The value of $999\frac{1}{7} + 999\frac{2}{7} + 999\frac{3}{7} + 999\frac{4}{7} + 999\frac{5}{7} + 999\frac{6}{7}$ is equal to :

Basic learning:

$1 + 0.5 = 1.5$

This can be written as a fraction:

$1.5 = 1 \frac{1}{2}$

This can also be written as:

$1 \frac{1}{2} = 1 + \frac{1}{2}$

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Hence, question states:

$999\frac{1}{7} + 999\frac{2}{7} + 999\frac{3}{7} + 999\frac{4}{7} + 999\frac{5}{7} + 999\frac{6}{7}$

This can be written as:

$999 + \frac{1}{7} + 999 + \frac{2}{7} + 999 + \frac{3}{7} + 999 + \frac{4}{7} + 999 + \frac{5}{7} + 999 + \frac{6}{7}$

We can add the fractions separately. Also, $999$ appears six times:

$\Rightarrow 6 \times 999 + \left( \frac{1}{7} + \frac{2}{7} + \frac{3}{7} + \frac{4}{7} + \frac{5}{7} + \frac{6}{7} \right)$

$\Rightarrow 5994 + \left( \frac{1}{7} + \frac{2}{7} + \frac{3}{7} + \frac{4}{7} + \frac{5}{7} + \frac{6}{7} \right)$

$\Rightarrow 5994 + \frac{21}{7}$

$\Rightarrow 5994 + 3 = \boxed{5997}$