Question : Find the value of the given expression:
$\frac{(4\frac{1}{3}+3\frac{1}{3}\times 1\frac{4}{5}\div 3\frac{3}{4}\times (1\frac{1}{2}+1\frac{1}{3}))}{(\frac{2}{3}\div \frac{5}{6}\times \frac{2}{3})}$
Option 1: $11 \frac{3}{8}$
Option 2: $10\frac{1}{8}$
Option 3: $14\frac{3}{8}$
Option 4: $16\frac{5}{8}$
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Correct Answer: $16\frac{5}{8}$
Solution :
Given, $\frac{(4\frac{1}{3}+3\frac{1}{3}\times 1\frac{4}{5}\div 3\frac{3}{4}\times (1\frac{1}{2}+1\frac{1}{3}))}{(\frac{2}{3}\div \frac{5}{6}\times \frac{2}{3})}$
Converting mixed fractions to improper fractions,
= $\frac{(\frac{13}{3}+\frac{10}{3}\times \frac{9}{5}\div \frac{15}{4}\times (\frac{3}{2}+\frac{4}{3}))}{(\frac{2}{3}\div \frac{5}{6}\times \frac{2}{3})}$
= $\frac{(\frac{13}{3}+\frac{10}{3}\times \frac{36}{75}\times \frac{17}{6})}{(\frac{4}{5}\times \frac{2}{3})}$
= $\frac{(\frac{13}{3}+\frac{68}{15})}{(\frac{8}{15})}$
= $\frac{(\frac{133}{15})}{(\frac{8}{15})}$
= $\frac{133}{8}$
= $16\frac{5}{8}$
Hence, the correct answer is $16\frac{5}{8}$.
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