A group of students decide to attend a concert. The cost of renting a bus to take them to the concert is $450, which is to be shared equally among the students. The concert promoters offer discounts to groups arriving by bus. Tickets normally costs $59 each but are reduced $0.10 per ticket for each person in the group (up to the maximum capacity of the bus). How many students must be in the group for the total cost per student to be less than $54?

Question

A group of students decide to attend a concert. The cost of renting a bus to take them to the concert is $450, which is to be shared equally among the students. The concert promoters offer discounts to groups arriving by bus. Tickets normally costs $59 each but are reduced $0.10 per ticket for each person in the group (up to the maximum capacity of the bus). How many students must be in the group for the total cost per student to be less than $54?

Answer 1

5/0.1 = 50

Answer 2

Still need an equation for all the students, and the fixed cost for the bus.

Answer 3

To find the number of students needed for the total cost per student to be less than $54, we need to calculate the total cost and divide it by the number of students.

Let's break down the costs involved:

1. Bus rental cost: $450
The cost of renting the bus is $450, and this cost is shared equally among the students in the group.

2. Ticket cost: Discounted based on the number of students in the group
The normal cost of a ticket is $59. However, the cost is reduced by $0.10 for each person in the group, up to the maximum capacity of the bus. This means that for each additional student in the group, the cost of each ticket is reduced by $0.10.

To find the total cost per student, we will sum the bus rental cost and the total ticket cost, and then divide it by the number of students.

Let's calculate the ticket cost first:
- For the first student, the ticket cost is $59.
- For the second student, the ticket cost is reduced by $0.10, so it becomes $59 - $0.10 = $58.90.
- For the third student, the ticket cost is further reduced by $0.10, so it becomes $58.90 - $0.10 = $58.80.

This pattern continues, and we see that for each additional student, the ticket cost decreases by $0.10. We can express this as follows:
- For the n-th student, the ticket cost is $59 - $0.10 * (n - 1) = $59 - $0.10n + $0.10 = $59 - $0.10n + $0.10 = $59 - $0.10n + $0.10n - $0.10 = $59 - $0.10.

Now that we have the ticket cost for each student, let's calculate the total cost and determine the number of students needed:

Total cost = Bus rental cost + Ticket cost
Total cost = $450 + ($59 - $0.10n)

To find when the total cost per student is less than $54, we will divide the total cost by the number of students, and set it to be less than $54:

Total cost / n < $54

Substituting the expressions for the total cost and the ticket cost into the equation:

($450 + ($59 - $0.10n)) / n < $54

To solve this equation, we can multiply both sides by n to eliminate the fraction:

$450 + ($59 - $0.10n) < $54n

Now, let's simplify the equation:

$450 + $59 - $0.10n < $54n

$509 - $0.10n < $54n

$509 < $54n + $0.10n

$509 < $54.10n

Now, let's solve for n:

$509 / $54.10 < n

n > 9.41

Since n represents the number of students and cannot be a fraction, we round up to the nearest whole number:

n > 10

Therefore, to ensure that the total cost per student is less than $54, there must be at least 11 students in the group.