A company owns two dealerships, both of which sell cars and trucks. The first dealership sells a total of 164 cars and trucks. The second dealership sells twice as many cars and half as many trucks as the first dealership, and sells a total of 229 cars and trucks.
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?
How many trucks did the dealership B sell?
Let's represent the number of cars sold by the first dealership as "C1" and the number of trucks as "T1".
Similarly, let's represent the number of cars sold by the second dealership as "C2" and the number of trucks as "T2".
From the problem, we have the following information:
C1 + T1 = 164 ..........(equation 1)
C2 + T2 = 229 ..........(equation 2)
We also know that the second dealership sells twice as many cars as the first dealership and half as many trucks as the first dealership. This can be expressed as:
C2 = 2C1 ..........(equation 3)
T2 = 0.5T1 ..........(equation 4)
To find the equation for the total cars and trucks for dealership A (C1 + T1), we can substitute equations 3 and 4 into equation 2:
2C1 + 0.5T1 = 229
Simplifying this equation, we get:
2C1 + 0.5T1 = 229
To find the equation for the total cars and trucks for dealership B (C2 + T2), we can substitute equations 3 and 4 into equation 2:
2C1 + 0.5T1 = 229
So, the equation for the total cars and trucks for dealership A is 2C1 + 0.5T1 = 229.
To find the number of cars sold by dealership A, we can substitute equations 3 and 4 into equation 1:
C1 + T1 = 164
Simplifying this equation, we get:
C1 + 0.5T1 = 164
We have two equations with two variables (C1 and T1), so we can solve them simultaneously.
Now, we need to solve the system of equations:
2C1 + 0.5T1 = 229 ..........(equation 5)
C1 + 0.5T1 = 164 ..........(equation 6)
We can eliminate T1 by multiplying equation 6 by -0.5 and adding it to equation 5.
-0.5(C1 + 0.5T1) + 2C1 + 0.5T1 = -0.5(164)
This simplifies to:
1.5C1 = -0.5(164)
1.5C1 = -82
Dividing both sides by 1.5 gives us:
C1 = -54
Since we cannot have a negative number of cars sold, this means there was an error in the problem statement or calculations.
As a result, we cannot determine the number of cars dealership A sold or the number of trucks dealership B sold based on the given information.
Let's break down the information given step-by-step and find the answers to your questions:
Step 1: Define variables:
Let's denote the number of cars sold by Dealership A as 'a' and the number of trucks sold by Dealership A as 'b'. Similarly, denote the number of cars sold by Dealership B as 'c' and the number of trucks sold by Dealership B as 'd'.
Step 2: Set up equations:
From the given information, we know:
- Dealership A sold a total of 164 cars and trucks, so we can write the equation: a + b = 164.
- Dealership B sold twice as many cars as Dealership A, and half as many trucks as Dealership A, so we can write the equations: c = 2a and d = (1/2)b.
Step 3: Solve the equations:
To find the number of cars sold by Dealership A, we need to solve the equation a + b = 164. Unfortunately, we don't have enough information to solve for the number of trucks sold by Dealership B.
Step 4: Substitute values:
To find the number of cars sold by Dealership A, we need to substitute the value of 'c' in terms of 'a' from the equation c = 2a. Substituting c = 2a into the equation a + b = 164, we have 2a + b = 164.
Step 5: Simplify and solve for 'a':
Now, we have the system of equations:
2a + b = 164 (Equation 1)
d = (1/2)b (Equation 2)
Since we don't have enough information to solve Equation 2, we will solve Equation 1 only.
Step 6: Solve Equation 1:
2a + b = 164
We can rewrite this equation as:
b = 164 - 2a
Step 7: Substitute value of 'b' in Equation 2:
We know d = (1/2)b.
So, d = (1/2)*(164 - 2a)
Simplifying, d = 82 - a.
Step 8: Answer the questions:
Now, we can answer your questions:
- The equation for the total cars and trucks sold by Dealership A is a + b = 164.
- The equation for the total cars and trucks sold by Dealership B is c + d = 229.
- To find how many cars Dealership A sold, we need more information or another equation.
- We don't have enough information to determine how many trucks Dealership B sold.
To solve this problem, let's break it down step by step.
Step 1: Define the variables
Let's define the variables:
C1 = the number of cars sold at dealership A
T1 = the number of trucks sold at dealership A
C2 = the number of cars sold at dealership B
T2 = the number of trucks sold at dealership B
Step 2: Set up the equations
The problem states that the first dealership sells a total of 164 cars and trucks:
C1 + T1 = 164 ------> Equation 1
The problem also states that the second dealership sells twice as many cars and half as many trucks as the first dealership and sells a total of 229 cars and trucks:
C2 + T2 = 229 ------> Equation 2
C2 = 2C1 ------> Equation 3
T2 = 0.5T1 ------> Equation 4
Step 3: Solve the system of equations
Now we need to solve the system of equations to find the values of C1, T1, C2, and T2.
Substituting Equation 3 and Equation 4 into Equation 2, we get:
2C1 + 0.5T1 = 229
Substituting Equation 1 into the above equation, we get:
2C1 + 0.5T1 = C1 + T1 = 229
Combining like terms, we have:
C1 + T1 = 229
Since this equation matches Equation 1, we can conclude that C1 + T1 must equal 164.
From Equation 1, we know that C1 + T1 = 164, so C1 = 164 - T1.
Substituting this expression for C1 into Equation 3, we get:
2(164 - T1) = C2
Simplifying, we have:
C2 = 328 - 2T1
Substituting the expressions for C2 and T2 into Equation 2, we get:
328 - 2T1 + 0.5T1 = 229
Combining like terms, we have:
-1.5T1 = -99
Dividing both sides by -1.5, we get:
T1 = 66
Substituting T1 = 66 into Equation 1, we get:
C1 + 66 = 164
Subtracting 66 from both sides, we have:
C1 = 98
Step 4: Answer the questions
From our calculations, we found that dealership A sold 98 cars (C1) and 66 trucks (T1). We also found that dealership B sold 196 cars (C2 = 2C1) and 33 trucks (T2 = 0.5T1).
Therefore, dealership A sold 98 cars and dealership B sold 33 trucks.