arjun purchased 5 tickets to a play,each with the same price. he was also changed an online service fee of $3.50 per ticket. Emilia purchased 3 tickets to the same play and paid twice as much for her tickets as Arjun.emilia was also charged a service fee of $2.75 per ticket. if they spent the same amount,what is the cost of each of the tickets Arjun purchased? let t represent the cost of each of arjun's tickets.

Let's solve the problem step by step.

### Step 1: Express the total cost for Arjun
Arjun purchased 5 tickets, each costing $ t $. He was also charged a service fee of $3.50 per ticket. Therefore, the total cost Arjun paid is:

\[
\text{Arjun's total cost} = 5t + 5 \times 3.50
\]

Simplifying the service fee portion:

\[
\text{Arjun's total cost} = 5t + 17.50
\]

### Step 2: Express the total cost for Emilia
Emilia purchased 3 tickets and paid twice as much for each ticket as Arjun. Therefore, the cost of each of Emilia's tickets is $ 2t $. She was also charged a service fee of $2.75 per ticket. Therefore, the total cost Emilia paid is:

\[
\text{Emilia's total cost} = 3(2t) + 3 \times 2.75
\]

Simplifying:

\[
\text{Emilia's total cost} = 6t + 8.25
\]

### Step 3: Set up the equation based on the fact that they spent the same amount
Since Arjun and Emilia spent the same amount of money, we can set their total costs equal to each other:

\[
5t + 17.50 = 6t + 8.25
\]

### Step 4: Solve for $ t $
Subtract $ 5t $ from both sides of the equation:

\[
17.50 = t + 8.25
\]

Next, subtract 8.25 from both sides:

\[
t = 9.25
\]

### Conclusion:
The cost of each ticket Arjun purchased is $ \boxed{9.25} $.

Would you like more details, or do you have any other questions? 

Here are some related questions you might find interesting:
1. How would the total cost change if Arjun purchased 7 tickets instead of 5?
2. If the service fee for Arjun was increased to $4.00 per ticket, how would that affect the cost of each ticket?
3. How much would Emilia have paid if the service fee was reduced to $2.00 per ticket?
4. What if Arjun's tickets were priced 25% higher? How would that change the outcome?
5. Can you create an equation that expresses Emilia's total cost if she bought $ n $ tickets?

**Tip:** When solving word problems, always break down the information step by step and translate it into mathematical expressions.

Learn how to solve the problem of determining the cost of each ticket Arjun purchased, given the conditions provided. This problem involves solving linear equations and understanding service fee impacts on total costs. Suitable for students in Grades 7-9.

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