Let's formulate the problem step by step.
a) An equation for the total cars and trucks for Dealership A:
Let \( c_A \) represent the number of cars sold by Dealership A, and \( t_A \) represent the number of trucks sold by Dealership A. We know that:
\[ c_A + t_A = 225 \]
b) An equation for the total cars and trucks for Dealership B:
Let \( c_B \) be the number of cars sold by Dealership B, and \( t_B \) be the number of trucks sold by Dealership B. The problem states that Dealership B sells twice as many cars and half as many trucks as Dealership A, and that the total sales for Dealership B is 300. Therefore, we have:
\[ c_B + t_B = 300 \]
Since \( c_B = 2c_A \) and \( t_B = \frac{1}{2}t_A \), we can substitute these into the equation for Dealership B:
\[ 2c_A + \frac{1}{2}t_A = 300 \]
c) How many cars did Dealership A sell?
We now have the system of equations:
- \( c_A + t_A = 225 \) (1)
- \( 2c_A + \frac{1}{2}t_A = 300 \) (2)
To solve these equations simultaneously, we can express \( t_A \) in terms of \( c_A \) from equation (1):
\[ t_A = 225 - c_A \]
Substituting \( t_A \) in equation (2):
\[ 2c_A + \frac{1}{2}(225 - c_A) = 300 \]
Multiply through by 2 to eliminate the fraction:
\[ 4c_A + 225 - c_A = 600 \]
Combine like terms:
\[ 3c_A + 225 = 600 \]
Subtract 225 from both sides:
\[ 3c_A = 375 \]
Finally, divide by 3:
\[ c_A = 125 \]
d) Given the fact that Dealership B sold half as many trucks as Dealership A, how many trucks did Dealership B sell?
Now that we know \( c_A = 125 \), we can find \( t_A \):
\[ t_A = 225 - c_A = 225 - 125 = 100 \]
Now we calculate the number of trucks sold by Dealership B:
\[ t_B = \frac{1}{2}t_A = \frac{1}{2} \times 100 = 50 \]
Summary of the Results:
- a) Equation for Dealership A: \( c_A + t_A = 225 \)
- b) Equation for Dealership B: \( 2c_A + \frac{1}{2}t_A = 300 \)
- c) Dealership A sold 125 cars.
- d) Dealership B sold 50 trucks.