Actually, let me make a modification to Brian's explanation. In the US, while we do use ratio scales, like 1:64 or 1:48, we will also write them as 1/64 or 1/48.
The other scale system is a common architectural scale and it is properly written out as, say, 3/16"=1'. It is often abbreviated as 3/16-inch scale or 3/16" scale. Some people incorrectly leave off the "inch" units and that causes a great deal of confusion, so it should always be included. So, in this example, it's not 3/16 scale, it's 3/16" scale.
Clare