1. Pressure excess in a soap bubble

For a soap bubble of radius $r$, the excess pressure inside over outside is

$\Delta P = \frac{4T}{r}$

where $T$ is the surface tension.

2. Pressure inside the outer bubble

The outer bubble has radius $6\text{ cm}$ and outside pressure is $P_0$. So pressure inside the outer bubble is

$P_1 = P_0 + \frac{4T}{6}$

3. Pressure inside the inner bubble

The inner bubble has radius $4\text{ cm}$ and it is surrounded by the air inside the outer bubble, whose pressure is $P_1$. Thus pressure inside the inner bubble is

$P_2 = P_1 + \frac{4T}{4}$

Substitute $P_1$:

$P_2 = P_0 + \frac{4T}{6} + \frac{4T}{4}$

So,

$P_2 - P_0 = \frac{4T}{6} + \frac{4T}{4}$

Take LCM:

$P_2 - P_0 = 4T\left(\frac{1}{6} + \frac{1}{4}\right) = 4T\left(\frac{2+3}{12}\right) = 4T\cdot \frac{5}{12} = \frac{5T}{3}$

4. Equivalent single soap bubble

Let the radius of another soap bubble be $R$. Its pressure difference between inside and outside is given as $P_2-P_0$.

Hence,

$\frac{4T}{R} = \frac{5T}{3}$

Cancel $T$:

$\frac{4}{R} = \frac{5}{3}$

$R = \frac{12}{5} = 2.4\text{ cm}$

5. Check options

• A: $12$ cm $\quad$ Incorrect
• B: $2.4$ cm $\quad$ Correct
• C: $6$ cm $\quad$ Incorrect
• D: $4.8$ cm $\quad$ Incorrect

So the correct option is B.