To solve the given expression $(13.8 \times 1.9 \div 5.7 + 11.2) \times \left(\frac{1}{16} - \frac{1}{20}\right)$, we need to follow the order of operations (PEMDAS/BODMAS).

1. Evaluate inside the parentheses:

   $13.8 \times 1.9 \div 5.7 + 11.2$

   First, perform the multiplication and division from left to right:

   $13.8 \times 1.9 = 26.22$

   Then divide by 5.7:

   $26.22 \div 5.7 \approx 4.6$

   Now add 11.2:

   $4.6 + 11.2 = 15.8$

2. Evaluate the fraction subtraction:

   $\frac{1}{16} - \frac{1}{20}$

   Find a common denominator, which is 80:

   $\frac{1}{16} = \frac{5}{80}, \quad \frac{1}{20} = \frac{4}{80}$

   Subtract the fractions:

   $\frac{5}{80} - \frac{4}{80} = \frac{1}{80}$

3. Multiply the results:

   $15.8 \times \frac{1}{80}$

   Simplify the multiplication:

   $15.8 \times \frac{1}{80} = \frac{15.8}{80} = 0.1975$

Thus, the final result is:

$0.1975$

Would you like more details or have any questions about this solution?

Here are 5 related questions:

1. How do you convert fractions to have a common denominator?
2. What is the order of operations in mathematics and why is it important?
3. How can you simplify complex fractions?
4. Can you show the detailed steps for multiplication and division with decimals?
5. How do you handle operations involving both fractions and decimals?

Tip: Always remember to follow the order of operations (PEMDAS/BODMAS) to ensure accurate results in calculations.