Question

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 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?

Skip to navigation

Answer 1

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