Problem 14 - Algebra 1

Question

A principal of \(\$ 200\) is deposited in an account that \(1-9\) pays \(4.2 \%\) interest compounded yearly. Find the balance after 5 years.

Step-by-Step Solution

Answer

The balance after 5 years is approximately \$243.51.

Step 1: Assign the provided values to the corresponding variables

The principal amount P = \$200, the annual interest rate r = 4.2%, or 0.042 in decimal form, it is compounded yearly so n = 1, and the time t = 5 years.

Step 2: Substitute these values into the formula

We substitute the values into the formula A = P(1 + r/n)^(nt), to get A = \$200(1 + 0.042/1)^(1*5)

Step 3: Simplify the expression

This gives us A = \$200(1 + 0.042)^5 = \$200*1.042^5

Step 4: Calculate the result

Calculate the value to get the total amount of money accumulated after 5 years.

Key Concepts

Principal

The principal is the original amount of money you deposit or invest. In this problem, the principal is \(\$ 200\). It's the base amount on which interest is calculated.
Understanding the principal is crucial because it's not just any starting amount.

• It's the initial sum before any interest gets added.
• This amount will grow over time based on the interest rate.
• For investments, it’s the starting point that investors aim to grow.

In financial terms, knowing your principal helps in setting realistic financial goals. When you know your principal, you have a clear picture of the foundation of your investment.
The principal can come from different sources like savings, earnings, or gifts, but it always remains the starting point for interest calculations.

Interest Rate

The interest rate is a percentage that tells us how much the principal will grow over a set period. In this example, the interest rate is 4.2% compounded yearly.
This tells us how earnings on the principal are calculated every year.
With an interest rate:

• You can determine how quickly your money will grow.
• It acts like a growth rate for your investment or savings.
• A higher rate means your money increases faster.

In our calculation, transforming the percentage into a decimal by dividing by 100 (so 4.2% becomes 0.042) allows us to use it in formulas easily. The "compounded yearly" means this rate applies once per year, making the effects of compounding more apparent over time.

Account Balance

Account balance is the total amount of money in the account after interest is added to the principal. After 5 years, this practical exercise shows how the account balance is computed using the formula:\[A = P(1 + \frac{r}{n})^{nt}\]Where:

• \(P\) is the principal amount (\(\$200\) in this scenario),
• \(r\) is the annual interest rate in decimal form,
• \(n\) refers to the number of times interest is compounded per year (which is 1 here),
• \(t\) is the time in years (5 years here).

After calculating, the formula gives us the account balance after 5 years. This balance reflects both the initial deposit and the interest earned. For anyone investing or saving, knowing your final account balance helps in assessing how much your money has grown over time. To further evaluate, it’s important to calculate and monitor account balances regularly to track financial growth.