem, we need to analyze the expressions for \( A \) and \( B \) given the definitions of \( p, q, r \) as decimal numbers.

1. **Understanding the Definitions**:
- \( A = [p + q + r] \) means we take the sum of \( p, q, \) and \( r \) and then apply the greatest integer function (floor function).
- \( B = [p] + [q] + [r] \) means we take the greatest integer less than or equal to each of \( p, q, \) and \( r \) individually and then sum those values.

2. **Expressing \( A - B \)**:
- We can express \( A - B \) as follows:
\[ A - B = [p + q + r] - ([p] + [q] + [r]) \]
- Let \( p = n_p + f_p \), \( q = n_q + f_q \), and \( r = n_r + f_r \) where \( n_p, n_q, n_r \) are the integer parts and \( f_p, f_q, f_r \) are the fractional parts of \( p, q, r \) respectively.

3. **Calculating \( A \)**:
- The sum can be expressed as:
\[ p + q + r = (n_p + n_q + n_r) + (f_p + f_q + f_r) \]
- Thus, applying the floor function:
\[ A = [n_p + n_q + n_r + f_p + f_q + f_r] = n_p + n_q + n_r + [f_p + f_q + f_r] \]
- The value of \( [f_p + f_q + f_r] \) can be either 0 or 1 depending on whether the sum of the fractional parts is less than 1 or greater than or equal to 1.

4. **Calculating \( B \)**:
- Since \( B = [p] + [q] + [r] = n_p + n_q + n_r \), we can write:
\[ B = n_p + n_q + n_r \]

5. **Finding \( A - B \)**:
- Therefore, we have:
\[ A - B = (n_p + n_q + n_r + [f_p + f_q + f_r]) - (n_p + n_q + n_r) = [f_p + f_q + f_r] \]
- The maximum value of \( [f_p + f_q + f_r] \) can be 1 (when \( f_p + f_q + f_r \geq 1 \)).

6. **Conclusion**:
- The maximum value of \( A - B \) is thus 1.

**Final Answer**: The maximum value of \( A - B \) is **1**.