Mean balance

If 

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 is less than 6, then adding another 

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 to the set would make the mean smaller.

If 

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 is more than 6, then adding another 

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 to the set would make the mean larger.

But adding another 

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 to the set does not change the mean, so 

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 must be exactly 6.

Alternatively:

If the mean of the seven numbers is 6, they must add up to 42.

If the mean of eight numbers is 6, they must add up to 48.

Therefore the eighth number is 48 - 42 = 6

So the mean of the set of numbers 8, 4, 5, 4, 

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, 6, 6 is 6.

That means that ( 8 + 4 + 5 + 4 + 

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 + 6 + 6 ) $\div$ 7 = 6,

so 8 + 4 + 5 + 4 + 

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 + 6 + 6 = 42,

so 33 + 

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 = 42, so 

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 = 9.

So 

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 = 9 and 

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 = 6.