# If H is the harmonic mean between P and Q then the values of (H/P)+(HQ) is equal to ?

##### 1 Answer
Sep 27, 2017

$\frac{H}{P} + H Q = 2 P$

#### Explanation:

In general, the harmonic mean is given by:

${\mu}_{H} = \frac{n}{{\sum}_{r = 1}^{n} \frac{1}{x} _ \left(i\right)}$

So for the two numbers $P$ and $Q$ we have:

$H = \frac{2}{\frac{1}{P} + \frac{1}{Q}}$

And if we use a common denominator, we can simplify:

$H = \frac{2}{\frac{Q + P}{P Q}} = \frac{2 P Q}{P + Q}$ ..... [A]

So the we seek:

$\frac{H}{P} + H Q = H \left(\frac{1}{P} + Q\right)$
$\text{ } = H \left(\frac{Q + P}{Q}\right)$
$\text{ } = \left(\frac{2 P Q}{P + Q}\right) \left(\frac{Q + P}{Q}\right) \setminus \setminus \setminus \setminus$ using [A]
$\text{ } = 2 P$