Problem 37 - Algebra 2 and Trigonometry

Question

Al invested \(\$ 3,000\) in a certificate of deposit that pays 5\(\%\) interest per year. What is the value of the investment at the end of each of the first four years?

Step-by-Step Solution

Answer

Year 1: $3,150; Year 2: $3,307.50; Year 3: $3,472.88; Year 4: $3,646.29.

Step 1: Understand the Problem

Al invested \( \$ 3,000 \) in a certificate of deposit with an annual interest rate of 5\(\%\). We need to find the value of the investment at the end of each of the first four years.

Step 2: Formula for Compound Interest

Since the interest is compounded annually, we will use the formula for compound interest: \[ A = P (1 + r)^n \] where: - \( A \) is the amount of money accumulated after n years, including interest.- \( P \) is the principal amount (initial investment).- \( r \) is the annual interest rate (decimal).- \( n \) is the number of years.

Step 3: Convert the Interest Rate to Decimal

The annual interest rate is 5\(\%\). To convert this to a decimal, divide by 100, so \( r = 0.05 \).

Step 4: Calculate the Value at the End of Year 1

Substitute \( P = 3,000 \), \( r = 0.05 \), and \( n = 1 \) into the formula:\[ A = 3,000 (1 + 0.05)^1 \]\[ A = 3,000 (1.05) \]\[ A = 3,150 \]The value at the end of year 1 is \( \$ 3,150 \).

Step 5: Calculate the Value at the End of Year 2

Repeat the calculation for \( n = 2 \):\[ A = 3,000 (1.05)^2 \]\[ A = 3,000 (1.1025) \]\[ A = 3,307.50 \]The value at the end of year 2 is \( \$ 3,307.50 \).

Step 6: Calculate the Value at the End of Year 3

Repeat the calculation for \( n = 3 \):\[ A = 3,000 (1.05)^3 \]\[ A = 3,000 (1.157625) \]\[ A = 3,472.88 \]The value at the end of year 3 is \( \$ 3,472.88 \).

Step 7: Calculate the Value at the End of Year 4

Repeat the calculation for \( n = 4 \):\[ A = 3,000 (1.05)^4 \]\[ A = 3,000 (1.21550625) \]\[ A = 3,646.29 \]The value at the end of year 4 is \( \$ 3,646.29 \).

Key Concepts

Understanding Annual Interest Rate

When dealing with financial mathematics, it's crucial to understand what an annual interest rate is. The annual interest rate is a percentage that tells you how much extra money you'll earn or owe on a principal amount over a year. For instance, in Al's case, he invested in a certificate of deposit with an annual interest rate of 5%. This means that every year, his investment grows by 5% of the initial or subsequent total.
The annual interest rate can also be converted into a decimal format for easy calculations. To do this, simply divide the percentage by 100. So, a 5% interest becomes 0.05 in decimal. This decimal is then used in compound interest calculations, making it fundamental for predicting investment growth.

Defining Principal Amount

The principal amount refers to the initial sum of money invested or loaned, excluding any interest earned or owed. In Al's situation, his principal amount is the \( \$ 3,000 \) he initially invested in the certificate of deposit.
Understanding the principal amount is key in financial mathematics because this figure serves as the foundation upon which interest accumulates in investment scenarios.

• If you increase the principal amount, you typically see a larger final amount due to more interest being earned.
• Conversely, a smaller principal results in a smaller accumulated amount over the same period and interest rate.

This core value is constant unless you make additional investments or withdrawals.

Calculating Investment Value Over Time

Investment value over time refers to how much investment is worth after accumulating interest over a specified period. Utilizing the compound interest formula, one can track this growth annually. The formula is: \[ A = P (1 + r)^n \] where:

• \( A \) is the amount accumulated after \( n \) years.
• \( P \) is the principal amount (initial investment).
• \( r \) is the annual interest rate as a decimal.
• \( n \) is the number of years.

For example, using Al's investment:

• End of Year 1: \( A = 3,000 (1.05)^1 = 3,150 \)
• End of Year 2: \( A = 3,000 (1.05)^2 = 3,307.50 \)
• End of Year 3: \( A = 3,000 (1.05)^3 = 3,472.88 \)
• End of Year 4: \( A = 3,000 (1.05)^4 = 3,646.29 \)

Each year, the investment's value grows due to compound interest.

Exploring Financial Mathematics

Financial mathematics is a vast field covering various quantitative methods to analyze financial markets and investment strategies. A core concept in this field is compound interest, which demonstrates how an investment can grow exponentially over time.
By leveraging the compound interest formula, individuals can understand the future value of their investments or loans. Through this understanding:

• Investors can make informed decisions about where to put their money.
• Borrowers can plan better by knowing potential debt growth over time.

With the aid of financial mathematics, one can employ strategies to maximize returns or minimize costs. It extends beyond simple interest calculations to include analyzing risks, setting goals, and ensuring sound financial planning.