Americans use 1/4":1' scale for 0 gauge.
This is the old, original, toymaker's scale.
When this was the case - over 100 years ago now, I would guess - gauge was measured from centre of rail to centre of rail by toymakers, and the rail was made of bar strip, placed on edge, about 1/16" wide by 1/4" high, maybe 3/16". The gauge worked out, therefore, at close to scale - 4'9" if you do the sums. Allowing for manufacturing tolerances (crude!) some bits of track may have been to correct gauge.Typically, "trains" were a live-steam piddler, running up and down straight track with no curves - pay attention this fact is important later on!
At the time, this error in measurement of the gauge was also prevalent in other scales, so what we now call gauge 3 was to 1/2" scale and if using rail that was 1/8" wide, also accurate. Similarly, 5" gauge would have been made to 1" scale, with 1/4" wide bar - and bar is often used still.
In the USA, 5" gauge is rare, it is usually 4.75"/4.8" gauge with a 1:12 scale, but along the way 0 gauge got bastardised. The gauge of 1 1/4" was kept by everyone but now measured to the inside faces to allow for gauge widening to get around curves and through points, and the Americans kept to the more sensible scale of 1:48. Many Europeans (Lima Mk 1s, possibly?) used 1:45 scale, which with 32mm gauge track produces a scaled-up track gauge only 5mm wider than it should be. In this country, some idiot somewhere (no names, no packdrill, but we all know who) decided that using millimetres was more scientific, and we end up with 7mm to the foot, which requires a track gauge of 32.96mm - a full "thou" less than 33mm, which is a lot easier to use and remember. The scale for 0 gauge in this country is therefore 1:43.5428571. In Europe it is 1:45 and as everything is metric over there, it isn't a problem, and in the US there is 1:48 scale, with 0 gauge using 1.25" (32mm) track gauge, but a Proto:48 movement which has whatever 56.5/48 turns out to be, 1.177" will do. Or, if you prefer to be silly, 1 17/96".
In practice, however, there are easier ays of dealing with this.
Firstly, buy or make some accurate track gauges, and use these religiously.
Secondly, buy a scale ruler, and stop messing about with mental arithmetic.
Or, do the first one but chose a more sensible scale, like One:32 or S...
Hope that helps everyone.
Here endeth the history lesson. Go in peace to love and serve the hobby.