Is the reciprocal of a fraction always a whole number

That is false.

To get the reciprocal of a number, we divide 1 by the number.

We can think of a whole number as being "number/1", so the reciprocal is just like "flipping it over"

For Fractions, Flip the Whole Fraction Over

Example: the reciprocal of 3/4 is 4 / 3.

No it is not. For example the reciprocal of 2/3 is 3/2 - 1.5 is not an integer since it is 1 1/2