<Question>
Daniel works at an electronics store, and he claims that the popularity of a television (measured in number of sales) is inversely proportional to its cost. If 15 customers buy a television that costs $\$$1500, according to Daniel's theory, how many customers would buy a television that costs $\$$2500?
</Question>

<think>
<Plan_1>
Okay, let's try to figure out this problem step by step. So, Daniel says that the popularity of a TV, which is the number of sales, is inversely proportional to its cost. Hmm, inverse proportionality. I remember that means if one thing goes up, the other goes down in such a way that their product is a constant. Right? So, if TV sales (let's call that S) is inversely proportional to cost (let's say C), then S = k/C, where k is the constant of proportionality.

<prompt_1.1> Set up the inverse proportionality equation for the number of sales (S) and cost (C) based on the given problem statement. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
Got that part.

Now, the problem says that 15 customers bought a TV that costs $1500. So, when C is $1500, S is 15. We can use this information to find the constant k. Let me write that down: 15 = k / 1500. To solve for k, I need to multiply both sides by 1500. So, k = 15 * 1500. Let me calculate that. 15 times 1500... Well, 15 times 1000 is 15,000, and 15 times 500 is 7,500. Adding those together gives 15,000 + 7,500 = 22,500. So, k is 22,500.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Now, the question is asking how many customers would buy a TV that costs $2500 according to Daniel's theory. So, we need to find S when C is 2500. Using the same formula, S = k / C. We already found k to be 22,500, so plugging in the new cost: S = 22,500 / 2500.

<prompt_2.1> Use the inverse proportionality equation and the constant k found earlier to calculate the number of sales for a TV priced at $2500. </prompt_2.1>
-----

Wait, but let me double-check the calculations to make sure I didn't make any mistakes. Let's verify the constant k first. If 15 TVs cost $1500, then k = 15 * 1500.

<prompt_2.2> Verify the calculations of the constant k and the number of sales for a TV priced at $2500. </prompt_2.2>

Alternatively, since it's inverse proportionality, the ratio of the two sales numbers should be the inverse ratio of the two costs. So, if S1 is 15, C1 is 1500, and C2 is 2500, then S2 = S1 * (C1 / C2).

<prompt_2.3> Use the inverse proportionality ratio method to verify the number of sales for a TV priced at $2500. </prompt_2.3>
</Plan_2>

<Execution_2>
<execution_2.1>
Let me compute that. Hmm, 22,500 divided by 2500. Let's see, 2500 goes into 22,500 how many times? Well, 2500 times 9 is 22,500 because 2500 times 10 is 25,000, which is 2,500 more than 22,500. So, 2500 times 9 is 22,500. Therefore, S = 9.
</execution_2.1>

<execution_2.2>
15 multiplied by 1500. Let me do that again. 15 * 1500. 1500 is 1.5 thousand, so 15 * 1.5 = 22.5, and then add three zeros, which makes 22,500. That's correct. Then, 22,500 divided by 2500. Let's see, 2500 goes into 22,500 how many times? 2500 * 9 = 22,500. So, yes, 9.
</execution_2.2>

<execution_2.3>
Let's compute that. 15 * (1500 / 2500). Simplify 1500/2500. Both numbers can be divided by 500: 1500 ÷ 500 = 3, 2500 ÷ 500 = 5. So, 3/5. Therefore, S2 = 15 * (3/5) = 15 * 0.6 = 9.
</execution_2.3>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Wait, that seems straightforward, but let me make sure I didn't skip any steps or make a mistake in the proportionality.

Inverse proportionality means S ∝ 1/C, so S = k/C.

<prompt_3.1> Verify the solution by revisiting the inverse proportionality equation and the calculations. </prompt_3.1>
-----

Based on execution_2.1, execution_2.2, execution_2.3:
So, whether I compute it through the constant k or use the ratio directly, I get 9 customers. That gives me more confidence that the answer is correct. I don't see any errors in the reasoning here.
So, the answer should be 9 customers.

<prompt_3.2> Conclude the solution by confirming the number of customers for a TV priced at $2500 and stating the final answer. </prompt_3.2>
</Plan_3>

<Execution_3>
<execution_3.1>
We found k by plugging in the initial values: 15 = k / 1500, so k = 15 * 1500 = 22,500. Then, for the new cost of 2500, S = 22,500 / 2500. Let me do that division again. 22,500 divided by 2500. Let's simplify the numbers by dividing numerator and denominator by 100 first: 225 / 25. 25 times 9 is 225, so yes, that's 9. So, S = 9.
</execution_3.1>

<execution_3.2>
The key was understanding that inverse proportionality implies S * C = k, so the product is constant. Then using the given values to find k and apply it to the new cost. Both methods lead to the same answer, so I think that's solid.
</execution_3.2>
</Execution_3>

<Final_answer>
Therefore, according to Daniel's theory, the number of customers who would buy a television costing $\$$2500$ is $\boxed{9}$.
</Final_answer>
</think>