If H is the harmonic mean between p and q, then the value of (H/p) + (H/q) is: (A) 2 (B) (pq) / (p + q) (C) (p + q) / (pq) (D) None of these

Let's first identify the harmonic mean $H$ between $p$ and $q$. The formula for the harmonic mean is given by:

$$H = \frac{2pq}{p + q}$$

To find the value of $\frac{H}{p} + \frac{H}{q}$, we will substitute the expression for $H$ into the equation:

$$\frac{H}{p} + \frac{H}{q} = \frac{\frac{2pq}{p+q}}{p} + \frac{\frac{2pq}{p+q}}{q}$$

Simplify each term individually:

$$\frac{\frac{2pq}{p+q}}{p} = \frac{2pq}{p(p+q)} = \frac{2q}{p+q}$$

$$\frac{\frac{2pq}{p+q}}{q} = \frac{2pq}{q(p+q)} = \frac{2p}{p+q}$$

Now, combine the simplified terms:

$$\frac{H}{p} + \frac{H}{q} = \frac{2q}{p+q} + \frac{2p}{p+q}$$

Since the denominators are the same, we can add the numerators:

$$\frac{2q + 2p}{p+q} = \frac{2(q + p)}{p+q}$$

When we simplify this fraction, the $p+q$ terms cancel out:

$$\frac{2(p + q)}{p+q} = 2$$

Therefore, the value of $\frac{H}{p} + \frac{H}{q}$ is 2.

Correct answer: (A) 2