The simplest form of an intermdiate gear is simply a single gear (the intermediate gear) connecting two other gears like shown in the example diagram below:

In this case the intermediate gear is the 8 tooth gear connecting the 28 tooth gear and 24 tooth gear. This is really a simpler case of a gear train where the intermediate gear acts as the gear being driven and driving the next gear in the sequence. To calculate the overall gearing ratio it is simply the product of the gearing ratios in the gear train. In this case we have the 28 tooth gear driving the 8 tooth and then the 8 tooth driving the 24 tooth gear:

Overal Gearing Ratio = (28 / 8)  x (8 / 24) = 28 / 24  = 7 / 6

Note that since the intermediate gear is used in the denominator in the first gearing ratio and the numerator in the 2nd gearing ratio it always cancels out and does not affect the overall gearing ratio. This means that the size of the intermediate gear is irrelevant to the overall gearing ratio.

In general you can have more than one intermediate gear as for example shown in the next diagram:

Here there are two intermediate gears (the 24 tooth and 40 tooth gears on the yellow axles) between the 12 tooth gear on the left and the 20 tooth gear on the right. Again the overall gearing ratio is not affected by the intermediate gears and is simply the ratio of the first gear to the last gear in the sequence. In this case:

12 / 20 = 3 / 5 = 0.6