Given Information:
- We have two arithmetic progressions (AP) with the sum of the first n terms in the ratio (7n+1):(4n+27).

To Find:
- We need to find the ratio of their 9th terms.

Approach:
1. Let's assume the first AP has a common difference of 'a' and the sum of its first n terms is S1.
2. Similarly, let's assume the second AP has a common difference of 'd' and the sum of its first n terms is S2.
3. According to the given information, we have the ratio of S1 to S2 as (7n+1):(4n+27).
4. Using the formula for the sum of an AP, we have S1 = (n/2) * [2a + (n-1)*d1] and S2 = (n/2) * [2a + (n-1)*d2].
5. We can simplify the given ratio as (S1/S2) = [(n/2) * [2a + (n-1)*d1]] / [(n/2) * [2a + (n-1)*d2]].
6. The 'n/2' terms on both sides cancel out, and we are left with [(2a + (n-1)*d1)] / [(2a + (n-1)*d2)] = (7n+1)/(4n+27).

Calculating the Ratio:
1. Let's consider the 9th term of the first AP as a9 and the 9th term of the second AP as b9.
2. Using the formula for the nth term of an AP, we have a9 = a + (9-1)*d1 and b9 = a + (9-1)*d2.
3. We need to find the ratio a9/b9.
4. Substitute the values of a9 and b9 in the ratio expression: (a + 8*d1) / (a + 8*d2).
5. We can rewrite this ratio using the given ratio equation: [(2a + (n-1)*d1) + 6*d1] / [(2a + (n-1)*d2) + 6*d2].
6. Substitute the value of n as 9 in the ratio expression: [(2a + 8*d1) + 6*d1] / [(2a + 8*d2) + 6*d2].
7. Simplify the expression: (2a + 14*d1) / (2a + 14*d2).

Conclusion:
The ratio of the 9th terms of the two arithmetic progressions is (2a + 14*d1) / (2a + 14*d2), where 'a' is the common difference of the first AP and 'd1' and 'd2' are the common differences of the second AP.