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 let \( t \) be the number of tickets needed for each hair tie.

According to the first scenario, if Liz buys the kickball for 2,200 tickets, she will have enough left for 13 hair ties. This can be expressed as:
\[
T - 2200 = 13t \quad \text{(1)}
\]

In the second scenario, if she buys the toy airplane for 2,500 tickets, she will have enough left for 7 hair ties. This can be expressed as:
\[
T - 2500 = 7t \quad \text{(2)}
\]

Now we have a system of two equations:
1. \( T - 2200 = 13t \)
2. \( T - 2500 = 7t \)

We can solve this system by eliminating \( T \). From equation (1), we can express \( T \) as:
\[
T = 2200 + 13t \quad \text{(3)}
\]

Substituting equation (3) into equation (2):
\[
2200 + 13t - 2500 = 7t
\]
Simplifying gives:
\[
2200 - 2500 + 13t = 7t
\]
\[
-300 + 13t = 7t
\]
\[
13t - 7t = 300
\]
\[
6t = 300
\]
\[
t = 50
\]

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

To recap, the equation used to find \( t \) is derived from the two scenarios given in equations (1) and (2):
- The total tickets she has can be expressed in terms of the remaining number of tickets after buying each item, related to \( t \).

So the final answer is:
**Liz needs 50 tickets for each hair tie.**