Introduction:
In this problem, we are given two scenarios where two metals are mixed together to form alloys. The specific gravity of the alloys is provided for both scenarios. We need to determine the specific gravity of each metal in both cases.

Scenario 1: Mixing metals in equal volumes to form an alloy with specific gravity 4.
In this scenario, let's assume we have two metals, Metal A and Metal B, which are mixed together in equal volumes. The specific gravity of the resulting alloy is 4.

Scenario 2: Mixing metals in equal masses to form an alloy with specific gravity 3.
In this scenario, we mix the same two metals, Metal A and Metal B, but this time in equal masses. The specific gravity of the resulting alloy is 3.

Explanation:
To solve this problem, we need to understand the concept of specific gravity. Specific gravity is the ratio of the density of a substance to the density of a reference substance (usually water). Mathematically, it is expressed as:

Specific Gravity = Density of Substance / Density of Reference Substance

Scenario 1: Mixing metals in equal volumes
Since the specific gravity of the resulting alloy is 4, we can write the equation as:

4 = (V_A * ρ_A + V_B * ρ_B) / (V_A + V_B)

Where V_A and V_B are the volumes of Metal A and Metal B respectively, and ρ_A and ρ_B are the densities of Metal A and Metal B respectively.

Since the metals are mixed in equal volumes, we can assume V_A = V_B = V. Substituting this into the equation, we get:

4 = (V * ρ_A + V * ρ_B) / (2V)
4 = (ρ_A + ρ_B) / 2

Therefore, ρ_A + ρ_B = 8

Scenario 2: Mixing metals in equal masses
In this case, since the specific gravity of the resulting alloy is 3, we can write the equation as:

3 = (m_A * ρ_A + m_B * ρ_B) / (m_A + m_B)

Where m_A and m_B are the masses of Metal A and Metal B respectively.

Since the metals are mixed in equal masses, we can assume m_A = m_B = m. Substituting this into the equation, we get:

3 = (m * ρ_A + m * ρ_B) / (2m)
3 = (ρ_A + ρ_B) / 2

Therefore, ρ_A + ρ_B = 6

Specific gravity of each metal:
We have two equations:
1. ρ_A + ρ_B = 8 (Scenario 1)
2. ρ_A + ρ_B = 6 (Scenario 2)

By solving these equations, we can find the specific gravity of each metal.

Subtracting equation 2 from equation 1, we get:
(ρ_A + ρ_B) - (ρ_A + ρ_B) = 8 - 6
0 = 2

This implies that the two scenarios are contradictory and cannot be simultaneously true. Therefore, there is no solution to this problem.

Conclusion:
Based