Let’s analyze and solve the mathematical problems from the provided image.

Problem 1

Fill in the blanks to evaluate the expression:
$45 + ((22 - 7) + 5) = 45 + ( \_\_\_ + 5 )$
$45 + \_\_\_ = \_\_\_$

Step-by-Step Solution:

1. Simplify $22 - 7$:
   $22 - 7 = 15$

2. Substitute $15$ into the equation:
   $45 + ((15) + 5)$

3. Simplify $15 + 5$:
   $15 + 5 = 20$

4. Now the equation becomes:
   $45 + 20 = 65$

Answer:
Fill in the blanks as follows: $45 + ((22 - 7) + 5) = 45 + (15 + 5)$
$45 + 20 = 65$

Problem 2

Evaluate:
$4 \times [ ((12 - 7) + (5 \times 5)) \div 6 ]$

Step-by-Step Solution:

1. Simplify $12 - 7$:
   $12 - 7 = 5$

2. Simplify $5 \times 5$:
   $5 \times 5 = 25$

3. Add $5 + 25$:
   $5 + 25 = 30$

4. Divide $30 \div 6$:
   $30 \div 6 = 5$

5. Multiply $4 \times 5$:
   $4 \times 5 = 20$

Answer:
$20$

Problem 3

Evaluate:
$4 \times \left[ \left( \frac{3}{5} \times \frac{2}{10} \right) - \left( \frac{1}{10} + \frac{1}{50} \right) \right]$

Step-by-Step Solution:

1. Simplify $\frac{3}{5} \times \frac{2}{10}$:
   $\frac{3}{5} \times \frac{2}{10} = \frac{6}{50} = \frac{3}{25}$

2. Simplify $\frac{1}{10} + \frac{1}{50}$ using a common denominator of 50:
   $\frac{1}{10} = \frac{5}{50}$
   $\frac{5}{50} + \frac{1}{50} = \frac{6}{50} = \frac{3}{25}$

3. Subtract $\frac{3}{25} - \frac{3}{25}$:
   $\frac{3}{25} - \frac{3}{25} = 0$

4. Multiply $4 \times 0$:
   $4 \times 0 = 0$

Answer:
$0$

Do you need detailed steps for any specific part? Let me know!

Follow-Up Questions:

1. How can you verify your answers to complex expressions?
2. What are the common mistakes to watch for when simplifying fractions?
3. Can parentheses affect the final outcome of an equation?
4. Why is finding a common denominator necessary in fraction addition or subtraction?
5. How do order of operations (PEMDAS) apply in these problems?

Tip: Always double-check your results by re-evaluating the simplified values step-by-step.