Question

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A company owns two dealerships, both of which sell 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.
(4 points)
a) An equation for the total cars and trucks for Dealership A:
b) An equation for the total cars and trucks for Dealership B:
c) How many cars did Dealership A sell?
d) Considering that Dealership B sold half as many trucks as Dealership A, how many trucks did
Dealership B sell?

Answer 1

Let's break down the problem step by step.

To solve for the number of cars and trucks sold by both dealerships, we can define:

- \( x \): the number of cars sold at Dealership A
- \( y \): the number of trucks sold at Dealership A

### a) An equation for the total cars and trucks for Dealership A:
The total number of vehicles sold at Dealership A is the sum of cars and trucks:

\[
x + y = 164
\]

### b) An equation for the total cars and trucks for Dealership B:
According to the problem, Dealership B sells twice as many cars as Dealership A and half as many trucks as Dealership A. Therefore:

- The number of cars sold at Dealership B is \( 2x \)
- The number of trucks sold at Dealership B is \( \frac{y}{2} \)

The total number of vehicles sold at Dealership B is then:

\[
2x + \frac{y}{2} = 229
\]

### c) Solve the equations:
Now we have a system of equations:

1. \( x + y = 164 \) (1)
2. \( 2x + \frac{y}{2} = 229 \) (2)

From equation (1), we can express \( y \) in terms of \( x \):

\[
y = 164 - x
\]

Substituting this value of \( y \) into equation (2):

\[
2x + \frac{164 - x}{2} = 229
\]

Now, to eliminate the fraction, multiply every term by 2:

\[
4x + 164 - x = 458
\]

Combine like terms:

\[
3x + 164 = 458
\]

Subtract 164 from both sides:

\[
3x = 294
\]

Divide by 3:

\[
x = 98
\]

So, Dealership A sold **98 cars**.

### d) Now, how many trucks did Dealership A and Dealership B sell:
Using the value of \( x \) in the equation for \( y \):

\[
y = 164 - x = 164 - 98 = 66
\]

So, Dealership A sold **66 trucks**.

Now, for Dealership B:
- The number of trucks sold at Dealership B is \( \frac{y}{2} = \frac{66}{2} = 33 \).

So, Dealership B sold **33 trucks**.

### Final Answers:
- c) Dealership A sold **98 cars**.
- d) Dealership B sold **33 trucks**.