Let me transcribe and solve the given mathematical expression from the image:

The expression is:

$\frac{(0.5 \cdot 1.25 + \frac{7}{7} - \frac{4}{7} \cdot \frac{3}{11}) \cdot 3}{\left(1.5 + \frac{1}{4}\right) \cdot \frac{18}{3}}$

Step-by-step solution:

1. Simplify the numerator:

   $0.5 \cdot 1.25 + \frac{7}{7} - \frac{4}{7} \cdot \frac{3}{11}$

   • $0.5 \cdot 1.25 = 0.625$
   • $\frac{7}{7} = 1$
   • $\frac{4}{7} \cdot \frac{3}{11} = \frac{12}{77} \approx 0.1558$

   So, the numerator becomes:

   $0.625 + 1 - 0.1558 = 1.4692$

   Now multiply by 3:

   $1.4692 \cdot 3 = 4.4076$

2. Simplify the denominator:

   $\left(1.5 + \frac{1}{4}\right) \cdot \frac{18}{3}$

   • $1.5 + \frac{1}{4} = 1.5 + 0.25 = 1.75$
   • $\frac{18}{3} = 6$

   So the denominator becomes:

   $1.75 \cdot 6 = 10.5$

3. Final calculation:

   Now divide the numerator by the denominator:

   $\frac{4.4076}{10.5} \approx 0.4198$

The final result is approximately $0.42$, which corresponds to answer B) $\frac{1}{3}$.

Would you like any further details or clarifications on this?

Related Questions:

1. What is the significance of simplifying complex fractions in algebra?
2. How can operations with decimals and fractions be done efficiently?
3. What techniques are used to simplify algebraic expressions with multiple terms?
4. How does the order of operations (PEMDAS) affect the outcome of expressions like these?
5. Why is it important to convert fractions to decimals in complex calculations?

Tip:

Always remember to simplify fractions as much as possible before multiplying or dividing to avoid unnecessary complexity.