an athletic department sold 454 tickets to a sporting event. adults paid $3.50 per ticket, and students paid $1.00. the total ticket sales were $1,154. how many adult tickets and how many
student tickets were sold to the event?

a 25 adult tickets and 429 student tickets were sold.

b 148 adult tickets and 306 student tickets were sold.

c 174 adult tickets and 280
student tickets were sold.

d 280 adult tickets and 174 student tickets were sold.

  D  280 adult tickets and 174 student tickets were sold

Step-by-step explanation:

When the answers differ by a lot as they do here, you can get an idea of the correct one by looking at the average ticket cost:

  $1154/(454 tickets) ≈ $2.54 per ticket

That is more than the average of adult and student ticket prices, (3.50+1.00)/2 = 2.25, so more than half of the tickets sold must be adult tickets.

Only one answer choice matches: choice D.

In problems such as this, the answer can be found using the ratio computed as above.

  adult tickets = (total tickets) · ((average ticket price) - (student price))/(price difference)

  = 454 · (2.54185 -1.00)/(2.50) = 280 . . . . . matches choice D

Usual solution

Let a represent the number of adult tickets. Then revenue is ...

  3.50a + 1.00(454-a) = 1154

  2.50a = 700 . . . . . eliminate parentheses, subtract 454

  a = 700/2.50 = 280 . . . adult tickets sold

the answer to the question is a  

it has the same slope and the same y-intercept.

a = 11 : 4

b = 38

c = 1 for every 6