Problem 31 - Prealgebra

Question

Find the simple interest to the nearest cent. \(\$ 3680\) at \(6.75 \%\) for \(2 \frac{1}{4}\) years

Step-by-Step Solution

Answer

The simple interest is \( \$558.90 \).

Step 1: Understand the Formula

The formula for simple interest is given by \( I = P imes r imes t \), where \( I \) is the interest earned, \( P \) is the principal amount, \( r \) is the rate of interest per year, and \( t \) is the time in years.

Step 2: Identify the Values

Here, the principal \( P \) is \( \$3680 \), the rate \( r \) is \( 6.75\% \) (which can be written as \( 0.0675 \) in decimal), and the time \( t \) is \( 2 \frac{1}{4} \) years (which is equivalent to \( 2.25 \) years).

Step 3: Substitute the Values

Substitute the values into the simple interest formula: \( I = 3680 \times 0.0675 \times 2.25 \).

Step 4: Calculate the Interest

Perform the calculations: \( 3680 \times 0.0675 = 248.4 \). Next, multiply this by \( 2.25 \): \( 248.4 \times 2.25 = 558.9 \).

Step 5: Round the Interest

Round the result to the nearest cent. Here, the interest \( 558.9 \) rounds to \( 558.90 \).

Key Concepts

Interest Calculation

When it comes to calculating simple interest, it's essential to first understand what simple interest actually is. Simple interest is a way to determine how much extra money, or interest, your initial amount or "principal" accrues over a certain period, using a pre-defined interest rate. It's calculated with this easy formula:

• I = P \( \times \) r \( \times \) t, where:
• I stands for the interest earned.
• P is the principal amount — the initial sum of money.
• r represents the annual interest rate, expressed as a decimal.
• t denotes the time, usually given in years.

Break down the calculation process by plugging in the values correctly into the formula. Without understanding the basic relationship between these variables, calculating interest can seem tricky. But with a clear grasp, you'll find it straightforward. Remember that this formula only works for simple interest, which differs from compound interest.

Financial Mathematics

Financial mathematics might sound complex at first, but it's just a set of calculations allowing us to make informed financial decisions, like determining how much interest we'll earn. This type of math includes various topics such as interest rates, loans, and investments.

In our exercise, we're focusing on simple interest — a fundamental concept. Financial mathematics helps us understand how our money grows or decreases over time.

The powerful yet simple principle here is that by knowing your principal and the terms of your interest (rate and time), you can determine the extra money you'll earn. For this, knowing

• how to convert interest rates to decimals,
• how to handle mixed numbers for time,
• and straight calculations

will greatly enhance your financial acumen.

Decimal Conversion

Decimal conversion might be a small part of the process, but it's crucial. In terms of interest calculation, rates are typically presented as percentages but must be converted into decimal form to use in formulas.

Consider our problem's rate: 6.75%. You'd convert this by dividing by 100, resulting in 0.0675 in decimal form.

Steps to convert:

• Take the percentage number.
• Divide by 100.
• Result is your decimal value.

Similarly, if you have mixed numbers in terms of time (like 2 1/4 years, which equals 2.25), it's best to convert them to decimals for ease of calculation. This ensures accuracy and simplifies the arithmetic, making your financial calculations more precise. Avoiding conversion errors is essential for correct interest calculations.

Rounding

Rounding is the final touch in any calculation but holds significance, especially in financial mathematics. We often need to round off numbers to the nearest cent or the nearest whole number, depending on the context. Proper rounding can prevent errors and overpayments.

Consider our exercise, where the interest calculated was $558.9. To make this a complete number, especially for financial purposes, we round off the last decimal, resulting in $558.90.

Rules for rounding:

• If the number after the decimal is 5 or more, round up.
• If it's less than 5, round down.
• Always round to the nearest specified precision — in this case, the nearest cent.

Rounding not only presents numbers more cleanly but also enhances the accuracy of your financial documentation and decisions. It’s a simple yet powerful tool in financial computations.