Question

If $$\$ 7500$$ is invested in an account paying \(3.25 \%\) interest compounded daily, how much money will be in the account after 5 years?

Short answer

The amount of money in the account after 5 years will be approximately $8891.06.

Step-by-step solution

Identify the Given Values

First, identify the principal amount (P), the annual interest rate (r), the number of times interest is compounded per year (n), and the number of years the money is invested (t). Here, P = 7500, r = 3.25%, n = 365 (since interest is compounded daily), t = 5.

Convert the Interest Rate to a Decimal

Convert the annual interest rate from a percentage to a decimal by dividing by 100. \( r = \frac{3.25}{100} = 0.0325 \)

Use the Compound Interest Formula

The formula for compound interest is \[ A = P \left(1 + \frac{r}{n} \right)^{nt} \]Substitute the known values into the formula: \[ A = 7500 \left(1 + \frac{0.0325}{365} \right)^{365 \times 5} \]

Calculate the Daily Interest Rate

First, calculate the daily interest rate: \( \frac{0.0325}{365} \approx 0.000089041 \)

Calculate the Exponent

Next, calculate the exponent: \( 365 \times 5 = 1825 \)

Compute the Amount

Substitute the values back into the formula and compute the amount: \[ A = 7500 \left(1 + 0.000089041 \right)^{1825} \]\[ \approx 7500 \left(1.000089041 \right)^{1825} \]

Final Calculation

Finally, compute the final amount using a calculator: \[ A \approx 7500 \times 1.185474 \approx 8891.06 \]

Key concepts

Principal Amount

The principal amount is the initial sum of money that is invested or loaned. It is the starting balance on which the interest will be calculated. In our example, the principal amount is \$ 7500\. Understanding the principal is crucial because it helps you determine how much interest you will earn over time. The larger the principal, the more interest you can accumulate.

Always keep in mind the principal is typically denoted by the letter \( P \) in mathematical formulas. So, when you see terms like \( P \) or 'initial investment', they are referring to the same thing — your starting amount.

Annual Interest Rate

The annual interest rate is the percentage of the principal that is paid as interest over the course of one year. It’s usually expressed as an annual percentage rate (APR). The rate can either earn you money in savings and investments, or cost you money in loans and debts.

In our case, the annual interest rate is \3.25 \%. To use this rate in our calculations, we have to convert it to a decimal form by dividing by 100, giving us \( 0.0325 \).

Remember:

• Higher interest rates lead to higher earnings on investments.
• Rates can vary based on the type of account or loan.
• Always compare interest rates when evaluating different financial options.

Compounded Daily

When interest is compounded daily, it means that the interest is calculated and added to the principal balance every day. This leads to interest being earned on interest, essentially compounding more frequently than monthly or annually.

For our example, the interest is compounded 365 times a year. This is why the formula includes \( n = 365 \) when calculating daily compounding. The daily interest rate, once you divide the annual rate by 365, becomes a very small fraction (around 0.000089041). Although this may seem insignificant, daily compounding dramatically increases the final amount over time.

Key points to remember:

• Daily compounding results in higher earnings compared to monthly or yearly compounding.
• It maximizes the interest earned on your investment.
• When selecting savings or investment accounts, consider the frequency of compounding.

Investment Period

The investment period is the length of time that your money is invested or loaned. It is a critical factor impacting the total amount of interest accrued. Usually represented by the letter \( t \), the longer the investment period, the more interest you will earn.

In our scenario, the investment period is 5 years. This means that the principal amount has a chance to grow through compound interest over five years.

Important details to remember about investment periods:

• Longer investment periods typically result in more interest earned.
• Consider your financial goals when determining how long to invest.
• Remember to evaluate the risks associated with long-term investments.