Hi Far_,
Do you have the whole book that the example is from? It should say in the text which standard the design check is based on.
The formulas for bending and shear permissible stresses are common. The 0.6 bending factor and 0.37 shear factor are the same as in the Australian Standard AS 3990, and similar in AISC as JedClampett said above (AISC rounded the 0.37 to 0.4). These provide about a 1.5 safety factor on the ultimate strength.
The formula for maximum design bending stress is M/Z, but they've rounded off both M & Z and converted them to units of Newtons (N) and millimetres (mm).
The formula for design shear stress is V/A, but again they've rounded off the numbers and converted the units which makes it a bit hard to follow. I'm not convinced that using the full section area is correct as it's usually just the web area. It would be good if we can work out which standard is being followed to check that use of the full section area is allowed by it.