A metallic sphere weighs `210g` in air, 180 g in water and 120 g in an unknown liquid. Find the density of metal and of liquid.

Explanation

To solve the problem, we need to find the density of the metallic sphere and the density of the unknown liquid using the given weights in air, water, and the unknown liquid.

### Step-by-step Solution:

1. **Understanding the given data**:
- Weight of the sphere in air (W_air) = 210 g
- Weight of the sphere in water (W_water) = 180 g
- Weight of the sphere in the unknown liquid (W_liquid) = 120 g

2. **Calculating the loss of weight in water**:
The loss of weight when the sphere is submerged in water is given by:
\[ \Delta W_{water} = W_{air} - W_{water} = 210 \, \text{g} - 180 \, \text{g} = 30 \, \text{g} \]

3. **Calculating the loss of weight in the unknown liquid**:
The loss of weight when the sphere is submerged in the unknown liquid is:
\[ \Delta W_{liquid} = W_{air} - W_{liquid} = 210 \, \text{g} - 120 \, \text{g} = 90 \, \text{g} \]

4. **Finding the density of the metal sphere**:
The density of the metal can be calculated using the formula for relative density. The relative density (specific gravity) is given by the ratio of the weight in air to the loss of weight in water:
\[ \text{Relative Density} = \frac{W_{air}}{\Delta W_{water}} = \frac{210 \, \text{g}}{30 \, \text{g}} = 7 \]
Since the density of water is 1 g/cm³, the density of the metal sphere (ρ_m) is:
\[ \rho_m = 7 \, \text{g/cm}^3 \]

5. **Finding the density of the unknown liquid**:
Using the relationship between the losses of weight in the two fluids, we can set up the following ratio:
\[ \frac{\Delta W_{liquid}}{\Delta W_{water}} = \frac{\rho_{liquid}}{\rho_{water}} \]
Substituting the known values:
\[ \frac{90 \, \text{g}}{30 \, \text{g}} = \frac{\rho_{liquid}}{1 \, \text{g/cm}^3} \]
Simplifying this gives:
\[ 3 = \rho_{liquid} \]
Thus, the density of the unknown liquid (ρ_liquid) is:
\[ \rho_{liquid} = 3 \, \text{g/cm}^3 \]

### Final Answers:
- Density of the metal sphere (ρ_m) = 7 g/cm³
- Density of the unknown liquid (ρ_liquid) = 3 g/cm³