Let's break down and solve the expression step by step:

Expression:
$2 \frac{1}{4} \times \frac{4}{5} \div \left( \frac{3}{4} - \frac{1}{2} \right)$

Step 1: Convert mixed number to improper fraction.

$2 \frac{1}{4} = \frac{9}{4}$ So, the expression becomes: $\frac{9}{4} \times \frac{4}{5} \div \left( \frac{3}{4} - \frac{1}{2} \right)$

Step 2: Subtract the fractions inside the parentheses.

To subtract $\frac{1}{2}$ from $\frac{3}{4}$, first find a common denominator: $\frac{1}{2} = \frac{2}{4}$ $\frac{3}{4} - \frac{2}{4} = \frac{1}{4}$ Now, the expression is: $\frac{9}{4} \times \frac{4}{5} \div \frac{1}{4}$

Step 3: Simplify the division.

Dividing by a fraction is the same as multiplying by its reciprocal: $\frac{9}{4} \times \frac{4}{5} \times \frac{4}{1}$

Step 4: Multiply the fractions.

First, simplify: $\frac{9}{4} \times \frac{4}{5} = \frac{36}{20}$ Next, multiply: $\frac{36}{20} \times \frac{4}{1} = \frac{144}{20}$

Step 5: Simplify the result.

Simplify $\frac{144}{20}$ by dividing both the numerator and the denominator by 4: $\frac{144 \div 4}{20 \div 4} = \frac{36}{5}$

So, the final result is: $\frac{36}{5} = 7 \frac{1}{5}$

Final Answer:

$7 \frac{1}{5}$

Would you like further details on any of the steps?

Here are 5 related questions:

1. How do you convert mixed numbers to improper fractions?
2. What is the rule for dividing by a fraction?
3. Can you simplify $\frac{9}{4} \times \frac{4}{5}$ directly without multiplying first?
4. How do you find a common denominator for subtracting fractions?
5. How can we simplify large fractions more efficiently?

Tip: When multiplying or dividing fractions, always simplify where possible before carrying out operations to make calculations easier.