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In the days of mechanical calculating machines, people used the "40
method" to add and subtract sexagesimal values. For example, 1°59' + 2'
would be added like this:

  159
+  2
----
  161
+ 40
----
  201 = 2°01'

After adding the two angles, the operator notes that the minutes part of
the sum is 61, an illegal value. Therefore, 40 minutes are added to make
the minutes correct and force a carry into the degrees column.

However, simply examining the sum does not always indicate when to add
40. For example, if you add 50' + 50', the machine says the sum is 100
(=  1°00'), which is not, per se, an illegal number.

We could think up some rules on when to add 40, but I think it's easier
to just inspect the summands and use common sense.

The 40 method can be extended to degrees and decimal minutes, or
degrees, minutes, and seconds. And it works for subtraction too. For
example, compute 2°03'04" - 1°23'45". This time I'll use the decimal
point to separate degrees and minutes:

  2.0304
-1.2345
-------
   .7959
- .4040
-------
   .3919 = 39'19"

Personally, I'm happy to use the sexagesimal math features of a
scientific calculator. But the 40 method is not too hard to remember,
and may be nice know someday if you have to work sexagesimals with a
simple calculator.

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