Okay, let's try to figure out this problem step by step. Daniel says that the popularity of a TV, which is the number of sales, is inversely proportional to its cost. Hmm, inversely proportional. So if one thing goes up, the other goes down in a specific way. Let me recall what inverse proportionality means. If two quantities A and B are inversely proportional, then A = k/B, where k is a constant. So in this case, the number of sales (let's call that S) is inversely proportional to the cost (let's call that C). So S = k/C, right?

Daniel claims that when the cost is $1500, 15 customers buy the TV. So we can use that information to find the constant k. Let's plug those numbers into the equation. S = k/C, so 15 = k/1500. To find k, we multiply both sides by 1500. That gives k = 15 * 1500. Let me calculate that. 15 times 1500. Hmm, 15 times 1000 is 15,000, and 15 times 500 is 7,500. So adding those together, 15,000 + 7,500 = 22,500. So k is 22,500.

Now, the question is asking how many customers would buy a TV that costs $2500. Using the same formula, S = k/C. We already found k is 22,500, and now C is 2500. So plugging those in: S = 22,500 / 2500. Let's compute that. 22,500 divided by 2500. Hmm, 2500 times 9 is 22,500. So 2500 * 9 = 22,500. Therefore, 22,500 divided by 2500 is 9. So S should be 9 customers.

Wait, let me double-check that. If the cost doubles from 1500 to 2500, then since it's inversely proportional, the number of sales should halve? Wait, 1500 to 2500 is not exactly doubling. 1500 times 1.666... is 2500. So actually, the cost is multiplied by 5/3 (since 2500 divided by 1500 is 25/15 which simplifies to 5/3). Therefore, the number of sales should be multiplied by 3/5. So 15 multiplied by 3/5 is 9. Yep, that matches. So 9 customers. That seems right.

Alternatively, maybe I can think of it as setting up a proportion. If S1/C1 = S2/C2, but wait, no, since it's inversely proportional, S1 * C1 = S2 * C2. Because if S = k/C, then S*C = k, which is constant. So S1*C1 = S2*C2. So 15*1500 = S2*2500. Then S2 = (15*1500)/2500. Which is the same as before. 15*1500 is 22,500. Divided by 2500 is 9. So that checks out.

Another way: Let's see, 1500 to 2500. The ratio of the new cost to the old cost is 2500/1500 = 5/3. Since they're inversely proportional, the ratio of sales should be the inverse of the ratio of costs. So 3/5. 15*(3/5) = 9. Yep, same answer. So all methods lead to 9. So the answer is 9 customers.
Therefore, according to Daniel's theory, $\boxed{9}$ customers would buy a television that costs $\$$2500$.
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