Problem 67 - College Algebra Essentials

Question

The formula \(S=C(1+r)^{t}\) models inflation, where \(C=\) the value today, \(r=\) the annual inflation rate, and \(S=\) the inflated value \(t\) years from now. Use this formula to solve. Round answers to the nearest dollar. If the inflation rate is \(6 \%,\) how much will a house now worth \(\$ 465,000\) be worth in 10 years?

Step-by-Step Solution

Answer

The house that's worth \(\$ 465,000\) now will be worth around \(\$ 877,447\) in 10 years, after adjusting for an annual inflation rate of 6%.

Step 1: Identify the given values

In the problem, they provide the current value of the house (C) as \(\$ 465,000\), the inflation rate (r) as \(6 \%\), and the period of time (t) as 10 years. As the given rate is percentage, convert it to a decimal by dividing by 100. Hence, the rate \(r = 0.06\).

Step 2: Substitute the given values into the formula

Now subtitute the values into the formula: \(S = C(1+r)^{t} = 465000(1+0.06)^{10}\).

Step 3: Solve the expression

Calculate the expression to find the inflated value of the house in 10 years. Remember to first solve the operation inside the parentheses then raise it to the power of 10. Multiply the result with the current value of the house \(C=465,000\).

Step 4: Round off the answer

The solution will be a decimal number. However, since the final answer needs to be rounded off to the nearest dollar, round off the calculated value.

Key Concepts

Exponential Growth

Understanding exponential growth is vital when estimating how investments or prices will increase over time. It describes a situation where the growth rate of a mathematical function becomes more rapid in proportion to the growing total number or size. It's akin to a snowball effect; the bigger the snowball becomes, the faster it can collect more snow.
With inflation, we often use exponential growth to model how the value of an item like a house increases over time. As inflation continues year after year, the house doesn't just gain value equal to the first year's inflation multiplied by the number of years. Instead, each year's added value also grows by inflation, leading to a larger and larger increase as time passes.
In the exercise, the formula for inflation models this concept by raising the sum of 1 plus the inflation rate to the power of the number of years. The inflation rate, acting as the growth rate in the formula, must be used as a decimal to accurately represent its proportional effect on the current value, leading to the exponential increase in price.

Future Value Formula

The future value formula is a fundamental equation in finance, particularly when it comes to understanding the impact of inflation on savings and investments. It calculates the future value of an asset based on its present value, the time period considered, and the rate of growth or interest. In this case, the formula is used to predict how much a house will be worth after a certain number of years, considering the constant annual inflation rate.
To break down the formula:

• \(C\) stands for the current value of the house,
• \(r\) is the annual inflation rate converted to a decimal, and
• \(t\) is the number of years into the future you want to calculate the value for.

So, by substituting our values into the future value formula, we get a precise estimate of the house's inflated value in 10 years. This allows homeowners, real estate investors, and economists to make informed decisions about the potential future cost or the right timing for investments.

Percent to Decimal Conversion

Converting a percent to a decimal is an essential skill that simplifies many mathematical calculations, including those involving the future value formula. Percentages are based on the concept that a whole is divided into 100 equal parts. So, when it comes to conversion, you are essentially moving the decimal place two places to the left. This is the same as dividing the percentage by 100.
For example, a 6% inflation rate means 6 out of every 100 units of currency. To convert this to a decimal, which is needed for formulas like the future value formula, you divide 6 by 100, which gives you 0.06. This conversion is crucial because it translates the percentage into the proportionate amount that is applied in the formula's calculations.
This step may appear minor, but it's critical for achieving accurate results. When dealing with financial formulas, forgetting to convert a percent to a decimal can lead to significant errors in your final calculations, as the entire formula depends on using proportions correctly.