Question

For Exercises \(1-10\), calculate the simple interest and final balance. \(\$ 2000\) at \(6.9 \%\) for 60 days

Short answer

Question: Calculate the simple interest and final balance for a $2000 investment with an annual interest rate of 6.9% after 60 days. Answer: The simple interest is approximately $22.69, and the final balance is approximately $2022.69.

Step-by-step solution

Convert the interest rate to a decimal

To convert the given annual interest rate of \(6.9 \%\) to a decimal, divide the percentage by \(100\). \(6.9\% = \frac{6.9}{100} = 0.069\)

Convert the time period to years

The time period is given in days, so we need to convert it to years. To do this, divide the number of days by the number of days in a year. \(60\ days = \frac{60}{365} = 0.1644\ years\)

Calculate the simple interest

Now that we have the principal amount (\(P = \$2000\)), the rate in decimal form (\(r = 0.069\)), and the time in years (\(t = 0.1644\)), we can calculate the simple interest using the formula: \(Simple\ Interest = P \times r \times t\) \(Simple\ Interest = \$2000 \times 0.069 \times 0.1644 \approx \$22.69\)

Calculate the final balance

To calculate the final balance, add the simple interest to the principal amount. \(Final\ Balance = Principal\ Amount + Simple\ Interest\) \(Final\ Balance = \$2000 + \$22.69 \approx \$2022.69\) Thus, the simple interest is approximately \(\$22.69\) and the final balance is approximately \(\$2022.69\).

Key concepts

Understanding Annual Interest Rate

The annual interest rate is expressed as a percentage. It represents the percentage of the principal that will be paid as interest over a year. To visualize, if your principal amount is $1000 with a 10% annual interest rate, you pay $100 in interest over a year.

• This rate is crucial because it influences how much you earn or pay over time.
• In this exercise, the annual interest rate is 6.9%.

It's important to convert this percentage into a decimal for calculations, such as in our example where 6.9% becomes 0.069. This conversion is critical when working with interest-related math formulas.

Converting Interest Rates

Converting an interest rate from a percentage to a decimal is a necessary step in interest calculations. This conversion simply makes it compatible with mathematical operations.

• For example, to convert 6.9% into a decimal, divide by 100, resulting in 0.069.

This decimal form allows you to multiply it directly with the principal and time in equations, as percentages aren’t suitable for raw calculations. Remember, always use this form in formulas to bring about correct results.

The Role of Principal Amount

The principal amount is the initial sum of money that interest calculations are based upon. This starting amount plays a crucial role in determining how much interest you earn or pay.

• In this exercise, the principal is $2000.

It serves as the base figure for all calculations involving interest. Knowing the principal helps in setting the stage for calculating interest through simple formulas, like the simple interest formula. Without the principal, other elements, like rate and time, can’t calculate financial growth or expense.

The Basics of Interest Calculation

Interest calculation involves using a specific formula to find how much additional money will be earned or owed over a period. With simple interest, the calculation remains straightforward thanks to the formula:

• \[ Simple\ Interest = P \times r \times t \]
• Where \( P \) is Principal, \( r \) is Rate, and \( t \) is Time.

Using our values—Principal: $2000, Rate (as a decimal): 0.069, and Time: 0.1644 years—we plug these into the formula to find the interest.
This approach ensures clear and accurate financial planning and management.

Time Conversion to Years

Converting time into years is pivotal for aligning with the standard practice of calculating annual interest rates. Often, time is not initially given in years, so conversion is necessary.

• To convert days to years, divide the number of days by 365, the number of days in a year.

In this exercise, 60 days convert to approximately 0.1644 years. This uniformity in time representation helps in the consistency of interest calculations, allowing for seamless application of the interest formula using the year-based rate.