The expression you have uploaded is quite complex. Let's break it down step by step and simplify it.

Here is the expression:

$\frac{2 \frac{3}{4}}{1 \frac{5}{6}} \div \frac{7}{8} \times \left( 1 \frac{1}{3} - \frac{3}{4} \right) + \frac{5}{7} \div \frac{3}{4} \text{ of } \frac{3}{7}$

Step 1: Convert all mixed numbers to improper fractions.

1. $2 \frac{3}{4} = \frac{11}{4}$
2. $1 \frac{5}{6} = \frac{11}{6}$
3. $1 \frac{1}{3} = \frac{4}{3}$

Now, the expression becomes:

$\frac{\frac{11}{4}}{\frac{11}{6}} \div \frac{7}{8} \times \left( \frac{4}{3} - \frac{3}{4} \right) + \frac{5}{7} \div \frac{3}{4} \times \frac{3}{7}$

Step 2: Simplify the complex fraction $\frac{\frac{11}{4}}{\frac{11}{6}}$.

When dividing fractions, multiply by the reciprocal of the denominator:

$\frac{11}{4} \times \frac{6}{11} = \frac{66}{44} = \frac{3}{2}$

So, the expression now becomes:

$\frac{3}{2} \div \frac{7}{8} \times \left( \frac{4}{3} - \frac{3}{4} \right) + \frac{5}{7} \div \frac{3}{4} \times \frac{3}{7}$

Step 3: Perform the division $\frac{3}{2} \div \frac{7}{8}$.

$\frac{3}{2} \times \frac{8}{7} = \frac{24}{14} = \frac{12}{7}$

So, the expression now becomes:

$\frac{12}{7} \times \left( \frac{4}{3} - \frac{3}{4} \right) + \frac{5}{7} \div \frac{3}{4} \times \frac{3}{7}$

Step 4: Simplify $\frac{4}{3} - \frac{3}{4}$.

Find the common denominator:

$\frac{4}{3} = \frac{16}{12}, \quad \frac{3}{4} = \frac{9}{12}$ $\frac{16}{12} - \frac{9}{12} = \frac{7}{12}$

Now the expression is:

$\frac{12}{7} \times \frac{7}{12} + \frac{5}{7} \div \frac{3}{4} \times \frac{3}{7}$

Step 5: Simplify $\frac{12}{7} \times \frac{7}{12}$.

$\frac{12}{7} \times \frac{7}{12} = 1$

So, the expression now becomes:

$1 + \frac{5}{7} \div \frac{3}{4} \times \frac{3}{7}$

Step 6: Simplify $\frac{5}{7} \div \frac{3}{4}$.

$\frac{5}{7} \times \frac{4}{3} = \frac{20}{21}$

Now the expression is:

$1 + \frac{20}{21} \times \frac{3}{7}$

Step 7: Simplify $\frac{20}{21} \times \frac{3}{7}$.

$\frac{20}{21} \times \frac{3}{7} = \frac{60}{147} = \frac{20}{49}$

Step 8: Final step.

$1 + \frac{20}{49} = \frac{49}{49} + \frac{20}{49} = \frac{69}{49}$

So, the simplified form of the given expression is:

$\frac{69}{49}$

Do you have any questions, or would you like further details?

5 Related Questions:

1. How do you convert a mixed number into an improper fraction?
2. How do you simplify complex fractions step by step?
3. What are the rules for dividing and multiplying fractions?
4. How do you find a common denominator for subtracting fractions?
5. How can you check your work when simplifying expressions involving fractions?

Tip:

When working with fractions, always remember to simplify at each step to avoid handling large numbers later on.