Two bulbs connected by a tube with two mercury threads

There is pressure in the gas between the two mercury columns.
The equilibrium equations should include this too.

 

Not when you take the ratio of the two heights. The ratio h1/h2 is 2:1 only if p=0.

You can calculate the difference though, h2-h1, and then compare with the answers.

 

... And only one of the offered answers is consistent with p > 0. That is all you are asked to do. There is not enough information to deduce the exact ratio.

 

However there is enough information to deduce the difference, h1-h2. And only one answer is compatible with that difference.

 

The ratio is h1 : h2 = 16000-p : 8000-p. p > 0. You cannot determine the ratio exactly, but only one of the offered ratios is consistent with these facts. That is why the question is worded this way. It does not ask you which of the offered answers is the ratio, it asks which could be the ratio.

 

Actually it does not say or ask anything about a ratio.
They give you the density of the mercury so you can calculate the actual value of the difference. (6 cm).

But even so, it seems that none of the pairs will work.
Taking the last one, h1=18cm and h2=12cm, for example.
In order to have the 18 cm column in equilibrium, and considering that 16000 Pa is about 12 cm Hg, the pressure p should be negative. Unless I made an error of calculation.

For the first pair, in order to have the 4 cm column in equilibrium with the 12 cm from the bulb, p should be 8 cm Hg.
But in order to have the 2 cm in equilibrium with the 6 cm from the other bulb, p should be 4 cm Hg.

Something is fishy.

 

You are right, you can calculate the difference and that does answer the question. But it is still the case that there is not enough information to determine the two heights - it is merely a matter of saying which pair of heights is feasible, and that can be resolved by either method.