Problem 6 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the formula

Start with the formula for simple interest, which is given by \( i = P \times r \times t \). Here, \( i \) is the interest, \( P \) is the principal amount, \( r \) is the interest rate, and \( t \) is the time in years.

Step 2: Substitute given values into the equation

Plug the given values into the formula: \( 126 = 700 \times r \times 2 \).

Step 3: Solve for \( r \)

Rearrange the equation to solve for \( r \). Start by simplifying the equation: \( 126 = 1400r \), then divide both sides by 1400 to isolate \( r \): \( r = \frac{126}{1400} \).

Step 4: Convert \( r \) to a percent

Simplify \( \frac{126}{1400} \) to obtain \( \frac{9}{100} \) as the simplified fraction. Convert this fraction into a percentage by multiplying by 100: \( \frac{9}{100} \times 100 = 9 \% \).

Key Concepts

Interest Rate Calculation

When dealing with simple interest, one of the crucial tasks is to calculate the interest rate. The formula for simple interest is:\[ i = P \times r \times t \]where:

• i is the interest earned or paid.
• P is the principal, or the initial amount of money.
• r is the interest rate.
• t is the time period the money is invested or borrowed, in years.

To find the interest rate \( r \), you need to rearrange the formula:\[ r = \frac{i}{P \times t} \]Utilizing this formula, you can calculate the interest rate by substituting the known values of interest, principal, and time.
This allows businesses and individuals to understand how much they would earn or owe in interest over a specified period.

Percentage Conversion

Once you have calculated the interest rate as a decimal, the next step is to convert this into a percentage. Percentages are more intuitive to understand for most people because they easily relate to parts per hundred.
To convert a decimal to a percentage, multiply the decimal by 100. For example, if you have computed the interest rate (r) as \(0.09\), you convert it as follows:\[ 0.09 \times 100 = 9\% \]
Always remember this step, as percentages provide a clearer picture, especially when you are communicating interest rates. They help in comparing different investment or loan options effectively by providing a uniform measurement.

Solving Equations

Solving the equation is a crucial part of finding the interest rate. After substituting the known values into the interest formula, you often arrive at an equation that needs to be simplified.
Consider the equation:\[ 126 = 700 \times r \times 2 \]
Start by simplifying:\[ 126 = 1400r \]
Then, isolate \( r \) by dividing both sides by 1400:\[ r = \frac{126}{1400} \]
Through simplification, you get:\[ r = \frac{9}{100} \]Solving equations step-by-step ensures accuracy and helps keep track of mathematical operations clearly.
It’s essential to follow through all operations systematically to avoid errors and reach the correct solution.