Question

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.


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

Answer 1

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

1. **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:
\[
c_A + t_A = 164 \quad \text{(Equation 1)}
\]

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

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**.