Question

I'm studying for my algebra exam and I came across a problem that I'm having trouble solving. The problem states that the number of concert tickets sold varies directly with the amount spent on advertising and inversely with the ticket price. If 4800 tickets were sold when $20,000 was spent on advertising and the ticket price was $40, how many tickets will be sold if the advertising budget is increased to $40,000?

Answer 1

To solve the problem, we start by expressing the relationship using variables. Let:

- \( T \) be the number of tickets sold,
- \( A \) be the amount spent on advertising,
- \( P \) be the ticket price.

According to the problem, the number of tickets sold varies directly with the amount spent on advertising and inversely with the ticket price. This can be expressed with the formula:

\[
T = k \cdot \frac{A}{P}
\]

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

Given the initial conditions, we know:

- \( T = 4800 \) tickets,
- \( A = 20000 \) dollars,
- \( P = 40 \) dollars.

We can substitute these values into the equation to find the value of \( k \):

\[
4800 = k \cdot \frac{20000}{40}
\]

Calculating \( \frac{20000}{40} \):

\[
\frac{20000}{40} = 500
\]

Now we substitute back into the equation:

\[
4800 = k \cdot 500
\]

To find \( k \), we can solve for \( k \):

\[
k = \frac{4800}{500} = 9.6
\]

Now that we have \( k \), we can use this value to determine how many tickets will be sold if the advertising budget is increased to \( A = 40000 \) dollars with the same ticket price \( P = 40 \) dollars.

Substituting into the formula:

\[
T = 9.6 \cdot \frac{40000}{40}
\]

Calculating \( \frac{40000}{40} \):

\[
\frac{40000}{40} = 1000
\]

Now substituting that value back into the equation for \( T \):

\[
T = 9.6 \cdot 1000 = 9600
\]

Thus, if the advertising budget is increased to $40,000, the number of tickets sold will be **9600**.