To solve the expression \((4 - \frac{5}{12}) : \frac{43}{36} + \frac{5}{24}\), we need to follow the order of operations (PEMDAS/BODMAS). Let's break it down step by step:

1. **Simplify inside the parentheses:**
\[
4 - \frac{5}{12}
\]
Convert 4 to a fraction with a denominator of 12:
\[
4 = \frac{48}{12}
\]
Now subtract:
\[
\frac{48}{12} - \frac{5}{12} = \frac{48 - 5}{12} = \frac{43}{12}
\]

2. **Division:**
\[
\left(\frac{43}{12}\right) : \frac{43}{36}
\]
Division by a fraction is the same as multiplication by its reciprocal:
\[
\frac{43}{12} \times \frac{36}{43}
\]
The \(43\)s cancel out:
\[
\frac{36}{12} = 3
\]

3. **Addition:**
\[
3 + \frac{5}{24}
\]
Convert 3 to a fraction with a denominator of 24:
\[
3 = \frac{72}{24}
\]
Now add:
\[
\frac{72}{24} + \frac{5}{24} = \frac{72 + 5}{24} = \frac{77}{24}
\]

So, the final result is:
\[
\frac{77}{24}
\]

This fraction can also be expressed as a mixed number:
\[
\frac{77}{24} = 3 \frac{5}{24}
\]

Thus, the simplified result of the expression \((4 - \frac{5}{12}) : \frac{43}{36} + \frac{5}{24}\) is \(\frac{77}{24}\) or \(3 \frac{5}{24}\).