Question

Use your graphing calculator: Desmos Graphing Calculator

A company owns two dealerships, both of which sell cars and trucks. Dealership A sells a total of 225 cars and trucks. Dealership B sells twice as many cars and half as many trucks as the Dealership A, and sells a total of 300 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) Given the fact that Dealership B sold half as many trucks as Dealership A, how many trucks did Dealership B sell?

Skip to navigation

Answer 1

Let's formulate the problem step by step.

### a) An equation for the total cars and trucks for Dealership A:

Let \( c_A \) represent the number of cars sold by Dealership A, and \( t_A \) represent the number of trucks sold by Dealership A. We know that:

\[
c_A + t_A = 225
\]

### b) An equation for the total cars and trucks for Dealership B:

Let \( c_B \) be the number of cars sold by Dealership B, and \( t_B \) be the number of trucks sold by Dealership B. The problem states that Dealership B sells twice as many cars and half as many trucks as Dealership A, and that the total sales for Dealership B is 300. Therefore, we have:

\[
c_B + t_B = 300
\]

Since \( c_B = 2c_A \) and \( t_B = \frac{1}{2}t_A \), we can substitute these into the equation for Dealership B:

\[
2c_A + \frac{1}{2}t_A = 300
\]

### c) How many cars did Dealership A sell?

We now have the system of equations:

1. \( c_A + t_A = 225 \) (1)
2. \( 2c_A + \frac{1}{2}t_A = 300 \) (2)

To solve these equations simultaneously, we can express \( t_A \) in terms of \( c_A \) from equation (1):

\[
t_A = 225 - c_A
\]

Substituting \( t_A \) in equation (2):

\[
2c_A + \frac{1}{2}(225 - c_A) = 300
\]

Multiply through by 2 to eliminate the fraction:

\[
4c_A + 225 - c_A = 600
\]

Combine like terms:

\[
3c_A + 225 = 600
\]

Subtract 225 from both sides:

\[
3c_A = 375
\]

Finally, divide by 3:

\[
c_A = 125
\]

### d) Given the fact that Dealership B sold half as many trucks as Dealership A, how many trucks did Dealership B sell?

Now that we know \( c_A = 125 \), we can find \( t_A \):

\[
t_A = 225 - c_A = 225 - 125 = 100
\]

Now we calculate the number of trucks sold by Dealership B:

\[
t_B = \frac{1}{2}t_A = \frac{1}{2} \times 100 = 50
\]

### Summary of the Results:

- a) Equation for Dealership A: \( c_A + t_A = 225 \)
- b) Equation for Dealership B: \( 2c_A + \frac{1}{2}t_A = 300 \)
- c) Dealership A sold **125 cars**.
- d) Dealership B sold **50 trucks**.