To solve this problem, we need to find a common denominator for all the fractions. The least common multiple (LCM) of 2, 8, 28, 77, 176, 352, and 638 is 1232.

Now, we can rewrite each fraction with the common denominator of 1232:

1/2 = 616/1232
2/8 = 308/1232
3/28 = 132/1232
4/77 = 64/1232
5/176 = 35/1232
6/352 = 18/1232
7/638 = 7/1232

Now, we can add all the fractions together:

616/1232 + 308/1232 + 132/1232 + 64/1232 + 35/1232 + 18/1232 + 7/1232

To add fractions, we need to have the same denominator. Since all the fractions have the same denominator of 1232, we can add the numerators together:

(616 + 308 + 132 + 64 + 35 + 18 + 7)/1232

Simplifying the numerator:

(1180)/1232

Now, we can simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 4:

(295)/308

Therefore, the sum of the given fractions is 295/308.