Problem 54 - Algebra and Trigonometry

Question

Compound Interest If \(\$ 4000\) is borrowed at a rate of 5.75\(\%\) interest per year, compounded quarterly, find the amount due at the end of the given number of years. $$ \begin{array}{llll}{\text { (a) } 4 \text { years }} & {\text { (b) } 6 \text { years }} & {\text { (c) } 8 \text { years }}\end{array} $$

Step-by-Step Solution

Answer

(a) $4985.05, (b) $5730.07, (c) $6586.82.

Step 1: Understand Compound Interest Formula

Compound interest can be calculated using the formula \( A = P \left(1 + \frac{r}{n}\right)^{nt} \), where \( A \) is the future value of the investment/loan, including interest, \( P \) is the principal investment amount (\(4000\) in this case), \( r \) is the annual interest rate (decimal), \( n \) is the number of times that interest is compounded per year, and \( t \) is the time the money is invested or borrowed for in years.

Step 2: Define Variables

Given: \( P = 4000 \), annual interest rate \( r = 5.75\% = 0.0575 \), compounded quarterly \( n = 4 \). We will solve for \( t = 4, 6, \) and \( 8 \) years.

Step 3: Calculate for 4 Years

Substitute the values for 4 years: \[ A = 4000 \left(1 + \frac{0.0575}{4}\right)^{4 \times 4} \]. Calculate step-by-step: \[ \frac{0.0575}{4} = 0.014375 \]. Then, \[ 1 + 0.014375 = 1.014375 \] and \[ A = 4000 (1.014375)^{16} \]. Calculate \( (1.014375)^{16} \approx 1.246263 \). Finally, \( A \approx 4000 \times 1.246263 = 4985.05 \).

Step 4: Calculate for 6 Years

Substitute the values for 6 years: \[ A = 4000 \left(1 + \frac{0.0575}{4}\right)^{4 \times 6} \]. Calculate step-by-step: \[ A = 4000 (1.014375)^{24} \]. Calculate \( (1.014375)^{24} \approx 1.432517 \). Finally, \( A \approx 4000 \times 1.432517 = 5730.07 \).

Step 5: Calculate for 8 Years

Substitute the values for 8 years: \[ A = 4000 \left(1 + \frac{0.0575}{4}\right)^{4 \times 8} \]. Calculate step-by-step: \[ A = 4000 (1.014375)^{32} \]. Calculate \( (1.014375)^{32} \approx 1.646706 \). Finally, \( A \approx 4000 \times 1.646706 = 6586.82 \).

Key Concepts

Interest Rate

The interest rate is a crucial component in the world of finance, particularly in calculating compound interest. This rate represents the percentage of the principal amount that is charged as interest over a specific period. For example, in our exercise, the interest rate is 5.75% per annum. Taking this percentage and converting it into a decimal format is essential for calculation; in this case, it is 0.0575.

• An interest rate can vary depending on the financial institution and the type of financial product.
• In compound interest scenarios, the rate at which it compounds can significantly impact the future value of the investment or loan.
• Understanding how interest rates affect growth can aid in making informed financial decisions.

Putting it simply, the interest rate is like a "speed limit" that dictates how fast your money can grow over time.

Future Value

The future value (FV) of an investment is the amount of money an investment will grow to over time at a given interest rate and compounding period. It's essentially the goal amount you hope to have after a certain period. In compound interest calculations, FV accounts for the initial principal and the accumulated interest over time.

• It helps to predict the financial outcome based on the current investment parameters.
• Future value calculations can assist individuals and businesses in planning their savings and investments.
• The more frequent the compounding, the greater the future value will generally be.

In our exercise, we calculate the future value for different years to understand how much the investment of $4000 will grow with the given rate and compounding frequency.

Principal Amount

The principal amount is the initial sum of money put into an investment or loan, upon which the interest is calculated. It is the starting point for any financial growth involving interest calculations. In our problem, the principal amount is $4000.

This principal plays a pivotal role as:

• It directly influences the amount of interest earned or paid. The larger the principal, the greater the potential for higher returns, given the same interest rate and time.
• When considering loans, it's the base sum borrowed that's to be repaid over time. With investments, it's the amount contributed at the start.

Understanding this concept is fundamental, as all interest calculations build upon this base figure.

Compounding Period

The compounding period refers to the frequency with which the interest is calculated and added to the principal balance within a year. It has a significant effect on the total amount of interest accrued because interest is being calculated on an ever-increasing amount.
For instance, compounding can be:

• Annually
• Semi-annually
• Quarterly
• Monthly
• Daily

In our exercise, interest is compounded quarterly, meaning it is calculated and added four times a year.

More frequent compounding intervals result in more frequent additions of interest to the principal balance, thereby accelerating the growth of the investment or loan. Understanding this concept is essential for accurately predicting how quickly your money can grow.