H.M. between `(1)/(28) and (1)/(10)` is

Options

A. `(1)/(19)`
B. `(1)/(18)`
C. `(1)/(17)`
D. `(1)/(16)`

Answer

To find the Harmonic Mean (H.M.) between the numbers \( \frac{1}{28} \) and \( \frac{1}{10} \), we can use the formula for the Harmonic Mean of two numbers \( A \) and \( B \):

\[ \text{H.M.} = \frac{2AB}{A + B} \]

### Step 1: Identify the values of A and B
Let:
- \( A = \frac{1}{28} \)
- \( B = \frac{1}{10} \)

### Step 2: Calculate the product \( AB \)
\[ AB = \left(\frac{1}{28}\right) \times \left(\frac{1}{10}\right) = \frac{1}{280} \]

### Step 3: Calculate the sum \( A + B \)
To add \( A \) and \( B \), we need a common denominator. The least common multiple (LCM) of 28 and 10 is 140.

Convert \( A \) and \( B \) to have the same denominator:
\[ A = \frac{1}{28} = \frac{5}{140}, \quad B = \frac{1}{10} = \frac{14}{140} \]
Now, add them:
\[ A + B = \frac{5}{140} + \frac{14}{140} = \frac{19}{140} \]

### Step 4: Substitute \( AB \) and \( A + B \) into the H.M. formula
\[ \text{H.M.} = \frac{2 \times \frac{1}{280}}{\frac{19}{140}} \]

### Step 5: Simplify the expression
First, calculate \( 2 \times \frac{1}{280} \):
\[ 2 \times \frac{1}{280} = \frac{2}{280} = \frac{1}{140} \]

Now, substitute this back into the H.M. formula:
\[ \text{H.M.} = \frac{\frac{1}{140}}{\frac{19}{140}} = \frac{1}{140} \times \frac{140}{19} = \frac{1}{19} \]

### Final Answer
The Harmonic Mean between \( \frac{1}{28} \) and \( \frac{1}{10} \) is:
\[ \text{H.M.} = \frac{1}{19} \]

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