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Solving a Ticket Mixture Problem: Student and Adult Tickets
Apply the seven-step problem-solving strategy and the total-value model to a ticket mixture problem where each ticket type has a whole-dollar value and the relationship between ticket counts involves both multiplication and subtraction.
Problem: At a school concert, the total value of tickets sold was $1,506. Student tickets sold for $6 each and adult tickets sold for $9 each. The number of adult tickets sold was five less than three times the number of student tickets sold. How many student tickets and how many adult tickets were sold?
- Read the problem and identify the types involved: student tickets (worth $6 each) and adult tickets (worth $9 each). The total value of all tickets is $1,506.
- Identify what to find: the number of student tickets and the number of adult tickets.
- Name the unknowns using a single variable. Let = the number of student tickets. The phrase "five less than three times" combines multiplication by with subtracting , so the number of adult tickets is . Organize in a table:
| Type | Number | Value ($) | Total Value ($) |
|---|---|---|---|
| Student | 6 | ||
| Adult | 9 | ||
| 1,506 |
- Translate into an equation by adding the total values and setting the sum equal to the overall total:
- Solve the equation:
- Distribute :
- Combine like terms:
- Add to both sides:
- Divide both sides by :
Find the number of adult tickets: .
- Check: and . Adding:
- Answer: They sold 47 student tickets and 136 adult tickets.
This example demonstrates that ticket and stamp problems follow the same total-value model used for coin problems — each type of ticket has a fixed value, just as each type of coin does. Here the relationship "five less than three times" translates to , and distributing the $9 ticket price across produces whole-number coefficients ( and ) that are easier to work with than the decimal coefficients typical of coin problems. After distribution, combining into and solving the resulting two-step equation yields the answer.
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Ch.3 Math Models - Elementary Algebra @ OpenStax
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Learn After
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An event coordinator is using the variable to represent the number of 'Standard' tickets sold for a corporate gala. If the number of 'Premium' tickets sold was 'five less than three times' the number of Standard tickets, the algebraic expression used to represent the quantity of Premium tickets is ____.
In the seven-step problem-solving strategy for ticket mixture problems, the 'Check' step is successfully performed by confirming that the total number of tickets sold is equal to the total dollar amount of the revenue collected.
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