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

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=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

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
50 x plus 20 equals 100 x plus 10 ; x equals 5 ; after 5 days the prices are equal
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Answer 1

To find out after how many days the rental car prices for A and B will be equal, we can set up the equations based on the given rates.

For rental car A:
- Initial cost: $100
- Daily cost: $10 per day

Let \( x \) be the number of days. Therefore, the total cost for rental car A can be expressed as:
\[
\text{Cost of A} = 100 + 10x
\]

For rental car B:
- Initial cost: $50
- Daily cost: $20 per day

The total cost for rental car B can be expressed as:
\[
\text{Cost of B} = 50 + 20x
\]

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

Now, we can solve this equation for \( x \):
1. Subtract \( 10x \) from both sides:
\[
100 = 50 + 10x
\]
2. Subtract 50 from both sides:
\[
50 = 10x
\]
3. Divide by 10:
\[
x = 5
\]

**Interpretation of the Solution:**
The solution \( x = 5 \) means that after 5 days, the total rental prices for both car A and car B will be equal.

To verify this, we can calculate the costs for both cars after 5 days:
- Cost of car A after 5 days: \( 100 + 10(5) = 100 + 50 = 150 \)
- Cost of car B after 5 days: \( 50 + 20(5) = 50 + 100 = 150 \)

Thus, both costs are indeed equal at $150 after 5 days.

Therefore, the correct response is:
\( 50 + 20x = 100 + 10x; x = 5; \) after 5 days the prices are equal.