Simplify: $\left[\left(\frac{3}{4} ight) + \left(\frac{7}{8} \div \frac{5}{6} ight) ight] \div \left[\left(\frac{5}{3} ight) - \left(\frac{4}{9} imes \frac{3}{5} ight) ight]$

Question

Simplify: $\left[\left(\frac{3}{4}ight) + \left(\frac{7}{8} \div \frac{5}{6}ight)ight] \div \left[\left(\frac{5}{3}ight) - \left(\frac{4}{9} 	imes \frac{3}{5}ight)ight]$

Accepted Answer

Simplify: $\left[\left(\frac{3}{4}ight) + \left(\frac{7}{8} \div \frac{5}{6}ight)ight] \div \left[\left(\frac{5}{3}ight) - \left(\frac{4}{9} 	imes \frac{3}{5}ight)ight]$
Solution:
First, simplify the division in the first bracket:
$$ \frac{7}{8} \div \frac{5}{6} $$
$$ = \frac{7}{8} 	imes \frac{6}{5} $$
$$ = \frac{42}{40} $$
$$ = \frac{21}{20} $$
Now, the first bracket becomes:
$$ \frac{3}{4} + \frac{21}{20} $$
$$ = \frac{15}{20} + \frac{21}{20} $$
$$ = \frac{36}{20} $$
$$ = \frac{9}{5} $$
Now, simplify the multiplication in the second bracket:
$$ \frac{4}{9} 	imes \frac{3}{5} $$
$$ = \frac{12}{45} $$
$$ = \frac{4}{15} $$
Now, the second bracket becomes:
$$ \frac{5}{3} - \frac{4}{15} $$
$$ = \frac{25}{15} - \frac{4}{15} $$
$$ = \frac{21}{15} $$
$$ = \frac{7}{5} $$
Now, divide both simplified values:
$$ \frac{9}{5} \div \frac{7}{5} $$
$$ = \frac{9}{5} 	imes \frac{5}{7} $$
$$ = \frac{9}{7} $$
Therefore, the simplified value is:
$$ \boxed{\frac{9}{7}} $$
Final Answer: **$\frac{9}{7}$**

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