If H is the harmonic mean of P and Q, then the value of H/P + H/Q is
2
Understanding the Harmonic Mean
The question asks us to find the value of the expression \(\frac{H}{P} + \frac{H}{Q}\), where H is the harmonic mean of two numbers, P and Q.
Let's first define what the harmonic mean is. For two numbers, P and Q, the harmonic mean H is defined as the reciprocal of the arithmetic mean of the reciprocals of P and Q. Mathematically, this is expressed as:
\(H = \frac{1}{\frac{\frac{1}{P} + \frac{1}{Q}}{2}}\)
We can simplify this formula:
\(H = \frac{2}{\frac{1}{P} + \frac{1}{Q}}\)
To further simplify the denominator, we find a common denominator for \(\frac{1}{P} + \frac{1}{Q}\):
\(\frac{1}{P} + \frac{1}{Q} = \frac{Q}{PQ} + \frac{P}{PQ} = \frac{Q+P}{PQ}\)
Now substitute this back into the formula for H:
\(H = \frac{2}{\frac{P+Q}{PQ}}\)
Inverting the denominator and multiplying gives us the standard formula for the harmonic mean of two numbers:
\(H = \frac{2PQ}{P+Q}\)
Evaluating the Expression H/P + H/Q
We need to find the value of \(\frac{H}{P} + \frac{H}{Q}\). We have the formula for H, so we can substitute it into this expression.
Let's calculate \(\frac{H}{P}\) first:
\(\frac{H}{P} = \frac{\frac{2PQ}{P+Q}}{P}\)
Dividing by P is the same as multiplying by \(\frac{1}{P}\):
\(\frac{H}{P} = \frac{2PQ}{P+Q} \times \frac{1}{P} = \frac{2Q}{P+Q}\)
Next, let's calculate \(\frac{H}{Q}\):
\(\frac{H}{Q} = \frac{\frac{2PQ}{P+Q}}{Q}\)
Dividing by Q is the same as multiplying by \(\frac{1}{Q}\):
\(\frac{H}{Q} = \frac{2PQ}{P+Q} \times \frac{1}{Q} = \frac{2P}{P+Q}\)
Now, we add the two results:
\(\frac{H}{P} + \frac{H}{Q} = \frac{2Q}{P+Q} + \frac{2P}{P+Q}\)
Since the denominators are the same, we can add the numerators:
\(\frac{H}{P} + \frac{H}{Q} = \frac{2Q + 2P}{P+Q}\)
Factor out 2 from the numerator:
\(\frac{H}{P} + \frac{H}{Q} = \frac{2(Q + P)}{P+Q}\)
Since \(Q+P\) is the same as \(P+Q\), we can cancel out the \((P+Q)\) term from the numerator and the denominator (assuming \(P+Q \neq 0\)):
\(\frac{H}{P} + \frac{H}{Q} = 2\)
Conclusion
The value of \(\frac{H}{P} + \frac{H}{Q}\), where H is the harmonic mean of P and Q, is 2.
Revision Table: Harmonic Mean Properties
| Concept | Description | Formula (for two numbers P, Q) |
|---|---|---|
| Harmonic Mean (H) | Reciprocal of the arithmetic mean of reciprocals. | \(H = \frac{2}{\frac{1}{P} + \frac{1}{Q}} = \frac{2PQ}{P+Q}\) |
| Arithmetic Mean (A) | Sum of numbers divided by the count. | \(A = \frac{P+Q}{2}\) |
| Geometric Mean (G) | N-th root of the product of N numbers. | \(G = \sqrt{PQ}\) |
| Relationship (for positive P, Q) | Relationship between the three means. | \(A \ge G \ge H\) |
Additional Information on Harmonic Mean and Means
The harmonic mean is particularly useful when dealing with rates or ratios. For example, if you travel a certain distance at speed P and return the same distance at speed Q, the average speed for the entire round trip is the harmonic mean of P and Q, not the arithmetic mean.
The three common Pythagorean means are the Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM). For any set of positive numbers, the relationship between these means is always \(AM \ge GM \ge HM\). Equality holds only when all the numbers in the set are equal.
In the context of this problem, we demonstrated an interesting property of the harmonic mean related to the sum of the ratios of the mean to each number.
Important Questions from Harmonic Progressions
The nth terms of the two series 3 + 10 + 17 + ... and 63 + 65 + 67 + .... are equal, then the value of n is:
The value of n, for which \(\dfrac{a^{n+1} + b^{n+1}}{a^n+b^n}\) is the harmonic mean of a and b, is
Suppose that m and n are fixed numbers such that the mth term of an HP is equal to n and the nth term is equal to m, (m ≠ n). Then the (m + n)th term is:
If H is the Harmonic Mean of three numbers 10C4, 10C5, and 10C6, then what is the value of \(\frac{270}{H}\) ?
If the roots of the equation a (b - c) x 2+ b (c - a) x + c (a - b) = 0 are equal, then which one of the following is correct?
