Understanding Ratios
Given the ratio \( a:b = 3:4 \), we can express \( a \) and \( b \) in terms of a common variable. Let’s denote:
- \( a = 3k \)
- \( b = 4k \)
for some constant \( k \).

Substituting Values
Now, we need to compute \( (2a + 3b) : (3a + 4b) \).
1. **Calculate \( 2a + 3b \)**:
- \( 2a = 2(3k) = 6k \)
- \( 3b = 3(4k) = 12k \)
- Therefore, \( 2a + 3b = 6k + 12k = 18k \)
2. **Calculate \( 3a + 4b \)**:
- \( 3a = 3(3k) = 9k \)
- \( 4b = 4(4k) = 16k \)
- Thus, \( 3a + 4b = 9k + 16k = 25k \)

Forming the Ratio
Now, we can form the ratio:
\[
(2a + 3b) : (3a + 4b) = 18k : 25k
\]
Since \( k \) is a common factor, it cancels out:
\[
= 18 : 25
\]

Final Result
Thus, the value of \( (2a + 3b) : (3a + 4b) \) is:

18 : 25
This illustrates how to manipulate ratios effectively, a crucial skill in solving problems in competitive exams like UPSC.