Problem 2 - Algebra for College Students

Question

Solve \(i=P r t\) for \(i\), given that \(P=\$ 500, r=9 \%\), and \(t=3 \frac{1}{2}\) years.

Step-by-Step Solution

Answer

The interest ( i ) is $157.50.

Step 1: Identify the formula

The formula given is the simple interest formula involving the interest ( i ), principal ( P ), rate ( r ), and time ( t ). The formula is: i = P imes r imes t .

Step 2: Substitute known values

Replace each variable in the formula with the values provided: P = 500, r = 9 ext{ extdiscount} = 0.09, t = 3 rac{1}{2} ext{ years} = 3.5 .

Step 3: Calculate the interest

Apply the formula with substituted values: i = 500 imes 0.09 imes 3.5 .

Step 4: Multiply step by step

First, multiply 500 by 0.09: 500 imes 0.09 = 45 . Then multiply the result by 3.5: 45 imes 3.5 = 157.5 .

Key Concepts

Interest Calculation

Interest calculation is the process of determining the amount of interest added to the principal over a period of time. It's a fundamental concept in finance that helps people understand how much they can earn through investments or how much they owe on loans.
There are different types of interest calculations, but one of the simplest forms to understand is simple interest. This type of interest is calculated as a fixed percentage of the original principal over a specific period of time.

• Principal (P): This is the amount of money initially lent or invested.
• Rate (r): Represents the interest rate, usually in percentage form.
• Time (t): Indicates how long the money is invested or borrowed for.

The interest earned or paid doesn't accumulate or compound over time, which is what makes it simple. Once calculated, this interest is either added to the original sum or paid out in the case of an investment.

Simple Interest Formula

The simple interest formula provides an easy way to calculate interest over time:\[ i = P \times r \times t \]Where:

• \( i \) is the interest earned or paid.
• \( P \) is the principal amount.
• \( r \) is the rate of interest in decimal form.
• \( t \) is the time period in years.

To employ this formula, the rate must be converted from a percentage to a decimal by dividing by 100. For instance, a 9% interest rate becomes 0.09. The time period must be in years; if given in months or days, convert it accordingly.
The formula is applicable in various scenarios, such as calculating the interest on loans, savings, or bonds. In this example, we substitute the known values, including 3.5 years for the time and 0.09 for the rate. We then follow through with simple multiplication to find out that the interest, i, is \(157.5\).

Financial Mathematics

Financial mathematics is the application of mathematical methods to solve financial problems. It's essential in helping us make informed financial decisions, from personal finance to complex financial models used in business and commerce. Simple interest is one of the foundational concepts in financial mathematics.
By understanding the basics of simple interest, individuals can better manage loans and investments. This not only helps in predicting the growth of an investment but also in forecasting how much debt will accrue over time.

• Understanding Investments: Knowing how to calculate simple interest aids in evaluating different investment opportunities and understanding the growth of savings.
• Loan Management: Calculating interest payments can help borrowers understand their obligations and plan their finances more effectively.

Mastering simple calculations enables individuals to apply these principles in real-world scenarios, expanding to more complex financial products and strategies as needed. Financial mathematics thus acts as a vital tool that underscores many personal and professional economic transactions.