A company owns two dealerships, both of which sells cars and trucks. Dealership A sells a total of 164 cars and trucks. Dealership B sells twice as many cars and half as many trucks as Dealership A, and sells a total of 229 cars and trucks.

Question

an equation for the total cars and trucks for Dealership A
an equation for the total cars and trucks for Dealership B
how many cars did Dealership A sell
considering that Dealership B sold half as many trucks as Dealership A, how many trucks did Dealership B sell

GPT-4o mini · Answer

To solve the problem, we need to set up equations based on the information given.

Let:
- \( c_A \) = number of cars sold by Dealership A
- \( t_A \) = number of trucks sold by Dealership A
From the information for Dealership A, we can set up the following equation: \[ c_A + t_A = 164 \quad \text{(Equation 1)} \]
For Dealership B, we know it sells twice as many cars and half as many trucks as Dealership A:
- Number of cars sold by Dealership B = \( 2c_A \)
- Number of trucks sold by Dealership B = \( \frac{1}{2}t_A \)
Therefore, the total sales for Dealership B can be represented as: \[ 2c_A + \frac{1}{2}t_A = 229 \quad \text{(Equation 2)} \]

Now, we have two equations:

\( c_A + t_A = 164 \)
\( 2c_A + \frac{1}{2}t_A = 229 \)

We can solve these equations step by step.

Step 1: Solve Equation 1 for \( t_A \)

From Equation 1: \[ t_A = 164 - c_A \]

Step 2: Substitute \( t_A \) into Equation 2

Substitute \( t_A \) in Equation 2: \[ 2c_A + \frac{1}{2}(164 - c_A) = 229 \] Now simplify the equation: \[ 2c_A + \frac{164}{2} - \frac{1}{2}c_A = 229 \] This simplifies to: \[ 2c_A + 82 - \frac{1}{2}c_A = 229 \] Combining like terms: \[ \left(2 - \frac{1}{2}\right)c_A + 82 = 229 \] \[ \frac{3}{2}c_A + 82 = 229 \]

Step 3: Solve for \( c_A \)

First, isolate \( \frac{3}{2}c_A \): \[ \frac{3}{2}c_A = 229 - 82 \] \[ \frac{3}{2}c_A = 147 \]

Now, multiply both sides by \( \frac{2}{3} \): \[ c_A = 147 \times \frac{2}{3} = 98 \]

Step 4: Calculate \( t_A \)

Now we can find the number of trucks sold by Dealership A: \[ t_A = 164 - c_A = 164 - 98 = 66 \]

Step 5: Calculate Dealership B's sales

Now we calculate Dealership B's sales:

Cars sold by Dealership B: \[ 2c_A = 2 \times 98 = 196 \]
Trucks sold by Dealership B: \[ \frac{1}{2}t_A = \frac{1}{2} \times 66 = 33 \]

Final Answers

Cars sold by Dealership A: 98
Trucks sold by Dealership B: 33

To summarize:

Equation for total cars and trucks for Dealership A: \( c_A + t_A = 164 \)
Equation for total cars and trucks for Dealership B: \( 2c_A + \frac{1}{2}t_A = 229 \)
Dealership A sold 98 cars, and Dealership B sold 33 trucks.