Daniel works at an electronics store, and he claims that the popularity of a toaster (measured in number of sales) is inversely proportional to its cost. If 12 customers buy a toaster that costs $\$500$, according to Daniel's theory, how many customers would buy a toaster that costs $\$750$?
Let the popularity of a toaster (or the number of customers who buy the appliance) equal $p$, and let the cost of the toaster equal $c$. According to Daniel's theory, $p$ and $c$ are inversely proportional. Thus, $(p)(c)=k$ for some constant value $k$. If $p=12$ when $c=500$, then $k=(12)(500)=6000$. So when $c=750$, \begin{align*} (p)(c)&=k
\\\Rightarrow\qquad (p)(750)&=6000
\\\Rightarrow\qquad p&=\frac{6000}{750}
\\ &=\boxed{8}.
\end{align*}According to Daniel's theory, 8 customers would buy the $\$750$ toaster.
8
Level 2
Algebra
Problem: Daniel works at an electronics store, and he claims that the popularity of a toaster (measured in number of sales) is inversely proportional to its cost. If 12 customers buy a toaster that costs $\$500$, according to Daniel's theory, how many customers would buy a toaster that costs $\$750$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To determine the number of customers who would buy a toaster costing $750, we start with the relationship that the number of sales is inversely proportional to the cost. This means that if the cost increases, the number of sales decreases in a specific manner.

Let \( S \) represent the number of sales and \( C \) represent the cost of the toaster. The relationship can be expressed mathematically as:

\[ S \propto \frac{1}{C} \]

This can be rewritten as:

\[ S = k \cdot \frac{1}{C} \]

where \( k \) is a constant of proportionality. 

From the information given, when the cost \( C = 500 \), the number of sales \( S = 12 \). We can use this to find the value of \( k \):

\[ 12 = k \cdot \frac{1}{500} \]

Multiplying both sides by 500 gives:

\[ k = 12 \cdot 500 = 6000 \]

Now, we need to find the number of sales when the cost is \( C = 750 \):

\[ S = 6000 \cdot \frac{1}{750} \]

Calculating this gives:

\[ S = \frac{6000}{750} \]

To simplify \( \frac{6000}{750} \), we can divide both the numerator and the denominator by 750:

\[ S = \frac{6000 \div 750}{750 \div 750} = \frac{8}{1} = 8 \]

Thus, the number of customers who would buy a toaster that costs $750 is 8.<|END_OF_LATENT|> The final answer is $8$.