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If you follow garage sales, tag sales and the like, you'll find that the cellars and garages of America are full of old, cheap tandem bikes, which can often be purchased for next to nothing.
These are not serious tandems, they're built for very casual riding on flat terrain. Typically they will have one-piece cranks, skinny frames and cheap parts. Most of them will be one-speeds, though a few of them will have two-speed or 5-speed gearing.
Most folks in the industry sneer at these relics, but if you're handy, they can be fun projects.
These bikes are generally built to the old American standards, so parts interchangeability is sometimes a challenge. If you have one of these, you should check out both of these links:
The idea is that the coaster brake is the main brake, with the front caliper as a backup. This is generally a bad theory (See my Article on Braking and Turning.)
For a tandem to rely on a coaster brake is especially bad for a couple of reasons:
They typically use the obsolescent American 21.15 mm stem diameter, and they generally have pretty crummy handlebar stems. If you want to upgrade or just to get a stem that provides a more comfortable riding position, it will be hard to find anything that will fit the narrow steering column.
Older mountain bikes generally had very sturdy, stiff forks. Just swapping out the fork will usually make a major improvement in the handling of one of these old OPC tandems.
In addition, you'll get a nice powerful cantilever front brake instead of the pitiful long-reach caliper from the original bike.
You can also use the handlebars and stem from the mountain bike, or you can substitute a new stem if you want a different handlebar position. (Note: some '80s mountain bikes also used the skinny 21.15 stem size, instead of the more normal 22.2 mm (7/8") size, so check that first.)
There are differences in the headset parts of the OPC standard stuff vs more common ISO or J.I.S. sizes used on mountain bikes, so you will most likely wind up using most of the original headset parts, or else buying a "BMX" size headset.
See my Headset Cribsheet for relevant dimensions.
One of the nice things to do with these old bikes is to adapt them for a child stoker. Typically, they have a "mixte" style rear end, where there are three pairs of rear stays. if you pick up your hack saw and chop off the upper set of stays and cut the seat tube a couple of inches above the middle stays, you can adapt the bike to fit a much shorter stoker.
The handling deficiencies of these frames mostly disappear when you have a small child in back, especially low down. This is often a quite worthwhile modification.