If 1/4, 1/x and 1/10 are in HP, then what is the value of x ?

Options

A. 5
B. 6
C. 7
D. 8

Answer

To solve the problem, we need to find the value of \( x \) given that \( \frac{1}{4}, \frac{1}{x}, \frac{1}{10} \) are in Harmonic Progression (HP).

### Step-by-Step Solution:

1. **Understanding Harmonic Progression (HP)**:
- If three numbers \( a, b, c \) are in HP, then their reciprocals \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \) are in Arithmetic Progression (AP).
- In our case, \( a = 4 \), \( b = x \), and \( c = 10 \).

2. **Setting up the AP Condition**:
- Since \( 4, x, 10 \) are in AP, we can use the property of AP:
\[ 2x = 4 + 10 \]

3. **Simplifying the Equation**:
- Calculate the right side:
\[ 2x = 14 \]

4. **Solving for \( x \)**:
- Divide both sides by 2:
\[ x = \frac{14}{2} = 7 \]

5. **Conclusion**:
- The value of \( x \) is \( 7 \).

### Final Answer:
\[ x = 7 \]