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:
An equation for the total cars and trucks for Dealership B:
How many cars did Dealership A sell?
Given the fact that Dealership B sold half as many trucks as Dealership A, how many trucks did Dealership B sell?
a) Let's use C1 to represent the number of cars sold by Dealership A and T1 to represent the number of trucks sold by Dealership A. Since Dealership A sells a total of 225 cars and trucks, we can write the equation as:
C1 + T1 = 225
b) Dealership B sells twice as many cars as Dealership A, so let's use C2 to represent the number of cars sold by Dealership B. Dealership B sells half as many trucks as Dealership A, so let T2 represent the number of trucks sold by Dealership B. We can then write the equation for Dealership B as:
C2 + T2 = 300
c) From equation a), we can't determine the number of cars Dealership A sold because we have two variables (C1 and T1) and only one equation.
Given that Dealership B sold twice as many cars as Dealership A, we can say:
C2 = 2C1
Since Dealership B sold half as many trucks as Dealership A, we can say:
T2 = 0.5T1
We can now substitute these expressions into equation b):
2C1 + 0.5T1 = 300
From here, we still can't determine the exact number of cars and trucks sold by either dealership because we have two variables (C1 and T1) and one equation. To solve for the number of cars and trucks sold by each dealership, we would need another equation or more information.
a) Let's represent the number of cars sold by Dealership A as "C_A" and the number of trucks sold as "T_A".
From the information given, we know that:
C_A + T_A = 225
b) Now let's represent the number of cars sold by Dealership B as "C_B" and the number of trucks sold as "T_B".
From the information given, we know that:
C_B + T_B = 300
We also know that Dealership B sold twice as many cars as Dealership A, and half as many trucks as Dealership A. We can use this information to relate the variables:
C_B = 2 * C_A
T_B = 0.5 * T_A
c) To find the number of cars sold by Dealership A, we can substitute the value of C_B from equation b into equation a:
C_B = 2 * C_A
300 = 2 * C_A
C_A = 300 / 2
C_A = 150
Therefore, Dealership A sold 150 cars.
d) Given that Dealership B sold half as many trucks as Dealership A, we can substitute the value of T_A from equation a into equation b:
T_B = 0.5 * T_A
T_B = 0.5 * T_A
T_B = 0.5 * (225 - C_A)
T_B = 0.5 * (225 - 150)
T_B = 0.5 * 75
T_B = 37.5
Since the number of trucks sold must be a whole number, Dealership B sold 37 trucks.
To find the answers, let's denote the number of cars sold by Dealership A as "x" and the number of trucks as "y".
a) An equation for the total cars and trucks for Dealership A:
Since Dealership A sells a total of 225 cars and trucks, we can write the equation as:
x + y = 225
b) An equation for the total cars and trucks for Dealership B:
Dealership B sells twice as many cars as Dealership A and half as many trucks. So, the equation becomes:
2x + 0.5y = 300
Now, we have two equations:
1) x + y = 225
2) 2x + 0.5y = 300
To solve this system of equations, we can use substitution or elimination.
Using substitution:
Rearrange equation 1) to solve for x:
x = 225 - y
Substitute the value of x in equation 2):
2(225 - y) + 0.5y = 300
450 - 2y + 0.5y = 300
450 - 300 = 2y - 0.5y
150 = 1.5y
y = 100
Now, substitute the value of y back into equation 1) to find x:
x + 100 = 225
x = 225 - 100
x = 125
So, Dealership A sold 125 cars.
Given that Dealership B sold half as many trucks as Dealership A, we can find the number of trucks sold by Dealership B:
Dealership B sold 100 trucks.