In counting FTE remember that you are counting the labor normally available to the library - positions both temporarily empty and those filled.  So start with the positions the library had on June 30.

Say person A normally works 5 hours a week.

Person B normally works 2 hours a week.

Person C normally works 12 hours a week.

Method I-

You could calculate the FTEs separately and add them together. 

Divide the hours a person normally works a week by 40.

Person A works 5/40 of a full time week - or .125 FTE
Person B works 2/40 of a full time week - or .05 FTE

Person C works 12/40 of a full time week - or .30 FTE

Total is .475 FTE which rounds to .48 FTE

Method II

Add up everyones normal hours in a week. Divide the total by 40.

A + B + C=19/40   or .475 rounding to .48 FTE