Calculating R with 2 resistors parallel


I've ran into a slight confusion.

The equation for resistance in a parallel circuit should work for any number of resistors, I assumed.
So when I'd have 2 resistors in parallel, total Rt would be:
Rt = 1/G
G=1/Ra + 1/Rb

Now my book says, for 2 resistors in parallel, you may use this formula too:
Rt = (Ra * Rb) / (Ra + Rb)

When I fill in 100 Ohms for Ra, and 220 Ohms for Rb, the two equations produce different results. (68.75 for the first, and 66.67 for the second).
Now, I have tried it with several sets of values, but I couldn't reproduce this difference with those.

What is happening!? I'm lost!

Edit: Yeah, I've been miscalculating indeed. How embarrassing to find out this way. Thanks for all your answers 🙂

Best Answer

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:

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