Let's denote:
- \( x_A \): the number of cars sold at Dealership A
- \( y_A \): the number of trucks sold at Dealership A
- \( x_B \): the number of cars sold at Dealership B
- \( y_B \): the number of trucks sold at Dealership B
Based on the problem information, we have the following information:
-
Dealership A sells a total of 225 cars and trucks:
\[ x_A + y_A = 225 \] -
Dealership B sells twice as many cars and half as many trucks as Dealership A, and a total of 300 cars and trucks:
- Cars: \( x_B = 2x_A \)
- Trucks: \( y_B = \frac{1}{2} y_A \)
Therefore, for Dealership B: \[ x_B + y_B = 300 \]
Substituting \( x_B \) and \( y_B \) into the total for Dealership B: \[ 2x_A + \frac{1}{2}y_A = 300 \]
(a) Equation for the total cars and trucks for Dealership A:
\[ x_A + y_A = 225 \]
(b) Equation for the total cars and trucks for Dealership B:
Substituting for Dealership B's cars and trucks into the total: \[ 2x_A + \frac{1}{2}y_A = 300 \]
(c) To find how many cars Dealership A sold, we can solve these two equations:
- \( x_A + y_A = 225 \) (1)
- \( 2x_A + \frac{1}{2}y_A = 300 \) (2)
First, we can rearrange equation (1): \[ y_A = 225 - x_A \]
Substituting \( y_A \) into equation (2): \[ 2x_A + \frac{1}{2}(225 - x_A) = 300 \]
Now, simplify the equation: \[ 2x_A + \frac{225}{2} - \frac{1}{2}x_A = 300 \] \[ \frac{4x_A - x_A + 225}{2} = 300 \] \[ \frac{3x_A + 225}{2} = 300 \]
Multiply both sides by 2: \[ 3x_A + 225 = 600 \]
Now, subtract 225 from both sides: \[ 3x_A = 375 \]
Divide by 3: \[ x_A = 125 \]
So, the number of cars Dealership A sold is: \[ \boxed{125} \]
(d) Now we can find how many trucks Dealership A sold and then Dealership B's trucks:
Substituting \( x_A = 125 \) back into equation (1): \[ 125 + y_A = 225 \] \[ y_A = 100 \]
Now, using the relationship \( y_B = \frac{1}{2}y_A \): \[ y_B = \frac{1}{2} \times 100 = 50 \]
Therefore, the number of trucks Dealership B sold is: \[ \boxed{50} \]
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