Amc 10

We start with our original expression: $10(\frac{1}{2}+\frac{1}{5}+\frac{1}{10})^{-1}$ . First, we sum the fractions inside the parentheses, which gives us $\frac{4}{5}$ . Substituting this into our expression, we get: $10(\frac{4}{5})^{-1}$ . Notice that $x^{-n}=\frac{1}{x^{n}}$ . From this, we can say that $(\frac{4}{5})^{-1}=\frac{1}{\frac{4}{5}^{1}}=\frac{1}{\frac{4}{5}}=\frac{5}{4}$ . Substituting this into our expression, we get: $10(\frac{5}{4})$ . Simplifying this, we get: $\frac{10\cdot5}{4}=\frac{50}{4}=\frac{25}{2}=\boxed{12.5}$ .