Question

College Costs The average yearly cost of tuition, fees, and room and board at private four-year colleges in the United States was \(\$ 19,929\) for the academic year \(1998 / 1999\) and \(\$ 26,489\) for the academic year \(2004 / 2005\). Find the average yearly rate of inflation over this sevenyear period. (Source: U.S. National Center for Education Statistics)

Short answer

The average yearly rate of inflation over the seven-year period is approximately 4.7%

Step-by-step solution

Identify the Initial and Final Values

From the problem, the average cost for the academic year 1998/1999 which is the initial value is $19,929, while the average cost for the academic year 2004/2005, the final value, is $26,489.

Implement the Inflation Formula

Substitute the given values into the inflation formula: \[Inflation Rate = \frac{$26,489-$19,929}{$19,929} * \frac{100}{7}\]

Simplify the Expression

Do the operations as indicated. When the calculations are done, this gives an inflation rate of approximately 4.7%

Key concepts

Inflation Formula

Understanding the inflation formula is crucial when it comes to evaluating how prices rise over time due to economic changes. Inflation itself is the rate at which the general level of prices for goods and services is rising, and subsequently, purchasing power is falling. The formula for calculating the average yearly rate of inflation involves two main components: the final value (price at the end of the period), and the initial value (price at the start of the period).

The standard inflation formula is:
\[ \text{Inflation Rate} = \frac{(\text{Final Value} - \text{Initial Value})}{\text{Initial Value}} \times 100 \% \]
We express inflation as a percentage because it gives a clear sense of the relative change in price levels. To determine an average annual rate, we also need to take into account the time period over which the prices have changed. For multi-year periods, you would divide the result by the number of years to get the average annual rate. In our exercise, the time frame is seven years, so the formula has an additional division by 7.

Simplifying this concept, when looking to find out how much more expensive your college tuition is now than it was in the past, you'd use the inflation formula to quantify that change as an average percentage for each year.

Simplifying Expressions

Simplifying expressions is a foundational skill in algebra, vital for making complex calculations understandable and manageable. It involves reducing an expression to its most basic form without changing its value. Like solving a puzzle, simplifying can involve various steps such as combining like terms, using the distributive property, and canceling out terms that appear on both sides of the equation.

When we simplify an expression, we make the calculation clearer and easier to work with. This is particularly important when dealing with financial formulas, such as the inflation rate calculation covered in our previous section. For instance, once you've plugged in your initial and final values into the inflation formula, your job is to simplify the fraction and multiply by the percentage factor. This may involve subtracting the initial value from the final value, reducing the fraction, and carrying out the percentage calculation by multiplying by 100. In the context of our exercise, the simplifying process gets us closer to understanding the actual percentage that represents the average yearly rate of inflation over a given period.

Percentage Rate Calculation

Comprehending how to calculate percentage rates is crucial in different domains like finance, statistics, and even everyday shopping discounts. A percentage is a way of expressing a number as a fraction of 100. It is denoted using the percent symbol (%).

Calculating the percentage rate often involves a comparison between two different values to find out what percent one value is of another. The basic formula for finding a percentage is:
\[ \text{Percentage} = \frac{(\text{Part})}{(\text{Whole})} \times 100 \% \]
In our exercise, the calculation of the inflation rate is essentially a percentage rate calculation where the 'part' is the difference between the final and initial college costs, and the 'whole' is the initial cost. After simplifying the numerical expression as described before, the final step is to multiply the result by 100 to convert it into a percentage. This conveys the inflation rate in terms that are immediately recognizable and comparable. This kind of calculation is not just used in the context of inflation but also in finding discounts, bank interest rates, and demographic statistics, among other uses.