When equal volumes of two substance are mixed, the specific gravity of the mixurie is 4. When equal weights of the same substance are mixed, the specific gravity of the mixture is 3. The soecufuc gravities of the two substance could be

Options

A. 6 and 2
B. 3and 4
C. `2.5` and `3.5`
D. 5 and 3

Explanation

To solve the problem, we need to find the specific gravities (or densities) of two substances based on the information given about their mixtures.

### 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.
- If we denote the specific gravities of the two substances as \( SG_1 \) and \( SG_2 \), we can relate them to their densities as follows:
\[ SG_1 = \frac{D_1}{D_w}, \quad SG_2 = \frac{D_2}{D_w} \]
where \( D_w \) is the density of water.

2. **Mixing Equal Volumes**:
- When equal volumes of the two substances are mixed, the specific gravity of the mixture is given as 4.
- The formula for the specific gravity of the mixture when equal volumes are mixed is:
\[ SG_{mix1} = \frac{D_1 + D_2}{2D_w} = 4 \]
- Rearranging gives:
\[ D_1 + D_2 = 8D_w \quad \text{(Equation 1)} \]

3. **Mixing Equal Weights**:
- When equal weights of the two substances are mixed, the specific gravity of the mixture is given as 3.
- The formula for the specific gravity of the mixture when equal weights are mixed is:
\[ SG_{mix2} = \frac{2}{\frac{1}{D_1} + \frac{1}{D_2}} = 3 \]
- Rearranging gives:
\[ \frac{2D_1D_2}{D_1 + D_2} = 3D_w \quad \text{(Equation 2)} \]

4. **Substituting Equation 1 into Equation 2**:
- From Equation 1, we have \( D_1 + D_2 = 8D_w \).
- Substitute this into Equation 2:
\[ \frac{2D_1D_2}{8D_w} = 3D_w \]
- Cross-multiplying gives:
\[ 2D_1D_2 = 24D_w^2 \]
- Simplifying gives:
\[ D_1D_2 = 12D_w^2 \quad \text{(Equation 3)} \]

5. **Solving the Equations**:
- Now we have two equations:
- \( D_1 + D_2 = 8D_w \) (Equation 1)
- \( D_1D_2 = 12D_w^2 \) (Equation 3)
- Let \( D_1 \) and \( D_2 \) be the roots of the quadratic equation:
\[ x^2 - (D_1 + D_2)x + D_1D_2 = 0 \]
- Substituting from Equations 1 and 3:
\[ x^2 - 8D_w x + 12D_w^2 = 0 \]
- Solving this quadratic equation using the quadratic formula:
\[ x = \frac{8D_w \pm \sqrt{(8D_w)^2 - 4 \cdot 12D_w^2}}{2} \]
- Simplifying gives:
\[ x = \frac{8D_w \pm \sqrt{64D_w^2 - 48D_w^2}}{2} = \frac{8D_w \pm \sqrt{16D_w^2}}{2} = \frac{8D_w \pm 4D_w}{2} \]
- This results in:
\[ x = 6D_w \quad \text{or} \quad x = 2D_w \]
- Thus, the specific gravities are:
\[ SG_1 = 6, \quad SG_2 = 2 \]

### Conclusion:
The specific gravities of the two substances are 6 and 2.