Problem 16 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the accumulated amount \(A\) if the principal \(P\) is invested at the interest rate of \(r /\) year for \(t\) yr. $$ P=\$ 42,000, r=7 \frac{3}{4} \%, t=8, \text { compounded quarterly } $$

Step-by-Step Solution

Answer

The accumulated amount after 8 years, with the investment compounded quarterly, is approximately $65,493.93.

Step 1: 1. Convert the given values into the appropriate units

The given values are: - Principal amount P = $42,000 - Annual interest rate r = 7(3/4) % = 7.75% - Time t = 8 years - Compounded quarterly, so n = 4 times per year To use the formula, we need to convert the annual interest rate into decimal form. \[r = 7.75\% = 0.0775\]

Step 2: 2. Plug in the given values into the formula

Now, we will insert the given values into the formula: \[A = 42000\left(1 + \frac{0.0775}{4}\right)^{4 \times 8}\]

Step 3: 3. Calculate the accumulated amount

Now, solve the equation to find the accumulated amount (A) after 8 years: \[A = 42000\left(1 + \frac{0.0775}{4}\right)^{32}\] \[A = 42000\left(1 + 0.019375\right)^{32}\] \[A = 42000 \times 1.019375^{32}\] \[A \approx 65493.93\] The accumulated amount after 8 years, with the investment compounded quarterly, is approximately $65,493.93.

Key Concepts

Applied Mathematics

Applied Mathematics is an invaluable tool in understanding the patterns of financial growth and the calculations involved in compound interest. It's about taking mathematical theories and methods and using them to solve practical problems in business, finance, engineering, and beyond.

In the context of finance, it aids in creating models to predict future values of investments and to grasp the deeper implications of interest rates. As seen in our exercise, Applied Mathematics allows us to convert a yearly interest rate into a quarterly one, and to calculate the accumulated amount of an investment over time. Having a solid understanding of Applied Mathematics equips us to navigate the financial world more effectively by making well-informed decisions based on precise calculations.

Future Value of Investment

The Future Value of Investment is a concept that predicts how much a current investment will be worth at a specific time in the future, accounting for interest. This is directly tied to the idea that money available today can be invested to earn additional money over time.

To determine this future value, we use formulas that involve the current principal amount, the annual interest rate, the frequency of compounding, and the total time period. In the given exercise, knowing the future value after 8 years helps in planning for financial goals, such as retirement savings or educational funds. It serves as a cornerstone for financial planning and investment strategies.

Compound Interest Formula

The Compound Interest Formula is essential for calculating the interest earned on an investment where the interest is reinvested to earn additional interest. The formula typically used is:
\[A = P(1 + \frac{r}{n})^{n \times t}\]
where \(A\) is the amount of money accumulated after \(n\) years, including interest, \(P\) is the principal amount, \(r\) is the annual interest rate (in decimal), \(n\) is the number of times that interest is compounded per period, and \(t\) is the time the money is invested for.

In practical terms, the formula shows how investments grow over time due to the effects of compounding. When applied with proper values, as in our exercise, it yields the future value of the investment.

Time Value of Money

The Time Value of Money is a financial principle stating that a specific amount of money today has a different value than the same amount of money in the future due to its earning potential. This principle underlines the importance of earning interest and demonstrates why individuals might prefer to receive money today rather than the same amount in the future.

With compound interest, the time value of money exhibits exponential growth because not only does the initial investment earn interest, but the accumulated interest earns interest as well over time. This exponential growth, captured in our problem where quarterly compounding over 8 years significantly increases the value of the initial investment, illustrates the power of time when combined with the potential for earning interest.