Problem 55 - Introductory and Intermediate Algebra for College Students 4th

Question

The formula \(S=C(1+r)^{t}\) models inflation, where \(C=\) the value today, \(r=\) the anmual 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 will be worth rounded to the nearest dollar after 10 years of inflation at 6% per annum.

Step 1: Identify the values from the problem

From the problem, it can be seen that the current value of the house (C) is $465,000, the annual inflation rate (r) is 6% or 0.06 when expressed as a decimal, and the time period (t) is 10 years. The goal is to find the inflated value (S) of the house after 10 years.

Step 2: Substitute the values into the formula

Now that we have values for \(C\) , \(r\) and \(t\) we substitute these into the formula \(S=C(1+r)^{t}\) which gives \(S=465000*(1+0.06)^{10}\).

Step 3: Calculate the inflated value

Calculate the value of \(S\) by performing the operations in the correct order (brackets first, then exponentiation, then multiplication). This gives a final value for \(S\).

Key Concepts

Annual Inflation Rate

The annual inflation rate represents the percentage by which prices increase over a year. It indicates how much the cost of goods and services rises, typically expressed as a percentage. This key economic indicator shows the rate at which currency loses purchasing power.

To calculate the annual inflation rate, you take the increase in prices over a year and divide it by the starting price. For an accurate depiction, it is often transformed into a decimal when inserted into mathematical equations. For instance, a 6% inflation rate is written as 0.06.

• Importance: Helps businesses and individuals forecast future expenses.
• Example: If inflation is 6%, it means that on average, prices will increase by 6% over the year.

Understanding the annual inflation rate is crucial when predicting the future cost of investments, assets, or everyday goods.

Future Value Calculation

Future value calculation allows individuals and businesses to anticipate the worth of an asset or investment over time, considering inflation or interest rates. It helps make informed financial decisions by estimating how much a sum of money invested today will be worth in the future.

The future value formula used in the problem is an essential tool:
\[ S = C(1+r)^{t} \]

Where:

• \(S\) is the future value.
• \(C\) is the present or current value.
• \(r\) is the annual inflation rate.
• \(t\) is the number of years.

The formula demonstrates how the current value is adjusted for inflation over a period of time. For example, in the given exercise, calculating how much the house will be worth in the future considering a 6% inflation rate over ten years. This empowers individuals to evaluate long-term financial goals and the future impact of economic changes on their investments.

Compound Interest Formula

The compound interest formula is closely related to future value calculations, used to determine how much an investment or asset will grow over time. Although traditionally associated with calculating interest on savings or investments, its principles apply to inflation calculations too.

When using the compound interest formula in the context of inflation:
\[ S = C(1+r)^{t} \]

Here, the formula accounts for compound interest by compounding the growth for each year over the specified time horizon. This means each year, the interest (or in this case, inflation) is calculated not only on the initial principal but also on the accumulated interest.

• Significance: Helps understand the potential growth of investments or increase in expenses due to inflation.
• Application: Used in financial forecasting and planning.

Thus, understanding and using the compound interest formula is vital for anticipating how values grow exponentially over time.