Both are correct and natural (depending on the speaker and the context), but have completely different meanings. “Barely adequate” means that it was adequate, but just by a small margin; any less would have been inadequate. “Hardly adequate” means that it was absolutely not adequate at all. Simple.

There is no real difference, you can use them interchangeably. However, there is a slight difference in nuance: 1. If you say "The heating is barely adequate" the nuance changes according to your tone. If your tone is normal, it means 'the heating is meeting the bare minimum criteria at the very least' so you're somewhat satisfied. If your tone is sarcastic, it means 'the heating is not meeting the bare minimum criteria' and you're not satisfied. 2. If you say 'the heating is hardly adequate' regardless of what your tone is like, it comes across as 'the heating does not meet the bare minimum requirement' and it shows you are unsatisfied. 3. No one would ever really say 'the heating is scarcely adequate' in real life, it's not a common phrase. Scarce has the nuance of 'not enough of' and adequate has the nuance of 'barely enough' so saying scarcely adequate is strange because you're saying two things together with kind of opposite meanings. You could say 'The heating is scarce' which would mean 'There is some heating but it is almost the same as having no heating'. It would show dissatisfaction.

I wouldn't be surprised to hear either of these, but I suspect the "hardly" version has more than one interpretation. Firstly "heating adequacy" could mean the temperature, or the ability to heat, so I will assume the temperature, here, for convenience. The heating is barely adequate means that the heating,is only just able to heat, sufficiently, for comfort, or whatever criteria involved. The heating is hardly adequate refers to the heating not being sufficient. It means effectively the opposite thing, that it does not meet the mark, does not reach the criteria. . Hardly can be used to mean just reaches, but always seem to refer to the inadequacy, or near inadequacy. Here, I'd take it to mean does not reach the appropriate level. So the "barely" one say ok, but only just ok. The hardly one, I'd take to mean "not ok", but I'd recognise that some might use it for the same as "barely" if suggested from context. . There may be a regional dependence on the meaning and grammar. See what other answers you get.

Barely, hardly, scarcely: are synonyms. There is some nuance in their use. Barely can mean "not quite enough" Hardly : means with difficulty. The stove hardly heats this room. It heats the room but with difficulty. Barely: the stove barely heats this room. It heats the room but not quite enough. Scarcely: the stove scarcely heats this room. It heats the room not enough. In this case, I think that your grammar book is splitting hairs. "to split hairs": If you say that someone is splitting hairs, you mean that they are making unnecessary distinctions between things when the differences between them are so small they are not important. I hope this helps you.

Does the grammar book explain why?