Let's solve the given expression step-by-step:

The expression is:

$\left[ \frac{3}{4} + \left(2 - \frac{1}{5}\right) - \frac{7}{10} \right] \cdot \left(1 \div \frac{9}{10}\right) + 3$

Step 1: Simplify inside the brackets and parentheses

1. Inside the first square bracket:
   $\frac{3}{4} + \left(2 - \frac{1}{5}\right) - \frac{7}{10}$

   • Calculate $2 - \frac{1}{5}$: $2 - \frac{1}{5} = \frac{10}{5} - \frac{1}{5} = \frac{9}{5}$

   • Now we have: $\frac{3}{4} + \frac{9}{5} - \frac{7}{10}$

   • Find a common denominator for the fractions. The least common multiple of 4, 5, and 10 is 20.

   • Convert each fraction to have a denominator of 20: $\frac{3}{4} = \frac{15}{20}, \quad \frac{9}{5} = \frac{36}{20}, \quad \frac{7}{10} = \frac{14}{20}$

   • Add and subtract the fractions: $\frac{15}{20} + \frac{36}{20} - \frac{14}{20} = \frac{15 + 36 - 14}{20} = \frac{37}{20}$

2. Inside the second parenthesis: $1 \div \frac{9}{10}$

   • Dividing by a fraction is the same as multiplying by its reciprocal: $1 \div \frac{9}{10} = 1 \times \frac{10}{9} = \frac{10}{9}$

Step 2: Combine the simplified terms

Now, the expression is: $\frac{37}{20} \times \frac{10}{9} + 3$

Step 3: Perform the multiplication

• Multiply the fractions: $\frac{37}{20} \times \frac{10}{9} = \frac{37 \times 10}{20 \times 9} = \frac{370}{180}$

• Simplify the fraction by dividing the numerator and the denominator by their greatest common divisor (GCD), which is 10: $\frac{370 \div 10}{180 \div 10} = \frac{37}{18}$

Step 4: Add the result to 3

• Convert 3 to a fraction with a denominator of 18: $3 = \frac{54}{18}$

• Add the fractions: $\frac{37}{18} + \frac{54}{18} = \frac{91}{18}$

Step 5: Simplify or convert to a mixed number (if necessary)

• $\frac{91}{18}$ can be converted to a mixed number: $\frac{91}{18} = 5 \frac{1}{18}$

Final Answer

$5 \frac{1}{18}$

Or, as an improper fraction:

$\frac{91}{18}$

Would you like more details on any step, or do you have any questions?

Related Questions

1. How do you find the least common multiple (LCM) of a set of numbers?
2. What is the process of converting improper fractions to mixed numbers?
3. Why is multiplying by a reciprocal the same as dividing by a fraction?
4. How do you simplify fractions by finding the greatest common divisor (GCD)?
5. What are some strategies for working with fractions involving multiple operations?

Tip

When working with expressions that have multiple fractions and operations, always simplify step-by-step and convert fractions to a common denominator where necessary to make calculations easier.