(4)/(5)div [(1)/(5)+((3)/(4)-(1)/(3))cdot 2]

## Step 1: Simplify the expression inside the parentheses<br />### The expression inside the parentheses is \(\left( \frac{3}{4} - \frac{1}{3} \right)\). To simplify this, we need a common denominator. The least common multiple of 4 and 3 is 12. Convert each fraction:<br />\[<br />\frac{3}{4} = \frac{9}{12}, \quad \frac{1}{3} = \frac{4}{12}<br />\]<br />Subtract these fractions:<br />\[<br />\frac{9}{12} - \frac{4}{12} = \frac{5}{12}<br />\]<br /><br />## Step 2: Multiply the result by 2<br />### Now multiply the result from Step 1 by 2:<br />\[<br />\left( \frac{5}{12} \right) \cdot 2 = \frac{10}{12} = \frac{5}{6}<br />\]<br /><br />## Step 3: Add the result to \(\frac{1}{5}\)<br />### Add \(\frac{1}{5}\) to the result from Step 2. Find a common denominator for 5 and 6, which is 30:<br />\[<br />\frac{1}{5} = \frac{6}{30}, \quad \frac{5}{6} = \frac{25}{30}<br />\]<br />Add these fractions:<br />\[<br />\frac{6}{30} + \frac{25}{30} = \frac{31}{30}<br />\]<br /><br />## Step 4: Divide \(\frac{4}{5}\) by the result from Step 3<br />### The original expression is \(\frac{4}{5} \div \frac{31}{30}\). Dividing by a fraction is equivalent to multiplying by its reciprocal:<br />\[<br />\frac{4}{5} \times \frac{30}{31} = \frac{4 \times 30}{5 \times 31} = \frac{120}{155}<br />\]<br />Simplify \(\frac{120}{155}\) by finding the greatest common divisor, which is 5:<br />\[<br />\frac{120}{155} = \frac{24}{31}<br />\]