Two identical heater filaments are connected first in parallel and then in series. At the same applied voltage, the ratio of heat produced in same time for parallel to series will be:

Let each heater filament have resistance $$R$$.

Case 1: Parallel connection

Equivalent resistance: $$R_p = \frac{R}{2}$$

Heat produced: $$H_p = \frac{V^2}{R_p} \times t = \frac{V^2}{R/2} \times t = \frac{2V^2 t}{R}$$

Case 2: Series connection

Equivalent resistance: $$R_s = 2R$$

Heat produced: $$H_s = \frac{V^2}{R_s} \times t = \frac{V^2}{2R} \times t = \frac{V^2 t}{2R}$$

Ratio of heat (parallel to series):

$$\frac{H_p}{H_s} = \frac{2V^2 t/R}{V^2 t/(2R)} = \frac{2}{1/2} = 4$$

The ratio is $$4 : 1$$.