Two celestial bodies are separated by some distance. If the mass of any one of the bodies is doubled while the mass of other is halved then how far should they be taken so that the gravitational force between them becomes one-fourth ?

Explanation

To solve the problem, we will use the universal law of gravitation and analyze how changes in mass and distance affect the gravitational force between two celestial bodies.

### Step-by-Step Solution:

1. **Understanding the Gravitational Force**:
The gravitational force \( F \) between two masses \( M \) and \( m \) separated by a distance \( d \) is given by the formula:
\[ F = \frac{G \cdot M \cdot m}{d^2} \]
where \( G \) is the gravitational constant.

2. **Initial Conditions**:
Let the masses of the two celestial bodies be \( M \) and \( m \), and the initial distance between them be \( d \). The initial gravitational force is:
\[ F = \frac{G \cdot M \cdot m}{d^2} \]

3. **Changing the Masses**:
According to the problem, the mass of one body is doubled (new mass = \( 2M \)) and the mass of the other body is halved (new mass = \( \frac{m}{2} \)).

4. **New Gravitational Force**:
The new gravitational force \( F' \) when the masses are changed and the distance is \( r \) becomes:
\[ F' = \frac{G \cdot (2M) \cdot \left(\frac{m}{2}\right)}{r^2} \]
Simplifying this, we get:
\[ F' = \frac{G \cdot 2M \cdot \frac{m}{2}}{r^2} = \frac{G \cdot M \cdot m}{r^2} \]

5. **Setting the Condition for the New Force**:
We want the new gravitational force \( F' \) to be one-fourth of the original force \( F \):
\[ F' = \frac{F}{4} \]
Substituting the expressions for \( F \) and \( F' \):
\[ \frac{G \cdot M \cdot m}{r^2} = \frac{1}{4} \cdot \frac{G \cdot M \cdot m}{d^2} \]

6. **Canceling Common Terms**:
Since \( G \), \( M \), and \( m \) are common in both sides, we can cancel them out:
\[ \frac{1}{r^2} = \frac{1}{4d^2} \]

7. **Cross-Multiplying**:
Cross-multiplying gives us:
\[ 4d^2 = r^2 \]

8. **Taking the Square Root**:
Taking the square root of both sides, we find:
\[ r = 2d \]

### Conclusion:
The new distance \( r \) should be twice the initial distance \( d \). Therefore, the two celestial bodies should be taken to a distance of \( 2d \) for the gravitational force between them to become one-fourth of the original force.