How can I do time/hours arithmetic in Google Spreadsheet?

Google Sheets now have a duration formatting option. Select: Format -> Number -> Duration.

When the number being returned by your formula is being formatted as a time, and you want it formatted as a plain number, change the format of the cell to a plain number format: click the cell and then click Format, Number, Normal.

Time values in Google spreadsheet are represented as days and parts of days. For example, 36:00:00 is the formatted representation of the number 1.5 (a day and a half).

Suppose you divide 36:00:00 by 3:00:00, as in your example. Google Spreadsheet performs the calculation 1.5 divided by 0.125, which is 12. The result tells you that you have 12 3-hour intervals in a 36-hour time period. 12, of course, is not a time interval. It is a unitless quantity.

Going the other way, it is possible to format any number as a time. If you format 12 as a time, it's reasonable to expect that you will get 288:00:00. 12 days contain 288 hours.