If a body floats with `(m//n)^(th)` of its volume above the surface of water, then the relative density of the material of the body is

Options

A. `((n-m))/(n)`
B. `(m)/(n)`
C. `(n)/(m)`
D. `((n-m))/(n)`

Explanation

To solve the problem, we need to find the relative density of a body that floats with a fraction of its volume above the surface of water. Let's denote the following:

- Let \( V \) be the total volume of the body.
- The volume of the body above the surface of water is given as \( \frac{m}{n} V \).
- Therefore, the volume of the body submerged in water is \( V_{sub} = V - \frac{m}{n} V = V \left(1 - \frac{m}{n}\right) \).

### Step-by-Step Solution:

1. **Identify the Volume Submerged**:
\[ V_{sub} = V \left(1 - \frac{m}{n}\right) \]

2. **Apply Archimedes' Principle**:
According to Archimedes' principle, the weight of the water displaced by the submerged part of the body is equal to the weight of the body itself.
\[ \text{Weight of the body} = \text{Weight of the water displaced} \]
The weight of the body can be expressed as:
\[ W_{body} = \text{Density of the material} \times V \times g \]
The weight of the water displaced is:
\[ W_{displaced} = \text{Density of water} \times V_{sub} \times g \]

3. **Substitute the Volume Submerged**:
Substitute \( V_{sub} \) into the equation for the weight of the water displaced:
\[ W_{displaced} = \text{Density of water} \times \left(V \left(1 - \frac{m}{n}\right)\right) \times g \]

4. **Set the Weights Equal**:
Equate the two expressions for weight:
\[ \text{Density of material} \times V \times g = \text{Density of water} \times \left(V \left(1 - \frac{m}{n}\right)\right) \times g \]

5. **Cancel Out Common Terms**:
Since \( g \) and \( V \) are common in both sides, we can cancel them:
\[ \text{Density of material} = \text{Density of water} \times \left(1 - \frac{m}{n}\right) \]

6. **Calculate Relative Density**:
The relative density (specific gravity) is defined as the ratio of the density of the material to the density of water:
\[ \text{Relative Density} = \frac{\text{Density of material}}{\text{Density of water}} = 1 - \frac{m}{n} \]

### Final Answer:
The relative density of the material of the body is:
\[ \text{Relative Density} = 1 - \frac{m}{n} \]