Question

Liz won a bunch of tickets at the arcade. She plans to spend most of them on one large item and use the rest to buy hair ties. An employee at the arcade tells Liz that if she buys the kickball for 2,200 tickets, she will have exactly enough left for 13 hair ties. However, if she buys the toy airplane for 2,500 tickets, she will have exactly enough left for 7 hair ties.
Which equation can you use to find t, the number of tickets Liz needs for each hair tie?
How many tickets does Liz need for each hair tie?
tickets

Answer 1

Let \( T \) be the total number of tickets Liz has, and \( t \) be the number of tickets needed for each hair tie.

From the information provided, we can set up two equations based on the two items Liz is considering:

1. If Liz buys the kickball for 2,200 tickets and has enough left for 13 hair ties, we can express this as:
\[
T - 2200 = 13t
\]

2. If Liz buys the toy airplane for 2,500 tickets and has enough left for 7 hair ties, we can express this as:
\[
T - 2500 = 7t
\]

Now we have a system of two equations:

1. \( T - 2200 = 13t \) → \( T = 13t + 2200 \)
2. \( T - 2500 = 7t \) → \( T = 7t + 2500 \)

Since both expressions equal \( T \), we can set them equal to each other:
\[
13t + 2200 = 7t + 2500
\]

Now let's solve for \( t \):
1. Subtract \( 7t \) from both sides:
\[
6t + 2200 = 2500
\]
2. Subtract 2200 from both sides:
\[
6t = 300
\]
3. Divide by 6:
\[
t = 50
\]

Thus, Liz needs **50 tickets for each hair tie**.