Problem 1 - Algebra for College Students

Question

Solve \(i=\) Prt for \(i\), given that \(P=\$ 300, r=8 \%\), and \(t=5\) years.

Step-by-Step Solution

Answer

The simple interest is \$120.

Step 1: Identify the Formula

We begin by identifying the formula for simple interest, which is given by \(i = Prt\), where \(i\) is the interest, \(P\) is the principal amount, \(r\) is the rate of interest per year, and \(t\) is the time period in years.

Step 2: Substitute the Given Values

Next, we substitute the given values into the formula. Here, \(P = 300\), \(r = 8\% = 0.08\), and \(t = 5\). So we substitute these values into the formula as follows: \(i = 300 \times 0.08 \times 5\).

Step 3: Perform the Multiplication

Now, calculate the multiplication to find the interest. First, multiply \(300\) and \(0.08\) to get \(24\), and then multiply that result by \(5\) to find \(i = 120\).

Step 4: State the Result

After performing the multiplication, we find that \(i = 120\). Therefore, the simple interest is \$120.

Key Concepts

Interest Calculation

Understanding how to calculate interest is essential in financial mathematics. Simple interest allows us to determine the additional amount paid back or earned on a principal sum over a certain period.
Interest Calculation is often abbreviated as SI for Simple Interest, which is calculated using the formula:

• \( i = P \times r \times t \)

Where:

• \( i \) = interest earned or paid
• \( P \) = principal amount or initial sum
• \( r \) = rate of interest per year (expressed as a decimal)
• \( t \) = time period in years

To calculate simple interest:

• First, convert the interest rate from a percentage to a decimal by dividing by 100.
• Next, substitute the given values into the formula.
• Finally, multiply the principal, rate, and time to find the interest.

The simplicity of this calculation makes it an invaluable tool for basic financial assessments.

Mathematical Formulas

Mathematical formulas are like recipes: they tell us how to achieve specific results systematically. In financial mathematics, they guide us in consistently computing quantities like interest.
For simple interest, the formula \( i = Prt \) directs us to input known quantities (principal, rate, time) to determine the unknown (interest). Here is how each formula component functions:

• Principal (\( P \)): the initial sum of money, either invested or borrowed.
• Rate (\( r \)): the percentage of the principal charged or earned as interest, yearly.
• Time (\( t \)): the duration, in years, that affects how much interest accumulates.
• Interest (\( i \)): the money earned or owed based on these factors.

Transform any given percentage rate into a usable decimal by dividing by 100. Then systematically substitute each known value into the formula to compute your unknown, maintaining consistency and accuracy in your calculations.

Financial Mathematics

Financial Mathematics encompasses various tools and techniques used to analyze financial problems, with simple interest being one of the basic yet essential concepts. This area of mathematics helps in making informed decisions in loans, investments, and savings.
Simple interest is used primarily in cases where linear growth over time is expected. It's easy to calculate and understand, making it ideal for short-term financial dealings or straightforward investments.
Considerations in financial mathematics when dealing with simple interest include:

• Determining the viability of a loan or investment.
• Assessing the return on a short-term deposit.
• Budgeting for future interest payments or earnings.

By understanding how simple interest functions and applying it to various scenarios, one ensures better financial planning and management. Financial mathematics supports individuals and businesses in making decisions that are advantageous in the long term.