$\left(\frac{2}{9} \times \frac{3}{6}\right)+\left(\frac{3}{4} \div \frac{12}{5}\right)-\left[\frac{1}{4}-\left(\frac{1}{4} \times \frac{2}{3}\right)\right]$

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\left(\frac{2}{9} \times \frac{3}{6}\right)+\left(\frac{3}{4} \div \frac{12}{5}\right)-\left[\frac{1}{4}-\left(\frac{1}{4} \times \frac{2}{3}\right)\right]

Simplify Fractions: First, simplify the fractions in the multiplication and division parts of the expression. $\newline$ $(\frac{2}{9})\times(\frac{3}{6}) = \frac{2\times3}{9\times6} = \frac{6}{54} = \frac{1}{9}$
Simplify Division: Now, simplify the division part. $\newline$ $(\frac{3}{4})\div(\frac{12}{5}) = (\frac{3}{4})\times(\frac{5}{12}) = (\frac{3\times5}{4\times12}) = \frac{15}{48} = \frac{5}{16}$
Simplify Subtraction: Next, simplify the subtraction part inside the brackets. $\newline$ $(\frac{1}{4}) - ((\frac{1}{4})\times(\frac{2}{3})) = (\frac{1}{4}) - (\frac{2}{12}) = (\frac{1}{4}) - (\frac{1}{6}) = (\frac{3}{12}) - (\frac{2}{12}) = \frac{1}{12}$
Add and Subtract Fractions: Now, add and subtract the simplified parts. $\newline$ $(\frac{1}{9}) + (\frac{5}{16}) - (\frac{1}{12})$ $\newline$ To add and subtract fractions, find a common denominator. $\newline$ The common denominator for $9$ , $16$ , and $12$ is $144$ . $\newline$ $(\frac{1}{9}) = (\frac{16}{144})$ , $(\frac{5}{16}) = (\frac{45}{144})$ , $(\frac{1}{12}) = (\frac{12}{144})$
Final Simplified Result: Add and subtract the fractions with the common denominator. $\newline$ $(\frac{16}{144}) + (\frac{45}{144}) - (\frac{12}{144}) = (\frac{16 + 45 - 12}{144}) = \frac{49}{144}$
Final Simplified Result: Add and subtract the fractions with the common denominator. $\newline$ $(\frac{16}{144}) + (\frac{45}{144}) - (\frac{12}{144}) = \frac{16 + 45 - 12}{144} = \frac{49}{144}$ The final simplified result is $\frac{49}{144}$ .