One inch is made up of 16 1/16ths of an inch. If you divide one by 1/16, you get the number 16. When dividing a fraction, the divisor must be multiplied by the dividend’s reciprocal, so dividing by 1/16 is equivalent to multiplying by 16.
According to about.com, one of the “Golden Rules of Fractions” is that “any integer over itself is equal to one.” That means that the solution to the original issue is 16/16, which equals 1.
Additionally, there are three thirds of an inch in one inch, four quarters, and the pattern continues. A sixteenth of an inch, or 1/4 of a quarter of an inch, is an extremely tiny measurement.