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, $\mathrm{V}=\mathrm{IR} \Rightarrow \mathrm{R}=\mathrm{V} / \mathrm{I}=\frac{25}{5}=5 \Omega$