To solve the expression \( \frac{5}{24} + \frac{3}{85/24} + \frac{3}{8} \), we first need to clarify what the expression means since the second term appears to be written incorrectly. If we interpret \( \frac{3}{85/24} \) as \( \frac{3 \cdot 24}{85} \), we can proceed with this assumption.

Let's solve it step by step.

1. First, simplify \( \frac{3 \cdot 24}{85} = \frac{72}{85} \).

Now, we have:

\[
\frac{5}{24} + \frac{72}{85} + \frac{3}{8}
\]

2. Next, we need to find a common denominator for all three fractions. The denominators are \( 24 \), \( 85 \), and \( 8 \).

- The least common multiple (LCM) of \( 24 \), \( 85 \), and \( 8 \):
- The prime factorization gives \( 24 = 2^3 \cdot 3 \), \( 85 = 5 \cdot 17 \), and \( 8 = 2^3 \).
- The LCM takes the highest powers of each prime: \( 2^3 \), \( 3^1 \), \( 5^1 \), \( 17^1 \) = \( 8 \cdot 3 \cdot 5 \cdot 17 = 2040 \).

3. Convert each fraction to have the common denominator of \( 2040 \):

- For \( \frac{5}{24} \):
\[
\frac{5}{24} = \frac{5 \cdot 85}{2040} = \frac{425}{2040}
\]

- For \( \frac{72}{85} \):
\[
\frac{72}{85} = \frac{72 \cdot 24}{2040} = \frac{1728}{2040}
\]

- For \( \frac{3}{8} \):
\[
\frac{3}{8} = \frac{3 \cdot 255}{2040} = \frac{765}{2040}
\]

4. Now combine the fractions:

\[
\frac{425}{2040} + \frac{1728}{2040} + \frac{765}{2040} = \frac{425 + 1728 + 765}{2040} = \frac{2918}{2040}
\]

5. Simplify \( \frac{2918}{2040} \). First, find the GCD of \( 2918 \) and \( 2040 \), which is \( 1 \) (since they share no common prime factors).

Thus, \( \frac{2918}{2040} \) is already in its simplest form.

The final answer is:

\[
\frac{2918}{2040}
\]
Or, as a decimal, approximately \( 1.43 \).