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I know what "How do you do?" means but I am surprised by this exercise from "Basic Survival" (AmE). This is the whole exercise with all possible answers. I matched these sentences and that is it what came out.

Match the sentences:

Greeting Response
Hello ~Hello
How do you do? ~Fine, thanks. And you?
Please call me Tom. ~OK... Tom.
Welcome to Smith Imc. ~I'm glad to be here.
So you are the director? ~ That's right.
Please sit down. ~Thank you.


So, can we say in response to How do you do? ~`Fine, thanks and you ???
Or, meybe it is a typo :/

Thanks!
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Hi,

So, can we say in response to How do you do? ~`Fine, thanks and you ???

Definitely not. Just reply with 'How do you do?'

Perhaps the writer of the exercise was thinking of the greeting 'How are you?' For this, ~`Fine, thanks and you? is OK.

Best wishes, Clive
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Also, if someone says "Please call me Tom," the expected resonse is something like "Okay Tom. And please call me Dominik." If one person asks you to call him by his first name but you don't respond in kind, at least in the U.S., that could be awkward. Do you expect to call him Tom while he calls you Mr. Smith? Again, in the U.S., very few people use Mr. or Ms. at work.
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Comments  
Hi Clive, thanks for the reply.
I know that How are you? in this context is ok.
I don't like the book more and more. To many typos Emotion: sad

Regards
Dominik
 BarbaraPA's reply was promoted to an answer.
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Hi,
Barbara, I agree with you. It looks like, the book is half-baked.
I hope you don't mind calling you Barbara.

Regards,
Dominik
Of course not! Though my preferred form of address is "Hello Barbara. May I buy you a beer?"

Emotion: big smile