Why can you use fraction multiplication to check the answer to a fraction division problem? How do you use multiplication to check a division problem?

Fraction division is basically multiplication of a reciprocal.

For example (1/4) / (1/8) = (1/4) x (8/1) = 2

Multiplication of the answer and the piece that divides (in this case 2 x 1/8), you should get the piece that is being divided (in this case 1/4). If that is correct, then your division is correct.