Problem 47 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The Kwans are planning to buy a house 4 yr from now. Housing experts in their area have estimated that the cost of a home will increase at a rate of \(5 \%\) year during that period. If this economic prediction holds true, how much can the Kwans expect to pay for a house that currently costs $$\$ 210,000$$ ?

Step-by-Step Solution

Answer

The Kwans can expect to pay approximately \$255,306.31 for the house 4 years from now if the economic prediction holds true.

Step 1: Identify the given information

We have the following data: - The current cost of the house: \(P = \$ 210,000\) - The annual increase rate (interest rate): \(r = 5\% = 0.05\) - The number of years: \(t = 4\)

Step 2: Use the compound interest formula

We will use the compound interest formula to calculate the future cost of the house: \(A = P(1 + r)^t\) where: - \(A\) is the future cost of the house - \(P\) is the current cost of the house - \(r\) is the annual interest rate - \(t\) is the number of years

Step 3: Plug in the given values

Substitute the given values into the formula: \(A = \$210,000(1 + 0.05)^4\)

Step 4: Calculate the future cost

Perform the calculations: \(A = \$210,000(1.05)^4\) \(A = \$210,000(1.21550625)\) \(A ≈ \$255,306.31\) The Kwans can expect to pay approximately \$255,306.31 for the house 4 years from now if the economic prediction holds true.

Key Concepts

Future Value Calculation

When we talk about future value calculation, we are essentially determining how much an amount today will grow to over a period under certain conditions. This is particularly important in investments or planning, where you want to know the outcome of certain financial decisions over time.

To calculate future value, we often use the compound interest formula, because it accounts for interest being added not just on the initial amount (or principal) but also on any accumulated interest from previous periods. The formula is:

• The future value (\(A\)) depends on your current investment (\(P\)), the interest rate (\(r\)), and time (\(t\)).
• The formula is: \[A = P(1 + r)^t\]This reflects how investment grows year after year with interest compounding on itself.

For example, the Kwans want to know what their house will cost in four years. With a house currently priced at \(\$210,000\) and an estimated increase at \(5\%\) annually, the formula allows them to project the cost effectively, giving them a financial goal to prepare for.

Interest Rate

The interest rate is a critical factor in how investments grow over time. It is expressed as a percentage and is typically annualized. This means it tells you how much of your investment will grow every year.

Interest rates can compound, meaning that each year they apply not only to the original principal, but also to any interest that accumulates over the previous years. This is why even small interest rates can lead to significant growth over time.

• In our exercise, the rate is \(5\%\) per year, an increase applied annually to the house price.
• This rate compounds, affecting how much more you will need to save or prepare per year if you want to buy at a future date.

Understanding interest rates can help you make informed decisions whether you're saving, borrowing, or investing, as it directly influences how much you pay or earn.

Investment Growth

Investment growth is the outcome of applying a constant interest rate over time to an initial sum. It represents how much more money you will have in the future compared to now.

When an investment grows, it means that it's increasing in value. For many financial goals, understanding and predicting this growth is crucial. It lets you predict how much an asset will cost or how much an investment could yield.

• The growth in the Kwans’ house price is an example of a common form of investment growth, applying the concepts of future value and interest rate to understand financial outcomes.
• Even in personal finance, recognizing how your savings or costs grow can help shape savings strategies and budget preparations.
• It’s critical to acknowledge that investment growth depends heavily on consistent rates and time, both of which magnify value or cost projections over longer periods.

With these ideas, you can better plan for future financial conditions, ensuring that you are ready for any costs and expenses that arise.