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.

Question

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?

GPT-4o mini · Answer

Let $ c_A $ and $ t_A $ be the number of cars and trucks sold by Dealership A, respectively. From the information provided: ### a) An equation for the total cars and trucks for Dealership A: The total number of cars and trucks sold at Dealership A can be expressed as: $$ c_A + t_A = 225 $$ ### b) An equation for the total cars and trucks for Dealership B: Let $ c_B $ and $ t_B $ be the number of cars and trucks sold by Dealership B, respectively. According to the problem, Dealership B sells twice as many cars as A and half as many trucks as A, and sells a total of 300 cars and trucks. This can be expressed in equations: $$ c_B = 2c_A \quad \text{(twice as many cars as A)} $$ $$ t_B = \frac{1}{2}t_A \quad \text{(half as many trucks as A)} $$ And the total for Dealership B is: $$ c_B + t_B = 300 $$ Now substituting the expressions for $ c_B $ and $ t_B $: $$ 2c_A + \frac{1}{2}t_A = 300 $$ ### c) How many cars did Dealership A sell? Now we have the following system of equations: 1. $ c_A + t_A = 225 $ 2. $ 2c_A + \frac{1}{2}t_A = 300 $ We can solve this system. Starting with the first equation, we can express $ t_A $ in terms of $ c_A $: $$ t_A = 225 - c_A $$ Now we substitute $ t_A $ into the second equation: $$ 2c_A + \frac{1}{2}(225 - c_A) = 300 $$ Distributing the $\frac{1}{2}$: $$ 2c_A + \frac{225}{2} - \frac{1}{2}c_A = 300 $$ Combine the $ c_A $ terms: $$ \left(2 - \frac{1}{2}\right)c_A + \frac{225}{2} = 300 $$ $$ \frac{3}{2}c_A + \frac{225}{2} = 300 $$ Now, subtract $\frac{225}{2}$ from both sides to solve for $ c_A $: $$ \frac{3}{2}c_A = 300 - \frac{225}{2} $$ $$ 300 = \frac{600}{2} \quad \Rightarrow \quad 300 - \frac{225}{2} = \frac{600}{2} - \frac{225}{2} = \frac{375}{2} $$ Thus: $$ \frac{3}{2}c_A = \frac{375}{2} $$ Now solve for $ c_A $: $$ 3c_A = 375 \quad \Rightarrow \quad c_A = 125 $$ ### d) Given the fact that Dealership B sold half as many trucks as Dealership A, how many trucks did Dealership B sell? Using the first equation: $$ t_A = 225 - c_A = 225 - 125 = 100 $$ Now, since Dealership B sold half as many trucks: $$ t_B = \frac{1}{2}t_A = \frac{1}{2}(100) = 50 $$ ### Summary of Findings: - c) Dealership A sold **125 cars**. - d) Dealership B sold **50 trucks**.