In an experiment with a post office box, the radio of arms are `1000 : 10`. If the value of the third resistance is `999 Omega`, find the unknows resistance.

Explanation

To solve the problem of finding the unknown resistance in a post office box experiment, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Ratio**: We are given the ratio of the arms as \( P : Q = 1000 : 10 \). This can be simplified to \( P : Q = 100 : 1 \).

2. **Identify the Known Values**: We know that the value of the third resistance \( R = 999 \, \Omega \).

3. **Set Up the Proportionality**: According to the principles of the Wheatstone bridge (which is what the post office box is based on), we have the relationship:
\[ \frac{P}{Q} = \frac{R}{X} \]
where \( X \) is the unknown resistance.

4. **Substitute the Known Values**: We can substitute \( P/Q = 100 \) and \( R = 999 \, \Omega \) into the equation:
\[ 100 = \frac{999}{X} \]

5. **Rearrange the Equation**: To find \( X \), we can rearrange the equation:
\[ X = \frac{999}{100} \]

6. **Calculate the Unknown Resistance**: Now, perform the division:
\[ X = 9.99 \, \Omega \]

### Final Answer:
The unknown resistance \( X \) is \( 9.99 \, \Omega \).

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