The mean of 100 numerical observation is 51.18 what is the value of the sum of all 100 numbers

The "mean" is the same thing as the "average".

To get the average, you add up all the numbers on the list,
then divide the big sum by the number of items on the list.

If there are 100 numbers on the list, then in order to find their average,
you would add up all 100 numbers, then divide the big sum by 100.

If you wound up with an average of 51.18, then you must have had
5,118 just before you divided it by 100.  So the big sum is 5,118.