Gearing with 3 sprockets question

So I understand gearing with 2 sprockets just fine but my latest project will require a jackshaft between the bike engine and the car transmission input shaft I'm powering.

 I need the large sprocket to fit inside the bellhousing but it isn't as big as I think I'd need to lower the gearing enough.

 I need to get the gearing way short,I can drop 2 possibly 3 teeth on the counter shaft sprocket off the bike engine,it in turn drives a jackshaft which will then turn the large sprocket inside the bellhousing.

The question I have is the effect on gearing by playing with the middle sprocket.

 In my head increasing that sprocket diameter relative to the countershaft sprocket will reduce the gearing further then just the same size sprockets in all locations other then the large one inside the bellhousing.

 Hopefully thats clear enough despite my rambling.☺

 Thanks in advance.

Yes, as long as you only increase the second sprocket as a jackshaft will have 4 not 3. If you change both middle sprockets it offsets the change on either end

Think of it as if you are computing the ratio between the engine and rear tires on a vehicle.  In first gear, you have a 3:1 ratio, in the rear axle, you have a 4:1 ratio, so the total gear reduction would be 12:1 in first gear.  Between the bike engine and the jackshaft is the trans, jackshaft to input would be the final drive.

Edit:Now, you have the trans and diff ratios to add into the math, as well.

I hope I read your question correctly.

Edit again:  The jackshaft sprockets can't change the rpms of the jackshaft, but you can fudge the numbers.  If you really want to slow the rpm down, 10 teeth on the engine to 30 teeth on the jack is three to one, but then you can have the other sprocket on the jack be 10 teeth again, driving a 30 tooth sprocket for another 3:1, giving you 9:1 overall. The tooth ratio will be the same, but you can run a smaller sprocket on the trans by changing tooth count on the jack.

Seems simple?

You may also be able to understand this by thinking of it as an "inches per revolution" problem. If you use a big sprocket on the input of the jackshaft, and a sprocket 1/3 the size on the output, then it takes 3 inches of  chain movement in to get one inch out, with a proportional force  increase across the shaft. 

I only need to be able to hit 75mph in 6th gear with the bike trans.

 I will need to leave the car trans in probably 3rd to avoid overspinning the bearings etc.Its an old Justy awd trans,final drive is short and I doubt the trans is particularly robust.

 Standing start races(on ice) so I need to be able to pull away without losing positions to the usual hoard of subarus.

 So I could go from say a 14 tooth countershaft sprocket to say an 18 tooth on one end of the jackshaft to say a 24 tooth on the other end before driving the sprocket on the trans that will be somewhere in the high 40's and each dia increase results in a reduction?.

 Having 2 final drives,2 transmissions and 11 gears makes my head hurt.

 Damn rules.☺ 

If you need to visualize your options, buy a Spirograph, drill centered holes, pin some gears to a board and see what happens.

As you consider the changes, it will be better  if you can go to a larger sprocket on the driven, rather than changing to a smaller sprocket on the drive. 

Larger sprockets create less vibration since the chain moves less distance vertically during rotation.  

Have you looked at snowmobiles for the jack shaft and drive chain assembly?

Ski doo and Polaris offered a reverse gear option that was mechanically actuated in the chain case, and could handle 100+ HP,  high speeds, and the shock loading of the primary belt drive and centrifugal clutches.  

They were oil bath chains and bearings in the case, so if you go that route you will want a part sled so you can harvest any parts needed to make it work. 

In reply to No Time :

Thanks,I'm limited in size for the driven sprocket and for sure its not enough to get where I need to be so no choice go small on the drive sprocket.

Race only and the season miles total maybe 150,changing that chain as routine mauntaince isn't a big deal.

 No space for a sled chain case plus I can use the reverse in the car trans.