Problem 4 - Algebra for College Students

Question

Use the formula \(A=P\left(1+\frac{r}{n}\right)^{n t}\) to find the total amount of money accumulated at the end of the indicated time period for each of the following investments. $$ \$ 250 \text { for } 5 \text { years at } 7 \% \text { compounded annually } $$

Step-by-Step Solution

Answer

The accumulated amount is approximately $350.64.

Step 1: Identify Given Values

We have the principal amount \( P = 250 \), the rate \( r = 0.07 \) (given as 7%), the number of times interest is compounded per year \( n = 1 \) (since it's compounded annually), and the time period \( t = 5 \) years.

Step 2: Substitute Values into Formula

Substitute the given values into the compound interest formula: \[ A = 250 \left(1 + \frac{0.07}{1}\right)^{1 \times 5} \]

Step 3: Simplify Inside the Parenthesis

Calculate the expression inside the parenthesis: \[ 1 + \frac{0.07}{1} = 1.07 \]

Step 4: Calculate the Exponent

Raise the result to the power of \( n \times t \), which is \( 5 \) in this case: \[ (1.07)^5 \]

Step 5: Evaluate the Power

Calculate \( (1.07)^5 \) using a calculator:\[ (1.07)^5 \approx 1.402551 \]

Step 6: Multiply to Find the Total Accumulated Amount

Multiply this result by \( P \) to find \( A \): \[ A = 250 \times 1.402551 \approx 350.64 \]

Step 7: Conclude with the Result

Thus, the total amount of money accumulated after 5 years will be approximately \( \$350.64 \).

Key Concepts

Principal Amount

The principal amount is the initial sum of money that you invest or borrow. It's the starting point for your journey in growing your savings or paying off a loan. In the context of our exercise, the principal amount is $250. This is the base off of which your compounded interest will grow.
Understanding the principal amount is crucial because it directly influences how much interest you will gain or need to pay over time.

• A larger principal will earn more interest, making savings grow faster.
• Conversely, a larger principal will also mean more interest to pay if it's a loan.

Therefore, knowing your principal amount helps in planning your finances efficiently. Always keep an eye on it, as it's the foundation of all your interest calculations.

Interest Rate

The interest rate is a percentage that tells you how much extra you'll earn on your principal over a period of time, usually one year. In our formula, the interest rate is represented as a decimal, so 7% becomes 0.07.
No need to stress if you aren't familiar with these conversions; just remember to move the decimal point two places to the left.

• A higher interest rate means more earnings on your investments.
• Understanding how rates work helps you compare different investment or loan options.

By knowing how interest rates affect your principal, you can make better financial decisions that align with your goals.

Time Period

The time period is the length for which you let your investment grow or a loan accrue interest. In our example, the investment lasts for 5 years. Every year the money sits invested, it's busy working to make more money.
Time is an essential element in calculating compound interest because:

• Longer periods allow your investment to grow significantly through compounding.
• Extending the time lets the accumulated interest sum increase. Your money essentially earns interest on the previously earned interest in addition to the principal.

So, be patient with your investments, as they often grow more with time!

Compounded Annually

When interest is compounded annually, it means the interest is calculated and added to the principal once per year. This differs from other compounding frequencies, like quarterly or monthly. In this situation, because it's compounded annually, our annual interest rate remains the same throughout.
This approach benefits your investment:

• Interest builds up over time, gathering more interest each year it compounds.
• Providing a straightforward way to keep track of how your money grows.

Understanding this basics of compounding annually can assist you in optimizing your investments for faster growth, especially when paired with a suitable time and principal.