A uniform heating wire of resistance 36 `ohms` is connected across a potential difference of 240 V. The wire is then cut into half and a potential difference of 240 V is applied across each half separately. The ratio of power dissipation in first case to the total power dissipation in the second case would be 1 : x, where x is _________ .

Answer

To solve the problem, we need to find the ratio of power dissipation when a uniform heating wire of resistance 36 ohms is connected across a potential difference of 240 V, and then when the wire is cut in half and the same potential difference is applied across each half.

### Step-by-Step Solution:

1. **Calculate Power Dissipation in the First Case (P1)**:
- The resistance of the wire is \( R = 36 \, \Omega \).
- The potential difference applied is \( V = 240 \, V \).
- The power dissipated in the wire can be calculated using the formula:
\[ P_1 = \frac{V^2}{R} \]
- Substituting the values:
\[ P_1 = \frac{240^2}{36} \]

2. **Calculate Power Dissipation in the Second Case (P2)**:
- When the wire is cut in half, each half has a resistance of:
\[ R_{\text{half}} = \frac{36}{2} = 18 \, \Omega \]
- When a potential difference of 240 V is applied across each half, we need to calculate the power for each half.
- The power dissipated in each half (let's call it \( P_{\text{half}} \)):
\[ P_{\text{half}} = \frac{V^2}{R_{\text{half}}} = \frac{240^2}{18} \]
- Since there are two halves, the total power dissipation in the second case is:
\[ P_2 = 2 \times P_{\text{half}} = 2 \times \frac{240^2}{18} \]

3. **Calculate the Ratio \( \frac{P_1}{P_2} \)**:
- Substitute the expressions for \( P_1 \) and \( P_2 \):
\[ \frac{P_1}{P_2} = \frac{\frac{240^2}{36}}{2 \times \frac{240^2}{18}} \]
- Simplifying this expression:
\[ \frac{P_1}{P_2} = \frac{240^2}{36} \times \frac{18}{2 \times 240^2} = \frac{18}{72} = \frac{1}{4} \]

4. **Determine the Value of \( x \)**:
- The problem states that the ratio of power dissipation in the first case to the total power dissipation in the second case is given as \( \frac{1}{x} \).
- From our calculations, we found:
\[ \frac{P_1}{P_2} = \frac{1}{4} \]
- Therefore, we can equate:
\[ \frac{1}{x} = \frac{1}{4} \implies x = 4 \]

### Final Answer:
Thus, the value of \( x \) is **4**.