Introduction:
When water and ice are mixed together, heat transfer occurs between them until they reach a common final temperature. To determine the final temperature of the mixture, we need to consider the heat gained or lost by each substance.

Key information:
- Mass of water (m1) = 300 gm
- Initial temperature of water (T1) = 25°C
- Mass of ice (m2) = 100 gm
- Initial temperature of ice (T2) = 0°C

Calculating heat gained or lost:
1. Heat gained or lost by water (Q1) = m1 * c1 * ΔT1
- where c1 is the specific heat of water and ΔT1 is the change in temperature of water.
- c1 = 1 cal/g°C (specific heat of water)
- ΔT1 = final temperature - T1 (unknown)

2. Heat gained or lost by ice (Q2) = m2 * c2 * ΔT2
- where c2 is the specific heat of ice and ΔT2 is the change in temperature of ice.
- c2 = 0.5 cal/g°C (specific heat of ice)
- ΔT2 = final temperature - T2 (unknown)

Heat transfer between water and ice:
- When water and ice are mixed, heat transfer occurs until they reach a common final temperature.
- The heat lost by the water (Q1) will be equal to the heat gained by the ice (Q2).
- Therefore, Q1 = Q2.

Setting up the equation:
Equating the heat gained and lost:
m1 * c1 * ΔT1 = m2 * c2 * ΔT2

Substituting the known values:
(300 gm) * (1 cal/g°C) * (ΔT1) = (100 gm) * (0.5 cal/g°C) * (ΔT2)

Solving the equation:
300 * ΔT1 = 100 * 0.5 * ΔT2

Final temperature calculation:
- Rearranging the equation:
ΔT1 = (0.5 * ΔT2) / 3

- Substituting the equation for ΔT1 into the equation for ΔT2:
300 * (0.5 * ΔT2) / 3 = 100 * 0.5 * ΔT2

- Solving for ΔT2:
ΔT2 = 0°C

- Substituting the value of ΔT2 into the equation for ΔT1:
ΔT1 = (0.5 * 0) / 3 = 0°C

Conclusion:
The final temperature of the mixture is 0°C. This means that the water and ice will reach thermal equilibrium at the freezing point of water, resulting in the formation of a mixture of water and ice at 0°C.