Let's solve this step by step.
Step 1: Let's represent the number of cars sold by Dealership A as "C_A" and the number of trucks sold by Dealership A as "T_A".
Step 2: According to the given information, Dealership A sold a total of 225 cars and trucks. So, we can set up the equation: C_A + T_A = 225.
Step 3: Dealership B sold twice as many cars as Dealership A, so we can represent the number of cars sold by Dealership B as "C_B" = 2 * C_A.
Step 4: Dealership B sold half as many trucks as Dealership A, so we can represent the number of trucks sold by Dealership B as "T_B" = 0.5 * T_A.
Step 5: According to the given information, Dealership B sold a total of 300 cars and trucks. So, we can set up the equation: C_B + T_B = 300.
Step 6: Substitute the expressions for C_B and T_B from steps 3 and 4 into the equation from step 5: 2 * C_A + 0.5 * T_A = 300.
Step 7: Simplify the equation from step 6: 2 * C_A + T_A/2 = 300.
Step 8: Now, we have a system of two equations:
C_A + T_A = 225,
2 * C_A + T_A/2 = 300.
Step 9: To solve the system of equations, we can multiply the first equation by 2 to eliminate T_A: 2 * C_A + 2 * T_A = 450.
Step 10: Subtract the equation from step 9 from the equation from step 8 to eliminate 2 * C_A: 2 * C_A + T_A/2 - (2 * C_A + 2 * T_A) = 300 - 450.
Step 11: Simplify the equation from step 10: -3/2 * T_A = -150.
Step 12: Solve for T_A by multiplying both sides by -2/3: T_A = -150 * (-2/3).
Step 13: Simplify the equation from step 12: T_A = 100.
Step 14: Substitute the value of T_A into the equation from step 2 to find C_A: C_A + 100 = 225.
Step 15: Solve for C_A: C_A = 225 - 100.
Step 16: Simplify the equation from step 15: C_A = 125.
Step 17: Substitute the values of C_A and T_A into the equation from step 4 to find T_B: T_B = 0.5 * 100.
Step 18: Simplify the equation from step 17: T_B = 50.
Therefore, Dealership B sold 50 trucks.