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

Options

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

Explanation

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

### Step-by-step Solution:

1. **Understanding Harmonic Progression**:
If three numbers are in HP, their reciprocals are in Arithmetic Progression (AP). Therefore, we can consider the reciprocals of the given numbers:
\[ 4, x, 10 \]

2. **Setting up the AP Condition**:
For three numbers \( a, b, c \) to be in AP, the condition is:
\[ 2b = a + c \]
In our case, let \( a = 4 \), \( b = x \), and \( c = 10 \). Thus, we can write:
\[ 2x = 4 + 10 \]

3. **Calculating the Right Side**:
Simplifying the right side:
\[ 2x = 14 \]

4. **Solving for \( x \)**:
Now, divide both sides by 2 to find \( x \):
\[ x = \frac{14}{2} = 7 \]

5. **Final Answer**:
Therefore, the value of \( x \) is:
\[ \boxed{7} \]