Problem 57 - Precalculus

Question

If you put \(\$ 3,200\) in a savings account that earns \(2.5 \%\) interest per year compounded quarterly, how much would you expect to have in that account in 3 years?

Step-by-Step Solution

Answer

After 3 years, you would have approximately $3448.22 in the account.

Step 1: Identify the variables

The principal amount (initial investment) is \( P = 3200 \). The annual interest rate is \( r = 0.025 \) or \( 2.5\% \). The number of times the interest is compounded per year is \( n = 4 \) (quarterly). The number of years the money is invested for is \( t = 3 \).

Step 2: Understand the formula

We will use the compound interest formula: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where \( A \) is the amount of money accumulated after \( t \) years, including interest.

Step 3: Substitute the values into the formula

Plug in the values into the formula: \[ A = 3200 \left(1 + \frac{0.025}{4}\right)^{4 \times 3} \]Simplify inside the parenthesis: \( \frac{0.025}{4} = 0.00625 \).

Step 4: Calculate the compounded interest

Now calculate: \[ A = 3200 \left(1 + 0.00625\right)^{12} \]This simplifies to: \[ A = 3200 \times (1.00625)^{12} \]

Step 5: Compute the exponentiation

Calculate \( (1.00625)^{12} \): approximately \( 1.07757 \).

Step 6: Find the final amount

Multiply the principal by the compound factor: \[ A = 3200 \times 1.07757 \approx 3448.22 \]

Key Concepts

Savings Account

A savings account is a type of bank account that is designed to hold your money securely while earning interest. Unlike a checking account, where you may write checks and make frequent transactions, a savings account encourages you to save by offering interest earnings.
It's a popular choice for both short-term and long-term savings goals due to its safety and reliability. Banks offer a small but steady interest rate on the deposited funds.

• Great for saving towards future expenses, like vacations or emergency funds.
• Provides liquidity, allowing easy access to your money compared to other investment options.

However, it is important to note that while savings accounts protect your money, they might not provide high returns compared to other investments due to lower interest rates.

Quarterly Compounding

Quarterly compounding means that the interest is calculated and added to the account balance four times a year. This approach makes the savings grow faster compared to annual compounding due to more frequent accumulation of interest.
Every quarter, the interest earned is added to your principal, and in the next quarter, further interest is calculated on this new total amount. This compounding cycle helps your investment earn more over time.

• Compounding frequency plays a crucial role in how fast your money grows.
• The more frequently interest is compounded, the more interest you earn.

For instance, in the context of the exercise, interest being compounded quarterly significantly contributed to the increase from $3,200 to approximately $3,448.22.

Interest Rate

The interest rate is a percentage that expresses how much your savings will grow over a period of time. It is a critical component affecting the growth of your investment in a savings account.
In our exercise, the annual interest rate is 2.5%, which signifies the expected annual return on the deposited money without considering compounding.

• A higher interest rate yields a higher return on investment.
• The interest rate can be fixed or variable, with fixed rates remaining the same for a set period as opposed to variable rates which can fluctuate.

Understanding how the interest rate affects your savings helps in choosing the right savings product to meet your financial goals.

Compound Interest Formula

The compound interest formula is essential for calculating how much money will accumulate in a savings account over time, taking into account the effects of earning interest on interest.
The basic formula is: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]Where:

• \( A \) is the amount after time \( t \).
• \( P \) is the principal investment amount.
• \( r \) is the annual interest rate (decimal).
• \( n \) is the number of times interest is compounded per year.
• \( t \) is the number of years the money is invested.

This formula shows how different factors affect the final amount: the initial principal, the rate of interest, frequency of compounding, and time. In the exercise mentioned, this formula helped calculate that the initial \(3,200 grew to approximately \)3,448.22 after 3 years, thanks to quarterly compounding and a 2.5% annual interest rate.