To find the ratio of the 9th terms of two arithmetic progressions (APs) given the ratio of their sum, we need to follow a step-by-step process. Let's break it down into the following sections:

1. Understanding the problem:
We are given two APs, and 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.

2. Formulating the problem:
Let's denote the first term of the first AP as a₁ and the common difference as d₁. Similarly, for the second AP, let's denote the first term as a₂ and the common difference as d₂. We need to find the ratio of the 9th terms, which can be represented as (a₁ + 8d₁)/(a₂ + 8d₂).

3. Finding the sum of n terms:
The sum of the first n terms of an AP can be calculated using the formula Sn = (n/2)(2a + (n-1)d), where Sn represents the sum, a is the first term, d is the common difference, and n is the number of terms.

4. Calculating the ratio of the sum of n terms:
Using the formula from step 3, we can calculate the sum of the first n terms for both APs. The ratio of the sum of the nth terms is given as (7n + 1)/(4n + 27).

5. Equating and solving:
Setting up an equation by equating the ratios obtained in step 4 with the formula from step 2, we have:
(7n + 1)/(4n + 27) = (a₁ + 8d₁)/(a₂ + 8d₂)

6. Finding the value of n:
To find the value of n, we can cross-multiply the equation obtained in step 5 and solve for n. This will give us the value of n for which the ratio of the sum of n terms is (7n + 1)/(4n + 27).

7. Calculating the ratio of the 9th terms:
Once we have the value of n, we substitute it back into the formula from step 2 to get the ratio of the 9th terms, which is (a₁ + 8d₁)/(a₂ + 8d₂).

By following these steps, we can find the ratio of the 9th terms of the two given APs.