Problem 6 - Algebra for College Students

Question

6\. Solve \(i=P r t\) for \(r\), given that \(P=\$ 700, t=2\) years, and \(i=\$ 126 .\) Express \(r\) as a percent.

Step-by-Step Solution

Answer

The rate of interest \( r \) is 9\%.

Step 1: Understanding the Equation

The formula given is the simple interest formula, \( i = P \cdot r \cdot t \), where \( i \) is the interest earned, \( P \) is the principal amount, \( r \) is the rate of interest, and \( t \) is the time period over which the interest is calculated.

Step 2: Substitute Given Values

We are given that \( P = 700 \), \( t = 2 \), and \( i = 126 \). Substitute these values into the equation: \( 126 = 700 \cdot r \cdot 2 \).

Step 3: Simplify the Equation

Simplify the equation by calculating the multiplication on the right side: \( 700 \cdot 2 = 1400 \). This simplifies the equation to: \( 126 = 1400 \cdot r \).

Step 4: Solve for r

To isolate \( r \), divide both sides of the equation by 1400: \( r = \frac{126}{1400} \).

Step 5: Convert r to a Percentage

Calculate \( \frac{126}{1400} \) which simplifies to \( r = 0.09 \). To express \( r \) as a percentage, multiply by 100: \( r = 9\% \).

Key Concepts

Simple Interest Formula

The simple interest formula is a fundamental concept in finance that helps in calculating the interest earned or paid on a principal sum over a period. The formula itself is expressed as:
\[ i = P \cdot r \cdot t \]
Here:

• \( i \) stands for the interest amount.
  
• \( P \) represents the principal, which is the original sum of money.
  
• \( r \) is the interest rate, usually expressed as a decimal in the equation.
  
• \( t \) is the time period for which the interest is calculated, typically in years.
  

When using the simple interest formula, you can solve for any one variable if you have the values for the other three. Understanding this equation is crucial for managing loans, savings, and all sorts of financial calculations.

Expressing Interest Rate as a Percent

To express an interest rate in a more intuitive manner, like a percent, is crucial for financial comparisons among different investment or loan opportunities. Initially, the interest rate \( r \) is presented as a decimal when used in the simple interest formula. However, people commonly communicate interest rates in percentage form.
The process of converting a decimal to a percent is quite simple:

• First, compute or solve for \( r \) as a decimal.
  
• Then, multiply the decimal by 100 to transform it into a percentage.
  

For example, if \( r = 0.09 \), multiplying by 100 tells us that the interest rate is 9%. Using percentages makes it easier to comprehend and compare the rates across different financial products.

Substituting Given Values into an Equation

Substituting known values into an equation is an essential step in problem-solving within algebra and applied mathematics. It involves replacing the variables in a formula with specific numbers to perform calculations.
Here is how you typically perform substitution for the simple interest formula:

• Identify which quantities you need to substitute based on the information provided. For example, identify \( P \), \( t \), and \( i \) from the problem statement.
  
• Introduce these values into the formula. In this exercise, with \( P = 700 \), \( t = 2 \), and \( i = 126 \), you substitute them into the equation to get \( 126 = 700 \cdot r \cdot 2 \).
  
• Proceed with mathematical operations to solve for the remaining unknown variable. Here, once substituted, the equation simplifies through arithmetic operations to find the interest rate \( r \).
  

Substitution is an invaluable skill not only in mathematics but also in practical scenarios, like understanding financial calculations and predicting outcomes.