You're travelling to another country with your boyfriend. When you got to the airport, he wasn't able to find his passport. You both got worried, but after searching for a few minutes he found it. You're worried that he'll lose it again, so you offer:
Why don't I hang on to that for you.
This is a way to offer to do something:
Why don't I drop by your office at around three thirty.
Why don't I call around and see if there's anyone interested in buying it.
This expression sounds confident and in-control. So it's someone who's in a position of authority, or who's close to the person they're speaking to.
When you pronounce a sentence that uses "Why don't I ___", say it like a statement, not a question.
You can also use the expressions "Why don't we ___" and "Why don't you ___":
To "hang on to" something means to hold or keep it, without losing it. You use this phrase when you think that someone might lose something:
If you find a good man, hang on to him.
You should hang on to the warranty, and contact the manufacturer if there are any problems with it.
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