Asimple circuit consists of a 100.-ohm resistor
connected to a battery. a 25-ohm resistor is to
be connected in the circuit. determine the
smallest equivalent resistance possible when
both resistors are connected to the battery.

If you want to make the total equivalent resistance as small as possible,
then you should connect the new resistor in parallel with the original one. 

When resistors are in parallel, then no matter how many of them there are
or what their individual values are, the total equivalent resistance is always
less than the smallest individual one.

For resistors in parallel .  . .

         (1 / total resistance) = (1/R₁) + (1/R₂) + (1/R₃) + . . . however many there are.

When there are only two resistors in parallel, you can take that formula, stretch it,
massage it, exercise it, manipulate it, and you find that

     Total equivalent resistance = (product of the resistors) / (sum of the resistors) .

With 100 ohms and 25 ohms in parallel,

     Total equivalent resistance = (100 x 25) / (100 + 25) = 2,500/125 = 20 ohms.