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In the era of mechanical calculating machines, the "rule of 40" was a
trick to force a base 10 machine to add or subtract in base 60.

For instance, to calculate 1°40′ + 2°30′, add 140 and 230. The result,
370, is clearly wrong since minutes are not in the legal range. You fix
that by adding 40, which rolls the minutes over, forces a carry into
degrees, and shows the correct sum: 410 (4°10′).

Inspection of the sum alone doesn't always indicate when the result
needs adjustment, however. For instance, if you add 50.5′ + 55.5′ the
calculator says the result is 1060 (1°06.0′). The minutes are in the
legal range but the result is wrong! You must add 400 to obtain the
correct sum: 1460 (1°46.0′). This situation occurs whenever the sum of
the minutes is 100 or greater, and thus is easy to detect by a glance at
the operands.

That last example shows the rule of 40 is valid with degrees and decimal
minutes, if you take care to align the 40 with the proper columns.

The rule of 40 is applicable to subtraction too. You subtract 40 to
force a borrow into that column.

It may help avoid confusion if you use the decimal point to separate
degrees and minutes. That's what I do, so the rule of 40 becomes the
rule of .4.

   

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