Gauge is the distance between the rails. Standard gauge on the real railroads is 56-1/2 inches, derived from the wheel spacing on an ancient Roman chariot. Gauges wider than this standard are referred to as broad gauge, and narrow gauge is rails spaced closer than standard. Common sizes for narrow gauge are 2 feet, 3 feet, and meter.
Scale is defined as the proportion or ratio of the real train (often referred to as prototype) to the model.
Four and three-quarter inch to the foot is 1/12 scale. The prototype is 12 times bigger than the model. 56-1/2 inches divided by 12 equals 4.7083 inches.
One and one half inch to the foot scale (1:8) is often used to model three foot narrow gauge trains to run on 4-3/4 inch gauge tracks (36 inches divided by 8 = 4.5 inches - close enough).
The one inch scale track started as a dog-bone shaped loop of about 1500 feet, with a few grades, for one-inch to the foot trains. The distance between the rails (gauge) is 4-3/4 inches. We recently expanded this layout with an additional 900 feet of track that crosses over itself on a 20 foot long through truss bridge. This size of equipment is popular because it is of a manageable size for one person, but powerful enough to haul several people.
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