The resistance of a wire is 5 ohm at $50{ }^{0}C$ and 6 ohm at  $100{ }^{0}C$. The resistance of the wire at $0^{ 0}C$  will be 

Resistance of a metal conductor at temperature $t^{0}C$ is given by $R_t=R_0\left(1+\alpha t\right),R_0$ is the resistance of the wire at $0^{0}C$ and $\alpha$ is the temperature coefficient of resistance. 

Resistance at  ${50}^{0}C,R_{50}=R_0\left(1+{50}_{\alpha }\right)$           ……….. (i)
Resistance at ${100}^{0}C,R_{100}=R_0\left(1+{100}_{\alpha }\right)$        ……….. (ii)
From (i), $R_{50}-R_0={50}_{\alpha }{R}_0$                      …………. (iii)
From (ii), $R_{100}-R_0={100}_{\alpha }{R}_0$              ………… (iv)

Dividing (iii) by (iv), we get $\frac{R_{50}-R_0}{R_{100}-R_0}=\frac{1}{2}$
Here, $R_{50}=5\Omega$  and $R_{100}=6\Omega$

$\therefore \frac{5-R_0}{6-R_0}=\frac{1}{2}$ or, $6-R_0=10-2R_0$  or,  $R_0=4\Omega$