(frac(1)/(2))((2)/(5)-1+(2)/(3+1))

To solve the expression \( \frac{\frac{1}{2}}{\frac{2}{5}-1+\frac{2}{3+1}} \), let's break it down step by step:<br /><br />1. Simplify the denominator:<br /> \[<br /> \frac{2}{5} - 1 + \frac{2}{3+1}<br /> \]<br /><br />2. Simplify \( \frac{2}{3+1} \):<br /> \[<br /> \frac{2}{3+1} = \frac{2}{4} = \frac{1}{2}<br /> \]<br /><br />3. Substitute \( \frac{1}{2} \) back into the denominator:<br /> \[<br /> \frac{2}{5} - 1 + \frac{1}{2}<br /> \]<br /><br />4. Find a common denominator for the terms in the denominator:<br /> \[<br /> \frac{2}{5} = \frac{4}{10}, \quad 1 = \frac{10}{10}, \quad \frac{1}{2} = \frac{5}{10}<br /> \]<br /><br />5. Combine the terms:<br /> \[<br /> \frac{4}{10} - \frac{10}{10} + \frac{5}{10} = \frac{4 - 10 + 5}{10} = \frac{-1}{10}<br /> \]<br /><br />6. Now, substitute the simplified denominator back into the original expression:<br /> \[<br /> \frac{\frac{1}{2}}{\frac{-1}{10}}<br /> \]<br /><br />7. Dividing by a fraction is the same as multiplying by its reciprocal:<br /> \[<br /> \frac{1}{2} \times \frac{10}{-1} = \frac{10}{-2} = -5<br /> \]<br /><br />So, the value of the expression is \( -5 \).