Two ticket types and two totals signal a system of equations: two rules that must hold for the same ticket counts. Write one equation for the number of tickets and another for the money collected. Then graph both in Desmos and click where the lines cross. Keep each price with its own ticket type, and read the coordinate for the type the question asks about.
Hints
- Hint 1
The ticket count includes both kinds. If counts adults and counts children, what equation adds those counts to match the total number sold?
- Hint 2
A revenue equation adds money, not tickets. Adult revenue is times the adult count, and child revenue is times the child count. What total must those amounts equal?
- Hint 3
A system's intersection is the point that satisfies both equations. In , is the second coordinate. Which ticket type did you assign to ?
Step-by-step
Approach 1: Graph the ticket and revenue equations
- Step 1
Count both kinds of tickets
Let count adult tickets and count child tickets. Every ticket is one kind or the other, so their counts add to the 250 tickets sold:
- Step 2
Add the money from each kind
At $12 each, adult tickets bring in dollars. At $7 each, child tickets bring in dollars. The theater collected $2,410, so add those dollar amounts:
Keep the ticket total and the dollar total in separate equations.
- Step 3
Read the child count at the intersection
Type and on separate Desmos lines. Click their intersection, the point that satisfies both equations. Desmos labels it . Since counts child tickets, use the second coordinate: the theater sold child tickets. Choice C.
Approach 2: Start with an all-adult sale
- Step 1
Find the all-adult total
Imagine all 250 tickets were adult tickets. Type in Desmos; it prints . That is the all-adult total, before any adult tickets are replaced by child tickets.
- Step 2
Find the drop for each child ticket
Replacing one $12 adult ticket with one $7 child ticket lowers the revenue by $5: . Each child ticket accounts for one such drop.
- Step 3
Find the total drop
Type on the next Desmos line. It prints , the gap between the all-adult total and the money actually collected.
- Step 4
Count the child tickets
Type on the next line; Desmos prints . Each $5 drop represents one child ticket, so the theater sold child tickets. Choice C.