Problem 23 - 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. $$\$ 400$$

Step-by-Step Solution

Answer

The balance after 5 years will be \$520.35.

Step 1: Identify the given values

Identify and write down the given values from the problem. The principal amount P is \$400, the annual interest rate r is 5% or 0.05, the number of times interest is compounded per year n is 1, and the number of years t is 5.

Step 2: Substitute the values into the formula

Substitute the identified values into the compound interest formula. A = \$400(1 + 0.05/1)^(1*5) becomes A = \$400(1 + 0.05)^5.

Step 3: Simplify the expression

Simplify the expression inside the parentheses first, then raise it to the power of 5. This becomes A = \$400 * 1.05^5.

Step 4: Calculate the final value

Perform the final calculations to find the value of A (the balance after 5 years). This becomes A = \$520.35, when rounded to the nearest cent.

Key Concepts

Interest Rate

The interest rate is a crucial part of growing your investment or savings account over time. In the exercise, the interest rate is provided as 5% per annum.
But what does this mean? Essentially, it is the percentage of your initial investment or principal that you will earn as interest each year. Interest rates are presented in different ways:

• Annual percentage rate (APR): The rate earned over one year without compounding.
• Nominal interest rate: The stated rate before compounding is considered.
• Effective interest rate: Reflects the actual annual return, accounting for compounding.

Understanding the interest rate helps you predict how much your initial investment will grow. With a 5% interest rate, for example, you earn 5% of your account balance each year.

Compound Interest Formula

The compound interest formula lets you calculate how much your investment will grow over time with compounding.
The formula is: \[ A = P \left( 1 + \frac{r}{n} \right)^{nt} \]Where:

• A is the future value of the investment/loan, including interest.
• P is the initial principal balance (starting amount).
• r is the annual interest rate (as a decimal).
• n is the number of times that interest is compounded per year.
• t is the time in years.

In the given exercise, the interest is compounded yearly, so we plug in our values:
- Initial amount \( P = 400 \)
- Interest rate \( r = 0.05 \) (5% as a decimal)
- Compounding frequency \( n = 1 \)
- Years \( t = 5 \)After substituting these values correctly, you can calculate how your investment will grow over time.

Initial Principal Amount

The initial principal amount is the amount of money you start with when you deposit it in an account. In the context of the exercise, this value is $400.
It serves as the base amount upon which interest is calculated. Why is the initial principal important?

• Foundation of Earnings: This initial amount determines how much interest you will earn. The bigger the principal, the more you earn.
• Initial Investment: It reflects your starting point in any financial calculation.

Understanding its role allows you to see how your money will increase over time, especially when interest is compounded. The principal grows with interest, affecting subsequent compounding periods.