If `p < q,` then `p "@" q = p # q` else `p "@" q = q # p` where `a # b = a/b`. Then the value of `(4 "@" 5) "@" (6 "@" 5)` is :

Options

A. a. `24/25`
B. b. `2/3`
C. c. `3//4`
D. d. none of these

Answer

To solve the problem step by step, we need to evaluate the expression `(4 "@" 5) "@" (6 "@" 5)` using the defined operations.

### Step 1: Evaluate `4 "@" 5`
Given the condition:
- If `p < q`, then `p "@" q = p # q = p / q`
- If `p >= q`, then `p "@" q = q # p = q / p`

Here, we have `p = 4` and `q = 5`. Since `4 < 5`, we use the first condition:
\[ 4 "@" 5 = 4 / 5 \]

### Step 2: Evaluate `6 "@" 5`
Now we need to evaluate `6 "@" 5`:
- Here, `p = 6` and `q = 5`. Since `6 >= 5`, we use the second condition:
\[ 6 "@" 5 = 5 / 6 \]

### Step 3: Evaluate `(4 "@" 5) "@" (6 "@" 5)`
Now we substitute the results from Steps 1 and 2 into the expression:
\[ (4 "@" 5) "@" (6 "@" 5) = (4 / 5) "@" (5 / 6) \]

### Step 4: Determine the relationship between `4/5` and `5/6`
We need to compare `4/5` and `5/6`:
- To compare, we can cross-multiply:
\[ 4 \times 6 = 24 \quad \text{and} \quad 5 \times 5 = 25 \]
Since `24 < 25`, we have `4/5 < 5/6`.

### Step 5: Apply the operation `(4/5) "@" (5/6)`
Since `4/5 < 5/6`, we use the first condition again:
\[ (4/5) "@" (5/6) = (4/5) # (5/6) = (4/5) / (5/6) \]

### Step 6: Simplify `(4/5) / (5/6)`
To simplify:
\[ (4/5) / (5/6) = (4/5) \times (6/5) = (4 \times 6) / (5 \times 5) = 24 / 25 \]

### Final Answer
Thus, the value of `(4 "@" 5) "@" (6 "@" 5)` is:
\[ \frac{24}{25} \]