Problem 1 - Calculus Concepts

Question

Simple Interest Calculate the total amount due after two years on a loan of \(\$ 1500\) with a simple interest charge of \(7 \%\).

Step-by-Step Solution

Answer

The total amount due is $1710.

Step 1: Identify the Formula

The formula for calculating simple interest is: \( I = P \times r \times t \), where \( I \) is the interest, \( P \) is the principal amount, \( r \) is the rate of interest per year (as a decimal), and \( t \) is the time in years.

Step 2: Convert Percentage to Decimal

The interest rate given is 7%. To use this in the formula, convert it to a decimal by dividing by 100. Therefore, \( r = \frac{7}{100} = 0.07 \).

Step 3: Insert Values into Formula

Now, substitute the given values into the simple interest formula: \( P = 1500 \), \( r = 0.07 \), and \( t = 2 \). Compute the interest: \( I = 1500 \times 0.07 \times 2 \).

Step 4: Perform Calculations

First, multiply the interest rate by the time: \( 0.07 \times 2 = 0.14 \). Then multiply by the principal: \( 1500 \times 0.14 = 210 \). So, the interest \( I = 210 \).

Step 5: Calculate Total Amount

To find the total amount due after two years, add the interest to the principal: \( 1500 + 210 = 1710 \).

Key Concepts

interest calculation

When we talk about interest calculation, especially in the realm of simple interest, it is crucial to understand the components involved. Simple interest is computed using a straightforward formula:

• \( I = P \times r \times t \)

Here, \( I \) represents the interest earned or paid, \( P \) is the principal amount—the original sum of money, \( r \) stands for the rate of interest per year expressed as a decimal, and \( t \) is the time span for which the money is borrowed or lent, measured in years.
To determine the interest rate as a decimal, convert the percentage rate by dividing it by 100. For example, a 7% interest becomes 0.07 when converted into decimal form. This conversion is essential for accurate calculations.
Let's apply these components: using \( P = 1500 \), \( r = 0.07 \), and \( t = 2 \), we substitute into the formula, simplifying the expression step by step to find the interest amount. Understanding each term and operation in this formula paves the way for calculating the total expense or gain over specified time frames.

financial mathematics

Financial mathematics feels like a complex topic, but breaking down its principles can make it more understandable. Simple interest is just one part of financial mathematics, and it serves as a foundation for understanding more complex financial products.
What makes simple interest appealing is its linear nature—interest accrues evenly over time, based only on the initial principal, not affected by subsequent interest added each period. In our example, this would mean the same interest amount accrues each year on the \(\$1500\) principal. This forms a crucial contrast to compound interest, where interest is calculated on both the initial principal and previously accumulated interest.
In practical terms, knowing how to calculate simple interest is particularly useful in situations such as short-term loans, savings accounts, and some types of bonds. These applications demonstrate why grasping even basic financial maths can help individuals make more informed financial decisions.

math problem solving

Math problem solving, especially in financial contexts, is a step-by-step process. The key to efficiently solving a math problem like simple interest lies in understanding the formula and methodically substituting values.
Begin by clearly identifying each component of the formula: principal \( P \), rate \( r \), and time \( t \). Convert percentages into decimals to maintain consistency in calculations.

• Break down the problem: address each step separately.
• Calculate the intermediary results first, such as multiplying the rate by the time.
• Verify each step to ensure accuracy before moving on.

Once you've determined the interest, remember to add it back to the principal to obtain the total amount. For instance, after computing an interest of \(\\(210\) for a \(\\)1500\) loan, the total repayment amount becomes \(\$1710\).
This methodical approach simplifies the solution process, allowing you to tackle similar problems with confidence. Problem solving in math is essentially about clarity, strategy, and careful execution.