A theater group made appearances in two cities. The hotel charge before tax in the second city was $500 lower than in the first. The tax in the first city was 7.5%, and the tax in the second city was 4.5%. The total hotel tax paid for the two cities was $457.50. How much was the total charge in each city before tax?

Answer 1

Let \( x \) be the hotel charge before tax in the first city. Then, the hotel charge before tax in the second city would be \( x - 500 \).

Now, we can calculate the tax for each city:

1. First City Tax: \[ \text{Tax in First City} = 0.075x \]

2. Second City Tax: \[ \text{Tax in Second City} = 0.045(x - 500) \]

According to the problem, the total hotel tax paid for the two cities was $457.50. Thus, we can set up the following equation:

\[ 0.075x + 0.045(x - 500) = 457.50 \]

Now, let's simplify the equation:

First, distribute \(0.045\):

\[ 0.075x + 0.045x - 0.045 \times 500 = 457.50 \]

Calculating \(0.045 \times 500\):

\[ 0.045 \times 500 = 22.5 \]

Now, substitute this value back into the equation:

\[ 0.075x + 0.045x - 22.5 = 457.50 \]

Combine the \( x \) terms:

\[ 0.120x - 22.5 = 457.50 \]

Next, add \( 22.5 \) to both sides:

\[ 0.120x = 457.50 + 22.5 \]

Calculating the right-hand side:

\[ 457.50 + 22.5 = 480 \]

Now we have:

\[ 0.120x = 480 \]

To find \( x \), divide both sides by \( 0.120 \):

\[ x = \frac{480}{0.120} = 4000 \]

So, the hotel charge before tax in the first city is \( x = 4000 \).

Now, we can calculate the hotel charge for the second city:

\[ x - 500 = 4000 - 500 = 3500 \]

Summary:

• The total charge in the first city before tax is \( \boxed{4000} \).
• The total charge in the second city before tax is \( \boxed{3500} \).