What is the value of the following expression? – $\frac{4}{7}\times 4\frac{1}{2}\div {5\frac{1}{3}} \text {of}\: 2\frac{1}{2} – (7\frac{7}{8}\div 5\frac{1}{9} \text{of} \:9\frac{9}{20}) +\frac{1}{2}$

Solution / समाधान:
$\frac{4}{7}\times 4\frac{1}{2}\div 5\frac{1}{3} \text{ of } 2\frac{1}{2} - \left(7\frac{7}{8}\div 5\frac{1}{9} \text{ of } 9\frac{9}{20}\right) + \frac{1}{2}$

Step 1: Convert into improper fractions

$4\frac{1}{2} = \frac{9}{2}, \quad 5\frac{1}{3} = \frac{16}{3}, \quad 2\frac{1}{2} = \frac{5}{2}$

$7\frac{7}{8} = \frac{63}{8}, \quad 5\frac{1}{9} = \frac{46}{9}, \quad 9\frac{9}{20} = \frac{189}{20}$

Step 2: Solve “of” first

$\frac{16}{3} \text{ of } \frac{5}{2} = \frac{16}{3} \times \frac{5}{2} = \frac{40}{3}$

$\frac{46}{9} \text{ of } \frac{189}{20} = \frac{46}{9} \times \frac{189}{20} = \frac{322}{5}$

Step 3: First part

$\frac{4}{7} \times \frac{9}{2} = \frac{18}{7}$

$\frac{18}{7} \div \frac{40}{3} = \frac{18}{7} \times \frac{3}{40} = \frac{27}{140}$

Step 4: Second part

$\frac{63}{8} \div \frac{322}{5} = \frac{63}{8} \times \frac{5}{322} = \frac{315}{2576}$

Step 5: Final expression

$\frac{27}{140} - \frac{315}{2576} + \frac{1}{2}$

LCM = 12880

$\frac{27}{140} = \frac{2484}{12880}, \quad \frac{315}{2576} = \frac{1575}{12880}, \quad \frac{1}{2} = \frac{6440}{12880}$

Step 6: Final calculation

$= \frac{2484 - 1575 + 6440}{12880} = \frac{7349}{12880}$

Simplify:

$= \frac{853}{1610}$

Final Answer / अंतिम उत्तर:

$\frac{853}{1610}$