Problem 26 - Algebra 1: Concepts and Skills

Question

You deposit \(\$900\) in an account that compounds interest yearly. Find the balance after 10 years for the given interest rate. $$6 \%$$

Step-by-Step Solution

Answer

The balance after 10 years will be approximately \$1610.29.

Step 1: Decipher the given information

The principal amount P is \$900, the annual interest rate r is 6% or 0.06 in decimal form, time t is 10 years, and the interest is compounded yearly so n is 1.

Step 2: Substitute the values in the Compound Interest formula

By substituting the values into the compound interest formula we have A=\$900(1+0.06/1)^(1*10). Simplify this to get A=\$900(1.06)^10.

Step 3: Calculate the final amount

Calculate this to find the total bank balance after 10 years.

Key Concepts

Principal Amount

The principal amount is the starting sum of money you deposit into an account. In this case, it is the initial $900 that you place in a bank account to earn interest.

• This amount is important because it is the base on which interest is calculated.
• Any growth in the bank balance will be based on this principal amount.

Understanding the principal amount is crucial because your return on investment starts from here. When calculating interest, the principal directly impacts how much you earn over time.
Starting with a larger principal can result in a significantly larger balance after the same time period!

Annual Interest Rate

The annual interest rate is expressed as a percentage that reflects how much the bank will pay you to use your money over one year.

Here, the annual interest rate is 6%, which means for every $100, you earn $6 in interest without considering compounding effects.

• You should convert it to a decimal for calculations by dividing by 100, which gives 0.06.
• This rate influences how quickly your principal amount grows over time.

A higher interest rate means faster growth of your deposited money, assuming other factors stay the same. Always consider the annual interest rate when choosing a bank account or investment, as it heavily influences your returns.

Interest Compounded Yearly

Compounding is a key concept in finance that significantly affects your investment growth. With interest compounded yearly, the bank calculates interest on your initial deposit and already earned interest once every year.

Here's how it works:

• The interest you earn is added back to your principal annually.
• In the second year, you earn interest on both the original principal and the interest added from the previous year.

This cycle repeats each year, leading to potentially far greater growth than with simple interest. Compounding yearly is beneficial as it means more frequent updates to your balance, each time growing on top of the previous amount.

Bank Balance After 10 Years

After using the compound interest formula, you unveil just how much your investment has grown after a decade.
The formula you use is:\( A = P \left(1 + \frac{r}{n}\right)^{nt}\)Where:

• \(A\) is the amount of money accumulated after n years, including interest.
• \(P\) is the principal amount (\\(900).
• \(r\) is the annual interest rate (0.06 as a decimal).
• \(n\) is the number of times interest is compounded per year (1 in this case).
• \(t\) is the time the money is invested for (10 years).

For our scenario, substituting the given values results in:\(A = 900 \times (1 + 0.06)^ {10} = 900 \times (1.06)^{10} \)Calculating this, your balance at the end of 10 years will be approximately \)1,610.51. This figure shows how effective compounding can be over time, resulting in a significant increase from your original investment.