Two conducting wires of the same material and of equal lengths and...

Let the resistance of each wire be $R$.

When the wires are connected in parallel, the equivalent resistance is $R_p = \dfrac{R}{2}$.

When the wires are connected in series, the equivalent resistance is $R_s = 2R$.

The heat produced in a circuit across a potential difference $V$ in time $t$ is given by $H = \dfrac{V^2}{R_{eq}} t$.

The heat produced in the parallel combination is $H_p = \dfrac{V^2}{R_p} t = \dfrac{2V^2}{R} t$.

The heat produced in the series combination is $H_s = \dfrac{V^2}{R_s} t = \dfrac{V^2}{2R} t$.

The ratio of heat produced in parallel and series combinations is $\dfrac{H_p}{H_s} = \dfrac{\dfrac{2V^2}{R} t}{\dfrac{V^2}{2R} t} = 4$.

Thus, the required ratio is $4 : 1$.

Answer: 4 : 1