There are 16 1/16s of an inch in one inch. Dividing one by 1/16 gives the answer 16. Dividing fractions requires multiplying the divisor by the reciprocal of the dividend, so dividing one by 1/16 is the same as multiplying one by 16.
One of the "Golden Rules of Fractions" is that "any number over itself is equal to one," according to about.com. Applying that statement to the original problem, 16/16 is equal to one. There are also three 1/3s of an inch in one inch, four 1/4s and the trend continues. One-sixteenth of an inch is a very small measurement that is 1/4 of a quarter of an inch.