Let's solve each part step by step.

### a) \( \frac{11}{24} - \frac{5}{41} + \frac{13}{24} + 0.5 - \frac{36}{41} \)

First, combine the fractions with the same denominators:
\[ \frac{11}{24} + \frac{13}{24} = \frac{24}{24} = 1 \]

Next, combine the fractions with the denominator 41:
\[ -\frac{5}{41} - \frac{36}{41} = -\frac{41}{41} = -1 \]

Now, add the results:
\[ 1 - 1 + 0.5 = 0.5 \]

So, the result is:
\[ 0.5 \]

### b) \( (-12) : \left( \frac{3}{4} - \frac{5}{6} \right) \)

First, find the common denominator for the fractions inside the parentheses:
\[ \frac{3}{4} = \frac{9}{12} \]
\[ \frac{5}{6} = \frac{10}{12} \]

Now, subtract the fractions:
\[ \frac{9}{12} - \frac{10}{12} = -\frac{1}{12} \]

Now, divide -12 by \(-\frac{1}{12}\):
\[ (-12) \div \left( -\frac{1}{12} \right) = (-12) \times \left( -12 \right) = 144 \]

So, the result is:
\[ 144 \]

### c) \( (-6.5) \cdot 5.7 + 5.7 \cdot (-3.5) \)

First, multiply the numbers:
\[ (-6.5) \cdot 5.7 = -37.05 \]
\[ 5.7 \cdot (-3.5) = -19.95 \]

Now, add the results:
\[ -37.05 + (-19.95) = -57 \]

So, the result is:
\[ -57 \]

### d) \( 23 \frac{1}{4} - 13 \frac{1}{4} - \frac{5}{7} \)

First, convert the mixed numbers to improper fractions:
\[ 23 \frac{1}{4} = \frac{93}{4} \]
\[ 13 \frac{1}{4} = \frac{53}{4} \]

Now, subtract the fractions:
\[ \frac{93}{4} - \frac{53}{4} = \frac{40}{4} = 10 \]

Next, subtract \(\frac{5}{7}\) from 10:
\[ 10 - \frac{5}{7} = \frac{70}{7} - \frac{5}{7} = \frac{65}{7} \]

So, the result is:
\[ \frac{65}{7} \]

### e) \( 16 \cdot \frac{2}{7} \cdot \left( -\frac{3}{5} \right) + 28 \cdot \frac{2}{7} \cdot \frac{3}{5} \)

First, multiply the numbers:
\[ 16 \cdot \frac{2}{7} = \frac{32}{7} \]
\[ \frac{32}{7} \cdot \left( -\frac{3}{5} \right) = -\frac{96}{35} \]

Next, multiply the other set of numbers:
\[ 28 \cdot \frac{2}{7} = 8 \]
\[ 8 \cdot \frac{3}{5} = \frac{24}{5} \]

Now, convert \(\frac{24}{5}\) to a common denominator with \(-\frac{96}{35}\):
\[ \frac{24}{5} = \frac{168}{35} \]

Add the results:
\[ -\frac{96}{35} + \frac{168}{35} = \frac{72}{35} \]

So, the result is:
\[ \frac{72}{35} \]

### f) \( 4 : \left( 4 \frac{4}{3} - \frac{1}{2} \right) : \left( 6 \frac{2}{5} - \left( -\frac{7}{15} \right) \right) \)

First, convert the mixed numbers to improper fractions:
\[ 4 \frac{4}{3} = \frac{16}{3} \]
\[ 6 \frac{2}{5} = \frac{32}{5} \]

Now, subtract and add the fractions:
\[ \frac{16}{3} - \frac{1}{2} = \frac{32}{6} - \frac{3}{6} = \frac{29}{6} \]
\[ \frac{32}{5} + \frac{7}{15} = \frac{96}{15} + \frac{7}{15} = \frac{103}{15} \]

Now, divide 4 by the results:
\[ 4 \div \frac{29}{6} = 4 \times \frac{6}{29} = \frac{24}{29} \]
\[ \frac{24}{29} \div \frac{103}{15} = \frac{24}{29} \times \frac{15}{103} = \frac{360}{2987} \]

So, the result is:
\[ \frac{360}{2987} \]