Two bulbs are rated '60 W, 220 V' and '60 W, 110 V' respectively. Calculate the ratio of their resistances.

Given,

Voltage (V₁) = 220 V

Voltage (V₂) = 110 V

P = 60 W (both bulbs have same power)

Resistance (R) = ?

From relation

$P = \dfrac{V^2}{R} \\[0.5em]$

Case 1:

$60 = \dfrac{220^2}{R} \\[0.5em] \Rightarrow R_1 = \dfrac{220^2}{60} \\[0.5em]$

Case 2:

$60 = \dfrac{110^2}{R} \\[0.5em] \Rightarrow R_2 = \dfrac{110^2}{60} \\[0.5em]$

Ratio between the two is —

$R$1 : R2 = \dfrac{220^2}{60} : \dfrac{110^2}{60} \\[1em] R1 : R2 = \dfrac{\dfrac{220^2}{60}}{\dfrac{110^2}{60}} \\[1em] R1 : R2 = \dfrac{\dfrac{220^2}{\cancel{60}}}{\dfrac{110^2}{\cancel{60}}} \\[1em] R1 : R2 = \dfrac{220^2}{110^2} \\[1em] R1 : R2 = \dfrac{2^2}{1^2} \\[1em] R1 : R2 = \dfrac{4}{1}

Hence, ratio of resistances of the two bulbs is = 4 : 1