tickets for the valentine’s day dance cost $3 per person or $5 per couple. if $475 worth of tickets were sold and 180 people attended the dance, how many couples were there?

Answer 1

Let \( x \) be the number of couples and \( y \) be the number of single people.

According to the problem, we have two equations based on the information provided:

1. The total number of people who attended the dance: \[ 2x + y = 180 \] (since each couple counts as 2 people)

2. The total cost of the tickets sold: \[ 5x + 3y = 475 \] (where \( 5x \) is the total cost for couples and \( 3y \) is the total cost for single tickets)

Now we have a system of equations:

1. \( 2x + y = 180 \) (1)
2. \( 5x + 3y = 475 \) (2)

We can solve this system step by step. First, solve equation (1) for \( y \): \[ y = 180 - 2x \]

Substitute \( y \) in equation (2): \[ 5x + 3(180 - 2x) = 475 \] Expanding this gives: \[ 5x + 540 - 6x = 475 \] Combining like terms results in: \[ -x + 540 = 475 \] Subtracting 540 from both sides: \[ -x = 475 - 540 \] \[ -x = -65 \] Thus: \[ x = 65 \]

Now that we have the number of couples \( x = 65 \), we can substitute this value back into equation (1) to find \( y \): \[ 2(65) + y = 180 \] \[ 130 + y = 180 \] Subtracting 130 from both sides gives: \[ y = 50 \]

So, there were \( \boxed{65} \) couples at the dance.