To find the distance of the point of contraflexure P from support B in a continuous beam with a uniformly distributed load (UDL), we can use the equation for the bending moment at any point along the beam.

1. **Understanding the problem:**
- We have a continuous beam ABC with a UDL of 4.5 kN/m.
- The length AB is 3m and BC is 1m.
- The reaction at support A is 6 kN and at support B is 12 kN.
- We need to find the distance of the point of contraflexure P from support B.

2. **Calculating the reaction at support C:**
- To calculate the reaction at support C, we need to consider the equilibrium of forces.
- The total load on the beam is the UDL multiplied by the length of AB (4.5 kN/m * 3m = 13.5 kN).
- The total load on the beam is distributed between supports A and C.
- The sum of the reactions at A and C should be equal to the total load on the beam.
- Thus, the reaction at C can be calculated as 13.5 kN - 6 kN = 7.5 kN.

3. **Calculating the bending moment at point P:**
- To calculate the bending moment at point P, we need to consider the beam's response to the applied loads.
- The bending moment at any point on the beam can be calculated using the equation M = wL^2/8, where M is the bending moment, w is the UDL, and L is the length of the beam segment.
- The bending moment at point P can be calculated using the equation M_p = w_AB * L_AB^2/8 + w_BC * L_BC^2/8, where w_AB is the UDL on segment AB, L_AB is the length of segment AB, w_BC is the UDL on segment BC, and L_BC is the length of segment BC.
- Plugging in the values, we get M_p = 4.5 kN/m * 3m * 3m / 8 + 4.5 kN/m * 1m * 1m / 8 = 30.375 kNm.

4. **Calculating the distance of point P from support B:**
- The point of contraflexure occurs where the bending moment changes sign, i.e., from positive to negative or vice versa.
- At the point of contraflexure, the bending moment is zero.
- So, we can set M_p = 0 and solve for the distance of point P from support B.
- 30.375 kNm = 0.33m * 4.5 kN/m * 3m * 3m / 8 + 4.5 kN/m * 1m * 1m / 8
- Simplifying the equation, we get 30.375 kNm = 0.33m * 30.375 kNm
- Dividing both sides by 30.375 kNm, we get 0.33m = 1.
- Therefore, the distance of point P from support B is 0.33m.

5. **Conclusion:**
- The distance of the point of contraflexure P from support B in the continuous