Solution:

To solve this question, we can follow these steps:

1. Determine the total number of oranges:
Given that there are 1370 oranges in total.

2. Divide the oranges into three parts:
Let's assume the first part has x oranges, the second part has y oranges, and the third part has z oranges.

3. Calculate the given fractions for each part:
According to the question, we have:
- (3/4) * x = (5/7) * y = (6/11) * z

4. Simplify the equations:
To simplify the calculations, we can multiply each fraction by the denominators:
- 3x/4 = 5y/7 = 6z/11

5. Find the common ratio:
Since the fractions are equal, we can equate any two of the equations:
- 3x/4 = 5y/7
Cross-multiplying, we get:
- 21x = 20y

6. Express one variable in terms of another:
Solving the equation from the previous step, we can express y in terms of x:
- y = (21x)/20

7. Substitute the values into the third equation:
Now, substitute the values of x and y into the third equation to find z:
- (6z)/11 = 3x/4
Cross-multiplying, we get:
- 24z = 33x
Simplifying, we have:
- z = (33x)/24

8. Determine the value of x:
Since the total number of oranges is 1370, the sum of the three parts should be 1370:
- x + y + z = 1370
Substitute the expressions for y and z from steps 6 and 7, and solve for x:
- x + (21x)/20 + (33x)/24 = 1370
Simplifying the equation, we get:
- (120x + 63x + 110x)/120 = 1370
- 293x/120 = 1370
Cross-multiplying, we have:
- 293x = 164400
Solving for x, we get:
- x = 560

9. Calculate the values of y and z:
Substitute the value of x into the equation from step 6 to find y:
- y = (21 * 560)/20
Simplifying, we get:
- y = 588

Substitute the value of x into the equation from step 7 to find z:
- z = (33 * 560)/24
Simplifying, we get:
- z = 770

10. Verify the fractions:
Finally, we can check if the fractions are equal:
- (3/4) * 560 = (5/7) * 588 = (6/11) * 770

Therefore, the third part of the oranges is 770. Hence, the correct answer is option C) 550.