Problem 20 - Algebra 1: Concepts and Skills

Question

You deposit money in an account that pays 5% interest compounded yearly. Find the balance after 5 years for the given initial amount. $$\$ 250$$

Step-by-Step Solution

Answer

The balance after 5 years will be \$320.71.

Step 1: Understand the Compound Interest Formula

The compound interest formula is generally given as \(A=P(1+r)^n\), where: - \(A\) is the amount of money accumulated after \(n\) years, including interest. - \(P\) is the principal amount (the initial amount of money). - \(r\) is the annual interest rate (in decimal form). - \(n\) is the number of years the money is invested.

Step 2: Convert the Interest Rate to Decimal Form

To insert the given percentage rate in the formula, it must be converted to decimal form. A rate of \(5\%\) is the same as \(0.05\) in decimal.

Step 3: Substitute Values into the Formula

Substitute the given values into the formula. We have \(P=\$250\), \(r=0.05\), and \(n=5\). So, \(A=250(1+0.05)^5\).

Step 4: Solve the Expression

Solving the expression will get the accumulated amount after 5 years. So after simplifying, you will find that \(A=\$320.71\).

Key Concepts

Understanding Interest Rates

Interest rates are percentages used by banks and financial institutions to express how much interest you will earn or owe over time.
In the context of compound interest, an interest rate determines how much extra money is added to your initial sum each period.
Consider the following:

• **Annual Interest Rate**: This is often expressed as a percentage, for example, 5%. To use it in calculations, convert it to decimal by dividing by 100 (5% becomes 0.05).
• **Compound Interest**: Unlike simple interest, compound interest adds the interest to the original amount, then calculates interest on the new total in subsequent periods. This can lead to exponential growth over time.

In our exercise, the interest rate of 5% is compounded once per year, building on the previous year's amount.

The Principal Amount

The principal amount is the initial sum of money you deposit or invest.
It's the starting point from which any interest calculations are made.
Key points include:

• **Initial Deposit**: This is the amount you initially invest. In our problem, it is \(\\)250\$.
• **Role in Interest Calculations**: The principal directly impacts how much interest you earn. Larger principal amounts yield more interest in absolute terms.

Understanding your principal is essential since any interest earned is a percentage of this original amount. Over time, the principal might also grow if you choose to reinvest the interest.

The Investment Period

The investment period is the time during which your money is earning compound interest.
It's typically measured in years.
Here's why it matters:

• **Duration**: The length of time your money is invested significantly affects the final amount. More time typically results in more interest accumulated.
• **Compound Effect**: Longer investment periods mean that interest has more opportunities to compound, allowing growth to accelerate.

In the given exercise, the investment period is 5 years. During this time, the interest compounds each year, building on the existing total.