It is part of British 'understatement', that aside though

[Learning_English]

I know them quite well.

I know them well.

What is th difference in the meaning with the above sentences?
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It depends, often in BE there is no difference. It is part of British "understatement", that aside though "quite well" can also mean not to know the person as well as one could.

Could anyone rephrase the highlighted sentence so that I can understand that sentence?
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Please let me know if I have made any mistakes in any of my sentences.

[Conchita]

Here is my suggestion:

It is an example of the British way of making things seem less important than they really are. Apart from that, however, 'quite well' can also mean...

Is it a bit clearer now?

[Learning_English]

[Conchita]

[Diverhank]

May be the writer was thinking of " I don't know him quite that well" ? Maybe it's a British thing, to me, "quite well" means "very well", there's no other meaning that I can think of.

[Mister Micawber]

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I cannot really follow this thread, but I'll chime in with the information that 'quite well' in BrE can mean either (1) 'perfectly' or (2) 'reasonably well'.

From the online Oxford:

adverb 1 to the utmost or most absolute extent or degree; completely. 2 to a certain extent; moderately. 3 US very; really.

(Notice the more limited definition in AmE.)
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Native English teacher at Mister Micawber's