To solve this problem, let's define our variables:
- Let \( c_A \) be the number of cars sold by Dealership A.
- Let \( t_A \) be the number of trucks sold by Dealership A.
- Let \( c_B \) be the number of cars sold by Dealership B.
- Let \( t_B \) be the number of trucks sold by Dealership B.
a) An equation for the total cars and trucks for Dealership A:
The total number of cars and trucks sold by Dealership A is given by: \[ c_A + t_A = 164 \]
b) An equation for the total cars and trucks for Dealership B:
According to the problem, Dealership B sells twice as many cars and half as many trucks as Dealership A. So, we can express it as follows:
- Cars sold by Dealership B: \( c_B = 2c_A \)
- Trucks sold by Dealership B: \( t_B = \frac{1}{2}t_A \)
The total for Dealership B is given by: \[ c_B + t_B = 229 \] Substituting the expressions for \( c_B \) and \( t_B \): \[ 2c_A + \frac{1}{2}t_A = 229 \]
c) How many cars did Dealership A sell?
Now we have the system of equations:
- \( c_A + t_A = 164 \) (Equation 1)
- \( 2c_A + \frac{1}{2}t_A = 229 \) (Equation 2)
We can solve these equations simultaneously.
From Equation 1: \[ t_A = 164 - c_A \]
Substitute \( t_A \) in Equation 2: \[ 2c_A + \frac{1}{2}(164 - c_A) = 229 \] Expanding this: \[ 2c_A + 82 - \frac{1}{2}c_A = 229 \] Simplifying the equation: \[ (2 - \frac{1}{2})c_A + 82 = 229 \] \[ \frac{3}{2}c_A = 229 - 82 \] \[ \frac{3}{2}c_A = 147 \] Multiplying both sides by \( \frac{2}{3} \): \[ c_A = \frac{2}{3} \cdot 147 = 98 \]
So, Dealership A sold 98 cars.
d) Considering that Dealership B sold half as many trucks as Dealership A, how many trucks did Dealership B sell?
Now that we know \( c_A = 98 \), we can find \( t_A \): \[ t_A = 164 - c_A = 164 - 98 = 66 \]
Next, we can find \( t_B \): Since Dealership B sold half as many trucks as Dealership A: \[ t_B = \frac{1}{2}t_A = \frac{1}{2} \cdot 66 = 33 \]
Thus, Dealership B sold 33 trucks.
Summary:
- a) \( c_A + t_A = 164 \)
- b) \( 2c_A + \frac{1}{2}t_A = 229 \)
- c) Dealership A sold 98 cars.
- d) Dealership B sold 33 trucks.
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