The harmonic mean of 4,8,16 is

Options

A. 6.4
B. 6.7
C. 6.85
D. 7.8

Explanation

To find the harmonic mean of the numbers 4, 8, and 16, we can follow these steps:

### Step 1: Identify the formula for harmonic mean
The formula for the harmonic mean (HM) of n numbers \( A_1, A_2, A_3, \ldots, A_n \) is given by:

\[ HM = \frac{n}{\frac{1}{A_1} + \frac{1}{A_2} + \frac{1}{A_3} + \ldots + \frac{1}{A_n}} \]

### Step 2: Determine the number of terms
In this case, we have three numbers: 4, 8, and 16. Therefore, \( n = 3 \).

### Step 3: Substitute the values into the formula
Now, we can substitute the values into the harmonic mean formula:

\[ HM = \frac{3}{\frac{1}{4} + \frac{1}{8} + \frac{1}{16}} \]

### Step 4: Calculate the individual fractions
Next, we need to calculate \( \frac{1}{4} \), \( \frac{1}{8} \), and \( \frac{1}{16} \):

- \( \frac{1}{4} = 0.25 \)
- \( \frac{1}{8} = 0.125 \)
- \( \frac{1}{16} = 0.0625 \)

### Step 5: Find a common denominator and sum the fractions
To add these fractions, we can find a common denominator. The least common multiple (LCM) of 4, 8, and 16 is 16. We convert each fraction:

\[ \frac{1}{4} = \frac{4}{16}, \quad \frac{1}{8} = \frac{2}{16}, \quad \frac{1}{16} = \frac{1}{16} \]

Now, we can add them together:

\[ \frac{1}{4} + \frac{1}{8} + \frac{1}{16} = \frac{4}{16} + \frac{2}{16} + \frac{1}{16} = \frac{4 + 2 + 1}{16} = \frac{7}{16} \]

### Step 6: Substitute back into the harmonic mean formula
Now we substitute this sum back into the harmonic mean formula:

\[ HM = \frac{3}{\frac{7}{16}} = 3 \times \frac{16}{7} = \frac{48}{7} \]

### Step 7: Calculate the final result
To find the decimal value of the harmonic mean:

\[ HM = \frac{48}{7} \approx 6.8571 \]

Thus, the harmonic mean of 4, 8, and 16 is approximately 6.8571.

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