Joe rents a skateboard from Kelli's shop and his friend, Ming, rents a skateboard from David's shop. Is it possible for Joe and Ming to rent the skateboards for the same number of hours, and pay the same price? Write an equation and solve it to justify your answer.

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UpStudy AI · Accepted Answer

Yes, it is possible for Joe and Ming to rent skateboards for the same number of hours and pay the same price, even if they are renting from different shops with potentially different pricing structures. To determine under what conditions this is possible, let's set up and solve an equation.

### Step 1: Define Variables

1. **Let $$ h $$ be the number of hours both Joe and Ming rent the skateboards.**
2. **Let $$ P_K $$ be the price per hour at Kelli's shop (where Joe rents).**
3. **Let $$ P_D $$ be the price per hour at David's shop (where Ming rents).**

### Step 2: Set Up the Equation

To ensure that both Joe and Ming pay the same total price for renting their skateboards, their total costs must be equal.

- **Total cost for Joe at Kelli's shop:** $$ \text{Total}_K = P_K \times h $$
- **Total cost for Ming at David's shop:** $$ \text{Total}_D = P_D \times h $$

Set these two expressions equal to each other:

$$
P_K \times h = P_D \times h
$$

### Step 3: Solve the Equation

$$
P_K \times h = P_D \times h
$$

To find if there’s a solution for $$ h $$, we can simplify the equation:

$$
P_K \times h - P_D \times h = 0
$$
$$
h (P_K - P_D) = 0
$$

This equation holds true if either:

1. **$$ h = 0 $$:** Both Joe and Ming decide not to rent the skateboards at all, resulting in a total cost of $0. This is a trivial solution.
   
2. **$$ P_K - P_D = 0 $$:** This implies that $$ P_K = P_D $$, meaning both shops charge the same price per hour. In this case, for any number of hours $$ h $$, both Joe and Ming would pay the same total price.

### Conclusion

- **If the hourly rates at both shops are the same ($$ P_K = P_D $$), then Joe and Ming can rent the skateboards for any number of hours $$ h $$ and pay the same price.**

- **If the hourly rates are different ($$ P_K \neq P_D $$), the only solution where they pay the same price is when $$ h = 0 $$, meaning they don't rent the skateboards.**

Therefore, **Joe and Ming can rent the skateboards for the same number of hours and pay the same price only if both shops charge the same hourly rate**.

### Example

Suppose:
- **Kelli's shop charges \$5 per hour ($$ P_K = 5 $$).**
- **David's shop charges \$5 per hour ($$ P_D = 5 $$).**

For any number of hours $$ h $$:
$$
5h = 5h \quad \text{(True for all } h \text{)}
$$

Thus, they will always pay the same amount.

However, if:
- **Kelli's shop charges \$5 per hour ($$ P_K = 5 $$).**
- **David's shop charges \$7 per hour ($$ P_D = 7 $$).**

Setting the total costs equal:
$$
5h = 7h
$$
$$
5h - 7h = 0
$$
$$
-2h = 0 \quad \Rightarrow \quad h = 0
$$

Only when $$ h = 0 $$ (i.e., not renting at all) will their costs be the same.
Yes, Joe and Ming can rent the skateboards for the same number of hours and pay the same price if both shops charge the same hourly rate.

Joe rents a skateboard from Kelli’s shop and his friend, Ming, rents a skateboard from David’s shop. Is it possible for Joe and Ming to rent the skateboards for the same number of hours, and pay the same price? Write an equation and solve it to justify your answer.

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