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?

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Answer 1

Let's define the variables for both dealerships based on the problem statement:

- 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) Equation for the total cars and trucks for Dealership A:

From the problem, we know:

- Dealership A sells a total of 164 cars and trucks.

Thus, the equation for Dealership A is:
\[
c_A + t_A = 164
\]

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

From the problem, we learn that Dealership B sells twice as many cars and half as many trucks as Dealership A and has a total of 229 cars and trucks.

Thus, we have:
- \( c_B = 2c_A \)
- \( t_B = \frac{1}{2}t_A \)

The equation for Dealership B is:
\[
c_B + t_B = 229
\]
Substituting \( c_B \) and \( t_B \):
\[
2c_A + \frac{1}{2}t_A = 229
\]

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

Now we can solve the two equations simultaneously.

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

From Equation 1, we can express \( t_A \) in terms of \( c_A \):
\[
t_A = 164 - c_A
\]

Now substitute \( t_A \) into Equation 2:
\[
2c_A + \frac{1}{2}(164 - c_A) = 229
\]
Distributing \(\frac{1}{2}\):
\[
2c_A + 82 - \frac{1}{2}c_A = 229
\]

To eliminate the fraction, multiply each term by 2:
\[
4c_A + 164 - c_A = 458
\]
Combine like terms:
\[
3c_A + 164 = 458
\]
Subtract 164 from both sides:
\[
3c_A = 294
\]
Now, divide by 3:
\[
c_A = 98
\]

### d) How many trucks did Dealership B sell?

Using the value of \( c_A \) to find \( t_A \):
\[
t_A = 164 - c_A = 164 - 98 = 66
\]

Now, to find the number of trucks sold by Dealership B:
\[
t_B = \frac{1}{2} t_A = \frac{1}{2} \cdot 66 = 33
\]

### Summary of Answers:
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.**