Alana can spend $40 on carnival tickets. Ride tickets cost $4 each, and food tickets cost $2 each. Alana buys at least 16 tickets. The system of inequalities represents the number of ride tickets, $r$, and the number of food tickets, $f$, she buys.
[tex]
\begin{array}{r}
r+f \geq 16 \\
4 r+2 f \leq 40
\end{array}
[/tex]
What is the maximum number of ride tickets she can buy?
Answer (1)
Simplify the budget constraint: 2 r + f ≤ 20 .
Express the minimum ticket purchase as: f ≥ 16 − r .
Combine inequalities to find the upper bound for ride tickets: r ≤ 4 .
The maximum number of ride tickets Alana can buy is 4 .
Explanation
Understanding the Problem We are given the following system of inequalities: r + f ≥ 16 4 r + 2 f ≤ 40 where r is the number of ride tickets and f is the number of food tickets. We want to find the maximum number of ride tickets Alana can buy.
Simplifying the Inequalities First, let's simplify the second inequality by dividing both sides by 2: 2 r + f ≤ 20
Isolating f in the First Inequality Now, we can express f in terms of r for both inequalities. From the first inequality, we have: f ≥ 16 − r
Isolating f in the Second Inequality From the simplified second inequality, we have: f ≤ 20 − 2 r
Combining the Inequalities Combining these two inequalities, we get: 16 − r ≤ f ≤ 20 − 2 r
Setting up the Inequality for r This implies that: 16 − r ≤ 20 − 2 r
Solving for r Now, let's solve for r :
2 r − r ≤ 20 − 16 r ≤ 4
Checking the Solution Since r must be a non-negative integer, the maximum value for r is 4. Let's check if this value satisfies the conditions. If r = 4 , then: f ≥ 16 − 4 = 12 f ≤ 20 − 2 ( 4 ) = 20 − 8 = 12 So, f = 12 . This means Alana can buy 4 ride tickets and 12 food tickets. The total cost would be 4 ( 4 ) + 2 ( 12 ) = 16 + 24 = 40 , which is within her budget. Also, 4 + 12 = 16 , which satisfies the condition that she buys at least 16 tickets.
Final Answer Therefore, the maximum number of ride tickets Alana can buy is 4.
Examples
Imagine Alana is planning a school carnival. She needs to figure out how many ride tickets and food tickets she can buy with her budget. This problem helps her understand how to maximize the number of ride tickets while staying within her budget and meeting the minimum ticket requirement. By using inequalities, Alana can determine the optimal combination of ride and food tickets, ensuring she has the most fun at the carnival without overspending. This is a practical application of linear inequalities in everyday decision-making.
