Your sister is going to come visit you but hasn't bought her airplane tickets yet. She wants to get the cheapest tickets possible. You just advised her that she ought to go ahead and book her flight soon. But now you're remembering that there are also good deals on plane tickets in the last few days before the flight. You say:
Although, on the other hand, you might be better off waiting until the last minute.
The word "although" is similar in meaning to "but". There are a few differences in how they're used, though. "But" is used to connect ideas that conflict with each other, but these ideas are somehow related to each other:
She's moving to London, but she'll be back to visit.
When you use "although" in spoken English, it usually connects two conflicting ideas. But these ideas are each a completely separate thought. When you start a new sentence with "although", it sounds like you weren't planning on saying this thought when you finished your last sentence:
She's moving to London so we probably won't see her much any more. Although I'm sure she'll be back to visit.
Use "on the other hand" when giving an opinion that's different from one that was just said. You can contradict another person's opinion, or you can contradict your own like in the example at top.
When you use "on the other hand" in speech or writing, it makes you seem careful, intelligent, and thoughtful. It's not really the best choice when you want to disagree with someone angrily.
"Be better off ___ing" is a phrase that's used to express a better choice. You can use it to describe the choice that you think someone should take:
A: Maybe we can try to sell it?
B: I think we'd be better off just throwing it away.
The phrase "the last minute" is a common way of describing doing something just before the deadline. Of course, it doesn't literally mean the last 1 minute before the deadline. In the case of buying an airplane ticket, "the last minute" might mean 1 week before you leave.
If you really want to emphasize how close to the deadline you are, use "very" before "last minute":
I waited until the very last minute to turn it in.
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