In making an alloy, a substance of specific gravity `s_(1)` and mass `m_(1)` is mixed with another substance of specific gravity of the alloy is

Options

A. `[(s_(1)s_(2))/(m_(1)+m_(2))]`
B. `[(s_(1)+s_(2))/(m_(1)+m_(2))]`
C. `[(m_(1)l s_(1)+m_(2)l s_(2))/((m_(1)+m_(2)))]`
D. `[(m_(1)+m_(2))/((m_(1)l s_(1)+m_(2)l s_(2)))]`

Explanation

To find the specific gravity of the alloy formed by mixing two substances, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Specific Gravity**:
- Specific gravity (SG) is defined as the ratio of the density of a substance to the density of water.
- Mathematically, \( SG = \frac{\text{Density of substance}}{\text{Density of water}} \).

2. **Identify Given Values**:
- Let the specific gravity of the first substance be \( S_1 \) and its mass be \( m_1 \).
- Let the specific gravity of the second substance be \( S_2 \) and its mass be \( m_2 \).

3. **Calculate Densities**:
- The density of the first substance can be expressed as:
\[ \rho_1 = S_1 \cdot \rho_w \]
- The density of the second substance can be expressed as:
\[ \rho_2 = S_2 \cdot \rho_w \]
- Here, \( \rho_w \) is the density of water.

4. **Calculate Total Mass of the Alloy**:
- The total mass of the alloy \( m_{total} \) is:
\[ m_{total} = m_1 + m_2 \]

5. **Calculate Volume of Each Substance**:
- The volume of the first substance is:
\[ V_1 = \frac{m_1}{\rho_1} = \frac{m_1}{S_1 \cdot \rho_w} \]
- The volume of the second substance is:
\[ V_2 = \frac{m_2}{\rho_2} = \frac{m_2}{S_2 \cdot \rho_w} \]

6. **Calculate Total Volume of the Alloy**:
- The total volume \( V_{total} \) of the alloy is:
\[ V_{total} = V_1 + V_2 = \frac{m_1}{S_1 \cdot \rho_w} + \frac{m_2}{S_2 \cdot \rho_w} \]

7. **Calculate Density of the Alloy**:
- The density of the alloy \( \rho_{alloy} \) can be calculated as:
\[ \rho_{alloy} = \frac{m_{total}}{V_{total}} = \frac{m_1 + m_2}{\frac{m_1}{S_1 \cdot \rho_w} + \frac{m_2}{S_2 \cdot \rho_w}} \]

8. **Calculate Specific Gravity of the Alloy**:
- The specific gravity of the alloy \( S_{alloy} \) is given by:
\[ S_{alloy} = \frac{\rho_{alloy}}{\rho_w} \]
- Substituting the expression for \( \rho_{alloy} \):
\[ S_{alloy} = \frac{m_1 + m_2}{\frac{m_1}{S_1} + \frac{m_2}{S_2}} \]

### Final Formula:
The specific gravity of the alloy can be expressed as:
\[ S_{alloy} = \frac{m_1 + m_2}{\frac{m_1}{S_1} + \frac{m_2}{S_2}} \]