Baseball fans can buy tickets for seats in the lower deck or upper deck of the stadium. Tickets for the lower deck cost $42 each. Ticket prices for the upper deck are 75% of the co... Baseball fans can buy tickets for seats in the lower deck or upper deck of the stadium. Tickets for the lower deck cost $42 each. Ticket prices for the upper deck are 75% of the cost of tickets for the lower deck. Which inequality represents all possible combinations of x, the number of tickets for the lower deck, and y, the number of tickets for the upper deck, that someone can buy for no more than $800? A. 42x + 56y ≤ 800 B. 42x + 31.5y ≤ 800 C. 42x + 56y > 800 D. 42x + 31.5y > 800. Which of the following can represent a possible combination of the upper deck and lower deck tickets? A. 22 lower deck and 5 upper deck B. 16 lower deck and 6 upper deck C. 8 lower deck and 14 upper deck D. 10 lower deck and 18 upper deck.
Understand the Problem
The question requires us to formulate an inequality representing the total cost of buying tickets for both the lower and upper decks, with the upper deck tickets being 75% of the lower deck price. We also need to determine which option fits a feasible combination of ticket purchases within a budget of $800.
Answer
The inequality is $L(x + 0.75y) \leq 800$.
Answer for screen readers
The inequality representing the total cost is $$ L(x + 0.75y) \leq 800 $$
Steps to Solve
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Define Variables for Ticket Prices
Let the price of a lower deck ticket be denoted as $L$. Then the price of an upper deck ticket, which is 75% of the lower deck price, will be $U = 0.75L$. -
Express Total Cost
If we let $x$ represent the number of lower deck tickets and $y$ represent the number of upper deck tickets purchased, the total cost can be expressed as: $$ \text{Total Cost} = xL + yU $$ Substituting for $U$ gives: $$ \text{Total Cost} = xL + y(0.75L) $$ -
Set Up the Inequality
Since the total cost must be less than or equal to the budget of $800, we set up the inequality: $$ xL + 0.75yL \leq 800 $$ -
Factor Out L
To simplify the inequality, factor out $L$: $$ L(x + 0.75y) \leq 800 $$ -
Determine Feasibility
Next, we need to examine the combinations of $x$ and $y$ that satisfy this inequality for various values of $L$ within a total cost of $800.
The inequality representing the total cost is $$ L(x + 0.75y) \leq 800 $$
More Information
This inequality shows how the number of tickets purchased for both decks influences the total cost. It helps to visualize the combinations of ticket purchases that can stay within the budget of $800. By adjusting the values for $L$, $x$, and $y$, one can explore affordable ticket purchases.
Tips
- Forgetting to factor out the ticket price ($L$) can lead to an incorrect representation of the cost. Always simplify the inequality correctly.
- Not considering different values for $L$ while trying to find combinations of $x$ and $y$ that meet the budget constraint.
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