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Algebra Difficulty 2.0 Junior Prove it Canada

A company rents out various sized passenger vehicles according to the following table. For example, a group of $5,6,7$ , or 8 people would need to rent a sports utility vehicle (SUV), which has a total cost of $200.00. Unfortunately, the total cost to rent a van is missing from the table. In each case, the members of the group equally share the total cost to rent the vehicle.

Vehicle
Number of Passengers Required
Total Cost

Car
1 to 4
$180.00

SUV
5 to 8
$200.00

Van
9 to 12

If 4 people rent a car, what is the cost per person?
If a group rents an SUV, what is the maximum possible cost per person?
When a van is rented, the difference between the maximum cost per person and the minimum cost per person is $6.00. Determine the total cost to rent a van.

Solution

The total cost to rent a car is $180.00.
If 4 people rent a car, the cost per person is $\dfrac{\$180.00}{4}=\$45.00$ .
Since the members of the group equally share the total cost to rent the vehicle, the smaller the group, the greater the cost per person.

To rent an SUV, the smallest group size required is 5 passengers and the total cost is $200.00.
Thus, the maximum possible cost per person to rent an SUV is $\dfrac{\$200.00}{5}=\$40.00$ .
Let the total cost to rent a van be $v$ .

When renting a van, the maximum possible cost per person occurs when the number of

passengers is 9 (the fewest number possible), and so this maximum cost is $\dfrac{v}{9}$ .

The minimum possible cost per person occurs when the number of passengers is 12 (the

greatest number possible), and so this minimum cost is $\dfrac{v}{12}$ .

Then, $\dfrac{v}{9}-\dfrac{v}{12}=\$6.00$ or $\dfrac{4v-3v}{36}=\$6.00$ , and so $v=\$6.00\times36$ .
Thus, the total cost to rent a van is $216.00.

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.