As president of the student council, Brenna needs to decide between two venues for the homecoming dance. City Ballroom charges a $300 setup fee plus $240 per hour, which includes the services of a DJ. La Rosa Loft doesn't charge a setup fee, but the space costs $295 per hour and does not include a DJ. If Brenna chooses La Rosa Loft, she would need to hire an outside DJ that costs an additional $45 per hour.

Question

How long would the dance need to last for the cost to be the same at either venue?
Simplify any fractions.
hours

Answer 1

Let's denote the length of the dance in hours as \( h \).

Cost of City Ballroom: The cost at City Ballroom, which includes a setup fee and the cost per hour, is given by: \[ \text{Cost}_{\text{City}} = 300 + 240h \]

Cost of La Rosa Loft: The cost at La Rosa Loft, which charges per hour and requires hiring an outside DJ, is given by: \[ \text{Cost}_{\text{La Rosa}} = 295h + 45h = 340h \]

We need to find the value of \( h \) such that the costs are equal: \[ 300 + 240h = 340h \]

Now, we will solve for \( h \):

1. Subtract \( 240h \) from both sides: \[ 300 = 340h - 240h \] \[ 300 = 100h \]

2. Divide both sides by 100: \[ h = \frac{300}{100} = 3 \]

Thus, the dance would need to last 3 hours for the cost to be the same at either venue.