Shawn and Dorian rented bikes from two different rental shops. The prices in dollars, $y$, of renting bikes from the two different shops for $x$ hours is shown.
Shop Shawn used: $y=10+3.5 x$
Shop Dorian used: $y=6 x$
If Shawn and Dorian each rented bikes for the same number of hours and each paid the same price, how much did each pay for the rental?
Answer (2)
Set up equations for the cost of renting bikes from each shop: y = 10 + 3.5 x and y = 6 x .
Equate the two equations to find the number of hours for which the cost is the same: 10 + 3.5 x = 6 x .
Solve for x : x = 4 hours.
Substitute x = 4 into either equation to find the total cost: y = 6 ( 4 ) = 24 . The final answer is 24 .
Explanation
Problem Analysis Let's analyze the problem. We have two different bike rental shops with different pricing models. Shawn's shop charges a base fee plus an hourly rate, while Dorian's shop charges only an hourly rate. We need to find the number of hours for which the total cost is the same for both shops, and then determine that cost.
Setting up Equations Let x be the number of hours Shawn and Dorian rented the bikes. Let y be the price each paid for the rental. We have the following equations for the cost at each shop:
Shawn's shop: y = 10 + 3.5 x Dorian's shop: y = 6 x
Equating the Costs Since they both paid the same price, we can set the two equations equal to each other:
10 + 3.5 x = 6 x
Solving for the Number of Hours Now, let's solve for x :
10 = 6 x − 3.5 x 10 = 2.5 x x = 2.5 10 = 4
So, they rented the bikes for 4 hours.
Finding the Total Cost Now, we substitute x = 4 into either equation to find the price y . Let's use Dorian's shop:
y = 6 ( 4 ) = 24
Therefore, each paid $24 for the rental.
Final Answer So, Shawn and Dorian each paid $24 for renting the bikes.
Examples
Imagine you and a friend are deciding between two phone plans. One plan has a base fee and a per-minute charge, while the other has only a per-minute charge. By setting up equations like we did, you can determine the number of minutes you'd need to talk for the plans to cost the same. This helps you choose the most economical plan based on your usage habits. This type of problem is also applicable when comparing costs of different services or products with varying pricing structures.
Shawn and Dorian each paid $24 for renting bikes, as they rented them for 4 hours. The cost from Shawn's shop is given by the equation y = 10 + 3.5 x and from Dorian's shop by y = 6 x . Setting these equal allows us to solve for the hours and consequently, the rental cost.
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