Question

If $H$ is the harmonic mean between $p$ and $q$, then the value of $\frac{H}{p} + \frac{H}{q}$ is:

(A) 2 (B) $\frac{pq}{p+q}$ (C) $\frac{p+q}{pq}$ (D) None of these

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Answer

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


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