Calculating R with 2 resistors parallel
What is happening!?


You made a calculation error.  

$$\frac{100 \cdot 220}{100 + 220} = \frac{1}{\frac{1}{100} + \frac{1}{220}} = 68.75$$

It's easy to show the equivalence of the two formula:

$$\frac{1}{\frac{1}{R_1} + \frac{1}{R_2}} = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2}}\frac{R_1\cdot R_2}{R_1\cdot R_2}=\frac{R_1\cdot R_2}{R_1 + R_2}$$