If the combined harmonic mean of 18 observations is 3 and harmonic mean of first 10 observation is 27 find the harmonic mean of remaining eight observation?

Question

Answer

**Solution:**

Given,
- Combined harmonic mean of 18 observations = 3
- Harmonic mean of first 10 observations = 27

Let the sum of all 18 observations be S, then
- Combined harmonic mean = 18/S
- Harmonic mean of first 10 observations = 10/x, where x is the sum of first 10 observations.
- Harmonic mean of remaining 8 observations = 8/y, where y is the sum of remaining 8 observations.

**Finding the value of S:**

- Combined harmonic mean = 3
- So, 18/S = 3
- Therefore, S = 18/3 = 6

**Finding the value of x:**

- Harmonic mean of first 10 observations = 27
- So, 10/x = 27
- Therefore, x = 10/27

**Finding the value of y:**

- We know that sum of all 18 observations is S = 6
- Sum of first 10 observations is x = 10/27
- So, the sum of remaining 8 observations = S - x = 6 - 10/27 = 152/27

**Finding the harmonic mean of remaining 8 observations:**

- Harmonic mean of remaining 8 observations = 8/y
- Substituting the value of y in the above equation, we get:
- Harmonic mean of remaining 8 observations = 8/(152/27)
- Simplifying, we get:
- Harmonic mean of remaining 8 observations = 54/19

Therefore, the harmonic mean of remaining 8 observations is 54/19.

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