What was the undiscounted price of a ticket?

Question

300 tickets to a concert were sold. Of these, 35 were at a 20% discount. The total sales came to $512.75. What was the undiscounted price of a ticket?

Answer 1

Normally to find the price of an object, we just divide the total sales by the number of items sold. 

Here we have two prices to worry about, so we will set up an equation to think about the combined sales. 

 

Let's use a variable, P, to represent the full price of one ticket. The other price we're thinking about is a discounted price, 20% off the full price. If people are paying 20% less than full price- that is, 20% less than 100%, then they are paying (100-20 =) 80% of the full price. We can represent the price of one discounted ticket as 80% of P or 0.80P.

 

Now let's think about what was sold overall. The problem tells us that 35 tickets were sold at a discount. We also know that 300 total tickets were sold. 35 discounted tickets + "x" full price tickets = 300 total tickets sold.

35 + x = 300

x = 265 full price tickets sold

 

Now we can set up an equation for the total sales, equaling $512.75. That price came from 265 times the full price of a ticket, P, and 35 times the discounted price of a ticket, 0.80P:

265P + 35(0.8P) = 512.75

265P + 28P = 512.75

293P = 512.75

P = 1.75

 

Simplifying the equation and combining like terms tells us that P, which we defined as the full price of a ticket, is 1.75. The undiscounted price of a ticket is $1.75.