Problem 81 - College Algebra with Modeling and Visualization

Question

Suppose that tuition is initially \(\$ 100\) per credit and increases by \(6 \%\) from the first year to the second year. What is the cost of tuition the second year? Now suppose that tuition decreases by \(6 \%\) from the second to the third year. Is tuition equal to \(\$ 100\) per credit the third year? Explain.

Step-by-Step Solution

Answer

The third-year tuition is $99.64 per credit, which is not equal to $100.

Step 1: Calculate the First Year Increase

The tuition initially is \(100 per credit. The increase from the first year to the second year is 6%. To calculate the new tuition, we multiply the initial tuition by 1 plus the rate of increase:\[\text{New Tuition} = 100 \times (1 + 0.06) = 100 \times 1.06 = 106\]So, the cost of tuition the second year is \)106 per credit.

Step 2: Calculate the Second Year Decrease

Now, the tuition decreases by 6% from the second year to the third year. To calculate this decrease, we multiply the second-year tuition by 1 minus the rate of decrease:\[\text{Third Year Tuition} = 106 \times (1 - 0.06) = 106 \times 0.94\]Calculating this gives:\[\text{Third Year Tuition} = 99.64\]So, the cost of tuition the third year is $99.64 per credit.

Step 3: Compare to Initial Tuition

We now compare the tuition of the third year, which is $99.64, to the initial tuition which was $100. Clearly, $99.64 is not equal to $100.

Step 4: Conclusion

The tuition of $99.64 is not equal to the initial $100 per credit. Therefore, after a 6% increase followed by a 6% decrease, the price per credit at the third year is slightly less than the original tuition.

Key Concepts

Understanding Tuition Costs

Tuition refers to the money charged for teaching or instruction, typically at a college or university. In our example, the tuition was initially set at $100 per credit. This fee is what students pay for taking one credit of a course. It's important to note that several credits typically make up a complete course.

As tuition is a significant part of college expenses, understanding its dynamics, including potential hikes and reductions, can help in planning educational budgets. An increase or decrease in tuition rates can happen due to various factors, such as changes in institutional policies or economic conditions.

Calculating Credit Cost

Credit cost essentially refers to the price associated with taking a single credit of a course. In the context of higher education, courses are usually measured in terms of credits, with a standard course potentially being worth several credits.

In our case, the original credit cost was $100. When calculating the cost per credit after an increase or decrease, it's vital to apply the correct percentage changes correctly. For example:

• To calculate an increase, multiply the original cost by \(1 + ext{percentage increase as a decimal}\).
• To calculate a decrease, multiply the new cost by \(1 - ext{percentage decrease as a decimal}\).

These calculations help in understanding how much more or less will be paid per credit in subsequent years.

Impact of Increase and Decrease Percentage on Price

An increase or decrease in percentage affects the price significantly. Understanding this is crucial when dealing with tuition fees. Let's break it down:

The initial tuition was \(100, and it increased by 6%. To find the new tuition for the second year, apply the increase: \[ 100 \times (1 + 0.06) = 106 \]. So, the cost went up to \)106 per credit.

Following this, the tuition decreased by 6% for the third year. It's essential to note that the percentage decrease applies to the new value, not the original one. Thus, the third-year cost is calculated as: \[ 106 \times (1 - 0.06) = 99.64 \].

The decrease does not bring the tuition back to the original amount of \(100, but rather to \)99.64. This illustrates that an increase and then a decrease by the same percentage does not revert the value to the original number. Instead, it results in a small net change in the opposite direction of the initial change.