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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

3 answers

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.

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

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.