Question

Write an equation and interpret the solution to the following problem: Rental car A has a rate of an initial $100, plus $10 per day. Rental car B has a rate of an initial $50, plus $20 per day. After how many days are the rental car prices equal?(1 point) Responses 50+20x=100x+10; x=1; after 1 day the prices are equal 50 plus 20 x equals 100 x plus 10 ; x equals 1 ; after 1 day the prices are equal 20x−50=10x−100; x=−5; after 5 days the prices are equal 20 x minus 50 equals 10 x minus 100 ; x equals negative 5 ; after 5 days the prices are equal 50+20x=100+10x; x=5; after 5 days the prices are equal 50 plus 20 x equals 100 plus 10 x ; x equals 5 ; after 5 days the prices are equal 50x+20=100x+10; x=5; after 5 days the prices are equal

Answer 1

Write an equation and interpret the solution to the following problem: Rental car A has a rate of an initial $100, plus $10 per day. Rental car B has a rate of an initial $50, plus $20 per day. After how many days are the rental car prices equal?(1 point)

50+20x=100x+10; x=1; after 1 day the prices are equal

20x−50=10x−100; x=−5; after 5 days the prices are equal

50+20x=100+10x; x=5; after 5 days the prices are equal

50x+20=100x+10; x=5; after 5 days the prices are equal

Answer 2

To determine after how many days the costs of rental cars A and B will be equal, we can start by setting up equations for the total cost of each rental car.

1. For Rental Car A:
- Initial cost: $100
- Daily cost: $10 per day
- Total cost after \( x \) days:
\[
\text{Cost}_A = 100 + 10x
\]

2. For Rental Car B:
- Initial cost: $50
- Daily cost: $20 per day
- Total cost after \( x \) days:
\[
\text{Cost}_B = 50 + 20x
\]

To find when the costs are equal, we set the two equations equal to each other:
\[
100 + 10x = 50 + 20x
\]

Now, we can solve for \( x \):

1. Subtract \( 10x \) from both sides:
\[
100 = 50 + 10x
\]

2. Subtract 50 from both sides:
\[
50 = 10x
\]

3. Divide both sides by 10:
\[
x = 5
\]

The interpretation of the solution \( x = 5 \) means that after 5 days, the rental costs of both cars will be equal.

To verify, we can calculate the total cost for both cars after 5 days:

- For Car A:
\[
\text{Cost}_A = 100 + 10(5) = 100 + 50 = 150
\]

- For Car B:
\[
\text{Cost}_B = 50 + 20(5) = 50 + 100 = 150
\]

Both costs come out to $150, confirming that our solution is correct.

Answer 3

To find when the costs of Rental Car A and Rental Car B are equal, let's formulate the equation based on the provided rates:

1. **Rental Car A**:
- Initial cost: $100
- Daily cost: $10 per day
- Total cost after \( x \) days:
\[
\text{Cost}_A = 100 + 10x
\]

2. **Rental Car B**:
- Initial cost: $50
- Daily cost: $20 per day
- Total cost after \( x \) days:
\[
\text{Cost}_B = 50 + 20x
\]

Next, we set the two total costs equal to find \( x \):
\[
100 + 10x = 50 + 20x
\]

Now, let's solve this equation step by step:

1. **Rearranging the equation**:
- Subtract \( 10x \) from both sides:
\[
100 = 50 + 10x
\]

2. **Subtract 50 from both sides**:
\[
50 = 10x
\]

3. **Divide both sides by 10**:
\[
x = 5
\]

### Interpretation
The solution \( x = 5 \) means that after **5 days**, the rental costs for both car A and car B will be equal.

### Verifying the Solution
To ensure accuracy, we can compute the total costs for both cars after 5 days:

- **For Car A**:
\[
\text{Cost}_A = 100 + 10(5) = 100 + 50 = 150
\]

- **For Car B**:
\[
\text{Cost}_B = 50 + 20(5) = 50 + 100 = 150
\]

Both Car A and Car B will have a total cost of **$150** after 5 days, confirming that our calculated solution \( x = 5 \) is correct.

### Equation and Solution Reference
From the responses provided, the correct equation is:
- **50 + 20x = 100 + 10x; x = 5; after 5 days the prices are equal.**

This matches the numbers and solution we verified through the calculations.