Điện trở tương đương của mạch nối tiếp và song song là: \(\left\{ \begin{align} & {{R}_{nt}}={{R}_{1}}+{{R}_{2}} \\ & {{R}_{ss}}=\frac{{{R}_{1}}{{R}_{2}}}{{{R}_{1}}+{{R}_{2}}} \\ \end{align} \right.\) Theo bài ra ta có: \(\begin{align} & {{R}_{nt}}=4,5{{R}_{ss}}\Rightarrow {{R}_{1}}+{{R}_{2}}=4,5\frac{{{R}_{1}}{{R}_{2}}}{{{R}_{1}}+{{R}_{2}}} \\ & \Leftrightarrow {{\left( {{R}_{1}}+{{R}_{2}} \right)}^{2}}=4,5{{R}_{1}}{{R}_{2}} \\ & \Leftrightarrow R_{1}^{2}+R_{2}^{2}=2,5{{R}_{1}}{{R}_{2}} \\ & \Leftrightarrow \frac{{{R}_{1}}}{{{R}_{2}}}+\frac{{{R}_{2}}}{{{R}_{1}}}=2,5\,\,\,\,\left( * \right) \\ \end{align}\) Đặt : \(\begin{array}{l} \frac{{{R_1}}}{{{R_2}}} = x \Rightarrow \frac{{{R_2}}}{{{R_1}}} = \frac{1}{x}\\ \Rightarrow \left( * \right) \Leftrightarrow x + \frac{1}{x} = 2,5\\ \Rightarrow \left[ \begin{array}{l} x = 0,5\\ x = 2 \end{array} \right. \Rightarrow \left[ \begin{array}{l} \frac{{{R_1}}}{{{R_2}}} = 0,5\\ \frac{{{R_1}}}{{{R_2}}} = 2 \end{array} \right. \end{array}\)