A body P floats in water with half its volume immersed. Another body Q floats in a liquid of density `(3)/(4)`th of the density of water with two - third of the volume immersed. The ratio of density of P to that of Q is

Options

A. `1:2`
B. `1:1`
C. `2:3`
D. `3:4`

Answer

To solve the problem, we will use Archimedes' principle, which states that the buoyant force acting on a body immersed in a fluid is equal to the weight of the fluid displaced by the body.

### Step-by-Step Solution:

1. **Identify the given information:**
- Body P floats in water with half its volume immersed.
- Body Q floats in a liquid with density \( \frac{3}{4} \) of the density of water, with two-thirds of its volume immersed.

2. **Define the variables:**
- Let the volume of body P be \( V_P \).
- Let the density of body P be \( \rho_P \).
- Let the density of water be \( \rho_W \).
- Let the volume of body Q be \( V_Q \).
- Let the density of body Q be \( \rho_Q \).

3. **Apply Archimedes' principle for body P:**
- The weight of body P is equal to the buoyant force acting on it.
- The volume of body P that is immersed is \( \frac{V_P}{2} \).
- The buoyant force is given by:
\[ \text{Buoyant Force} = \text{Weight of liquid displaced} = \left(\frac{V_P}{2}\right) \rho_W g \]
- The weight of body P is:
\[ \text{Weight of P} = V_P \rho_P g \]
- Setting these equal gives:
\[ V_P \rho_P g = \left(\frac{V_P}{2}\right) \rho_W g \]
- Canceling \( V_P g \) from both sides (assuming \( V_P \neq 0 \) and \( g \neq 0 \)):
\[ \rho_P = \frac{\rho_W}{2} \]

4. **Apply Archimedes' principle for body Q:**
- The volume of body Q that is immersed is \( \frac{2V_Q}{3} \).
- The buoyant force is given by:
\[ \text{Buoyant Force} = \text{Weight of liquid displaced} = \left(\frac{2V_Q}{3}\right) \left(\frac{3}{4} \rho_W\right) g \]
- The weight of body Q is:
\[ \text{Weight of Q} = V_Q \rho_Q g \]
- Setting these equal gives:
\[ V_Q \rho_Q g = \left(\frac{2V_Q}{3}\right) \left(\frac{3}{4} \rho_W\right) g \]
- Canceling \( V_Q g \) from both sides (assuming \( V_Q \neq 0 \) and \( g \neq 0 \)):
\[ \rho_Q = \frac{2}{3} \cdot \frac{3}{4} \rho_W = \frac{1}{2} \rho_W \]

5. **Find the ratio of densities \( \rho_P \) and \( \rho_Q \):**
- From the previous steps, we have:
\[ \rho_P = \frac{\rho_W}{2} \]
\[ \rho_Q = \frac{\rho_W}{2} \]
- Therefore, the ratio of the densities is:
\[ \frac{\rho_P}{\rho_Q} = \frac{\frac{\rho_W}{2}}{\frac{\rho_W}{2}} = 1 \]

6. **Conclusion:**
- The ratio of the density of body P to the density of body Q is \( 1:1 \).