The twelve signs
If a result is off by exactly 30°, a sign's starting degree is wrong. If it is off by 360°, the wrap was forgotten.
Why sixty
The Babylonians counted in sixties, and every astronomer since has inherited it: a circle of 360°, a degree of 60 minutes, a minute of 60 seconds. Sixty divides evenly by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30, so fractions of a circle rarely leave a remainder. The zījes of Baghdad were written in this notation, in letters of the alphabet rather than numerals, and a chart drawn by hand today still is.
You will not often add longitudes on a printed chart. But Lots are distances between two points, a table of houses is entered by sidereal time, and a longitude becomes a time correction only by dividing it by fifteen — and then the calculator comes out.