Is there a simple method for determining whether a metal object contains precious metal?

There is no simple test and percentage is even harder to do. Your best bet is to take them to a real jeweler (not at a flea market) and have them evaluated. Beware test like that shown in the reference since they harm the item to do the test. A jeweler can do it with less harm.

Do what Archimedes did. If I'm not mistaken, precious metals are heavier than the common ones. Find the object's density by dividing its weight by the volume of water it displaces. If the density of the object is lower than it should be for the metal it's made of, then it must contain lighter (aka cheaper) metals.

[Edit] now that you've mentioned silver, the density of your objects should be the same or very close to the density of silver. See the links below.

Check its density which should be around 10.5 g/cm^3.

If it is an alloy of silver and copper it will be much less than this - around 9.