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 (densities) of two substances based on the conditions provided. Let's denote the specific gravities (or densities) of the two substances as \( d_1 \) and \( d_2 \).

### Step 1: Setting up the equations for equal volumes

When equal volumes of the two substances are mixed, the specific gravity of the mixture is given as 4.

The specific gravity (SG) is defined as the ratio of the density of a substance to the density of water. Thus, we can express the specific gravity of the mixture as follows:

\[ SG = \frac{\text{Total mass}}{\text{Total volume}} = 4 \]

Let the volume of each substance be \( V \). The total mass of the mixture is:

\[ \text{Total mass} = V \cdot d_1 + V \cdot d_2 = V(d_1 + d_2) \]

The total volume is:

\[ \text{Total volume} = V + V = 2V \]

Now, substituting these into the specific gravity equation:

\[ \frac{V(d_1 + d_2)}{2V} = 4 \]

This simplifies to:

\[ \frac{d_1 + d_2}{2} = 4 \]

Multiplying both sides by 2 gives:

\[ d_1 + d_2 = 8 \quad \text{(Equation 1)} \]

### Step 2: Setting up the equations for equal weights

When equal weights of the two substances are mixed, the specific gravity of the mixture is given as 3.

Let the weight of each substance be \( m \). The specific gravity of the mixture can be expressed as:

\[ SG = \frac{\text{Total mass}}{\text{Total volume}} = 3 \]

The total mass is:

\[ \text{Total mass} = m + m = 2m \]

The volume of each substance can be expressed as:

\[ \text{Volume of substance 1} = \frac{m}{d_1}, \quad \text{Volume of substance 2} = \frac{m}{d_2} \]

Thus, the total volume is:

\[ \text{Total volume} = \frac{m}{d_1} + \frac{m}{d_2} \]

Now substituting into the specific gravity equation:

\[ \frac{2m}{\frac{m}{d_1} + \frac{m}{d_2}} = 3 \]

Cancelling \( m \) from the numerator and denominator gives:

\[ \frac{2}{\frac{1}{d_1} + \frac{1}{d_2}} = 3 \]

Cross-multiplying leads to:

\[ 2 = 3\left(\frac{1}{d_1} + \frac{1}{d_2}\right) \]

This simplifies to:

\[ \frac{1}{d_1} + \frac{1}{d_2} = \frac{2}{3} \quad \text{(Equation 2)} \]

### Step 3: Solving the equations

Now we have two equations:

1. \( d_1 + d_2 = 8 \)
2. \( \frac{1}{d_1} + \frac{1}{d_2} = \frac{2}{3} \)

From Equation 1, we can express \( d_2 \) in terms of \( d_1 \):

\[ d_2 = 8 - d_1 \]

Substituting \( d_2 \) into Equation 2:

\[ \frac{1}{d_1} + \frac{1}{8 - d_1} = \frac{2}{3} \]

Finding a common denominator gives:

\[ \frac{(8 - d_1) + d_1}{d_1(8 - d_1)} = \frac{2}{3} \]

This simplifies to:

\[ \frac{8}{d_1(8 - d_1)} = \frac{2}{3} \]

Cross-multiplying gives:

\[ 24 = 2d_1(8 - d_1) \]

Expanding and rearranging leads to:

\[ 2d_1^2 - 16d_1 + 24 = 0 \]

Dividing by 2:

\[ d_1^2 - 8d_1 + 12 = 0 \]

Factoring gives:

\[ (d_1 - 6)(d_1 - 2) = 0 \]

Thus, \( d_1 = 6 \) or \( d_1 = 2 \).

Using \( d_1 + d_2 = 8 \):

- If \( d_1 = 6 \), then \( d_2 = 2 \).
- If \( d_1 = 2 \), then \( d_2 = 6 \).

### Conclusion

The specific gravities of the two substances are \( 6 \) and \( 2 \).

---