Simplify the following :

\(\rm \left(1\frac{1}{3}\times1\frac{4}{5}\div \frac{9}{10}\right) \times(2\frac{1}{6}\ of \ \frac{2}{3}\div 1\frac{1}{3})- (9\frac{4}{9} \div 11\frac{1}{3} \ of \ \frac{5}{6}) \)

Given:

\(\rm \left(1\frac{1}{3}\times1\frac{4}{5}\div \frac{9}{10}\right) \times(2\frac{1}{6}\ of \ \frac{2}{3}\div 1\frac{1}{3})- (9\frac{4}{9} \div 11\frac{1}{3} \ of \ \frac{5}{6}) \)

Concept used:

Calculation:

\(\rm \left(1\frac{1}{3}\times1\frac{4}{5}\div \frac{9}{10}\right) \times(2\frac{1}{6}\ of \ \frac{2}{3}\div 1\frac{1}{3})- (9\frac{4}{9} \div 11\frac{1}{3} \ of \ \frac{5}{6}) \)

⇒ \(({4 \over 3} \times {9 \over 5} \div {9 \over 10}) \times ({13 \over 6}\ of\ {2 \over 3} \div {4 \over 3}) - ({85 \over 9} \div {34 \over 3}\ of\ {5 \over 6})\)

⇒ \(({4 \over 3} \times {9 \over 5} \div {9 \over 10}) \times ({13 \over 9} \div {4 \over 3}) - ({85 \over 9} \div {85 \over 9})\)

⇒ \(({4 \over 3} \times {9 \over 5} \times {10 \over 9}) \times ({13 \over 9} \times {3 \over 4}) - ({85 \over 9} \div {85 \over 9})\)

⇒ \({8 \over 3} \times {13 \over 12} - 1\)

⇒ \({26 \over 9} - 1\)

⇒ \({17 \over 9}\)

⇒ \(1\frac{8}{9}\)

∴ The simplified value is \(1\frac{8}{9}\).