The ratio of sums of n terms of two A.P'. is (7n + 1) : (4n + 27). Find the ratio of their 11th terms.

Explanation

To solve the problem, we need to find the ratio of the 11th terms of two arithmetic progressions (APs) given the ratio of their sums of n terms.

### Step-by-Step Solution:

1. **Understanding the Sum of n Terms of an AP**:
The sum of the first n terms \( S_n \) of an arithmetic progression can be expressed as:
\[ S_n = \frac{n}{2} \left(2A + (n - 1)D\right) \]
where \( A \) is the first term and \( D \) is the common difference.

2. **Setting Up the Given Ratio**:
We are given that the ratio of the sums of n terms of two APs is:
\[ \frac{S_{n1}}{S_{n2}} = \frac{7n + 1}{4n + 27} \]
For the first AP, let the first term be \( A_1 \) and the common difference be \( D_1 \). For the second AP, let the first term be \( A_2 \) and the common difference be \( D_2 \).

3. **Expressing the Sums**:
The sums for the two APs can be expressed as:
\[ S_{n1} = \frac{n}{2} \left(2A_1 + (n - 1)D_1\right) \]
\[ S_{n2} = \frac{n}{2} \left(2A_2 + (n - 1)D_2\right) \]
Thus, the ratio becomes:
\[ \frac{2A_1 + (n - 1)D_1}{2A_2 + (n - 1)D_2} = \frac{7n + 1}{4n + 27} \]

4. **Cross Multiplying**:
Cross-multiplying gives us:
\[ (2A_1 + (n - 1)D_1)(4n + 27) = (2A_2 + (n - 1)D_2)(7n + 1) \]

5. **Finding the 11th Term**:
The 11th term of the first AP is given by:
\[ T_{11,1} = A_1 + 10D_1 \]
The 11th term of the second AP is:
\[ T_{11,2} = A_2 + 10D_2 \]
We need to find the ratio:
\[ \frac{T_{11,1}}{T_{11,2}} = \frac{A_1 + 10D_1}{A_2 + 10D_2} \]

6. **Using the Value of n**:
To find the specific values, we need to set \( n = 21 \) (since \( n - 1 = 20 \)). Substituting \( n = 21 \) into the ratio of sums gives:
\[ \frac{S_{21,1}}{S_{21,2}} = \frac{7(21) + 1}{4(21) + 27} = \frac{148}{111} \]

7. **Calculating the Ratio of 11th Terms**:
Now we can substitute \( n = 21 \) into our earlier derived equation for the sums:
\[ \frac{(2A_1 + 20D_1)}{(2A_2 + 20D_2)} = \frac{148}{111} \]
This gives us the ratio of the terms.

8. **Final Ratio**:
The ratio of the 11th terms can be calculated as:
\[ \frac{A_1 + 10D_1}{A_2 + 10D_2} = \frac{148}{111} \]

### Conclusion:
Thus, the ratio of the 11th terms of the two APs is:
\[ \frac{148}{111} \]