In the given circuit, if power rating of heater is 1000 Watt at 100...

Power rating of heater, $\mathrm{P}=1000 \mathrm{~W}$ Voltage rating of heater, $\mathrm{V}=100 \mathrm{~V}$ $\therefore$ Resistance of heater, $\mathrm{R}_1=\frac{\mathrm{V}^2}{\mathrm{P}}=\frac{(100)^2}{1000}=10 \Omega$ According to question, power dissipated in heater, $\mathrm{P}^{\prime}=62.5 \mathrm{~W}\therefore$ Voltage \left(\mathrm{V}^{\prime}\right.$) across heater can be calculated as$$\begin{aligned} & \mathrm{P}^{\prime}=\frac{\left(\mathrm{V}^{\prime}\right)^2}{\mathrm{R}_1} \Rightarrow\left(\mathrm{V}^{\prime}\right)^2=\mathrm{P}^{\prime} \times \mathrm{R}_1=62.5 \times 10 \\ & \Rightarrow \quad \mathrm{V}^{\prime}=25 \mathrm{~V} \text{ (across heater) } \\ & \end{aligned}$$\therefore$ Voltage across $10 \Omega$ resistor, $\mathrm{V}^{\prime \prime}=100-25=75 \mathrm{~V}$ Current in $10 \Omega$ resistor $=\frac{\mathrm{V}^{\prime \prime}}{10}=\frac{75}{10}=7.5 \mathrm{~A}$ Current in heater resistor $=\frac{\mathrm{V}^{\prime}}{10}=\frac{25}{10}=2.5 \mathrm{~A}$ So, current in $\mathrm{R} \Omega=7.5-25=5 \mathrm{~A}$ Now, $\mathrm{V}=\mathrm{IR} \Rightarrow$\mathrm{R}=\mathrm{V}$/$\mathrm{I}$=\frac{25}{5}=5 \Omega