The given expression to solve is:

$\left( \frac{2}{15} + \frac{7}{12} \right) \cdot \frac{30}{103} - \left( 2 : 2 \frac{1}{4} \right) \cdot \frac{9}{32}$

Let's break down the steps to solve this:

1. Evaluate $\frac{2}{15} + \frac{7}{12}$:

   • Find the least common denominator (LCD) for $\frac{2}{15}$ and $\frac{7}{12}$, which is 60.
   • Convert each fraction: $\frac{2}{15} = \frac{8}{60}, \quad \frac{7}{12} = \frac{35}{60}$
   • Add the fractions: $\frac{8}{60} + \frac{35}{60} = \frac{43}{60}$

2. Multiply $\frac{43}{60}$ by $\frac{30}{103}$: $\frac{43}{60} \cdot \frac{30}{103} = \frac{1290}{6180} = \frac{43}{206}$

3. Evaluate $2 : 2 \frac{1}{4}$:

   • Convert $2 \frac{1}{4}$ to an improper fraction: $2 \frac{1}{4} = \frac{9}{4}$
   • Divide $2$ by $\frac{9}{4}$: $2 : \frac{9}{4} = 2 \cdot \frac{4}{9} = \frac{8}{9}$

4. Multiply $\frac{8}{9}$ by $\frac{9}{32}$: $\frac{8}{9} \cdot \frac{9}{32} = \frac{72}{288} = \frac{1}{4}$

5. Final Expression:

   • Now, substitute back into the original expression: $\frac{43}{206} - \frac{1}{4}$
   • To subtract these fractions, find the LCD, which is 412. $\frac{43}{206} = \frac{86}{412}, \quad \frac{1}{4} = \frac{103}{412}$
   • Subtract: $\frac{86}{412} - \frac{103}{412} = \frac{-17}{412} = -\frac{17}{412}$

Final Answer:

$-\frac{17}{412}$

Would you like further clarification on any of these steps?

Related Questions:

1. How do you find a common denominator when adding fractions?
2. What are the steps to convert a mixed number to an improper fraction?
3. How do you simplify fractions after multiplication?
4. What are some strategies for simplifying complex fractions?
5. Why is it important to follow the order of operations in expressions?

Tip: When working with complex fractions, break down each operation step-by-step, simplifying as you go to avoid large numbers.