The given problem involves two arithmetic progressions (A.P.s) and their sum up to the nth term. We are required to find the ratio of their 9th terms based on the given ratio of the sum of the first n terms of the A.P.s.

To solve this problem, we will follow the steps outlined below:

**Step 1: Understanding the problem**
- We are given two A.P.s.
- The ratio of the sum of their first n terms is (7n + 1) : (4n - 27).
- We need to find the ratio of their 9th terms.

**Step 2: Defining the A.P.s**
Let the first A.P. have a common difference of d1 and the second A.P. have a common difference of d2.

**Step 3: Finding the sum of the first n terms of the A.P.s**
The sum of the first n terms of an A.P. is given by the formula:
Sn = (n/2)(2a + (n-1)d)
where a is the first term and d is the common difference.

For the first A.P., the sum of the first n terms is:
S1 = (n/2)(2a1 + (n-1)d1)
For the second A.P., the sum of the first n terms is:
S2 = (n/2)(2a2 + (n-1)d2)

**Step 4: Finding the ratio of the sum of the first n terms**
Given that the ratio of the two sums is (7n + 1) : (4n - 27), we can write the equation:
(7n + 1)/(4n - 27) = S1/S2

**Step 5: Simplifying the equation**
By substituting the expressions for S1 and S2 from Step 3, we can simplify the equation:
(7n + 1)/(4n - 27) = [(n/2)(2a1 + (n-1)d1)] / [(n/2)(2a2 + (n-1)d2)]

**Step 6: Cancelling out the common terms**
Since the ratio is independent of n, we can cancel out the common terms:
(7n + 1)/(4n - 27) = (2a1 + (n-1)d1) / (2a2 + (n-1)d2)

**Step 7: Solving for the ratio of the 9th terms**
To find the ratio of the 9th terms, we substitute n = 9 into the equation and solve for the ratio:
(7(9) + 1)/(4(9) - 27) = (2a1 + (9-1)d1) / (2a2 + (9-1)d2)

Simplifying the equation:
(63 + 1)/(36 - 27) = (2a1 + 8d1) / (2a2 + 8d2)
64/9 = (2a1 + 8d1) / (2a2 + 8d2)

Thus, the ratio of the 9th terms of the two A.P.s is 64/9.