Problem 71 - Algebra for College Students

Question

Solve \(i=\) Prt for \(r\), given that \(i=\$ 159.50, P=\$ 2200\), and \(t=0.5\) of a year. Express \(r\) as a percent.

Step-by-Step Solution

Answer

The rate \( r \) is 14.5\%.

Step 1: Understand the Formula

We start with 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, and \( t \) is the time in years. Our goal is to solve for \( r \).

Step 2: Substitute Known Values

Substitute the given values into the formula: \( i = 159.50\), \( P = 2200\), and \( t = 0.5 \). This gives us: \\[ 159.50 = 2200 imes r imes 0.5 \]

Step 3: Rearrange the Equation

Rearrange the equation to solve for \( r \). Divide both sides by the product of \( P \) and \( t \): \\[ r = \frac{159.50}{2200 imes 0.5} \]

Step 4: Calculate the Value of r

Calculate the value of \( r \) using the rearranged equation: \\[ r = \frac{159.50}{1100} \] \Perform the division: \\[ r = 0.145 \]

Step 5: Convert r to a Percentage

Convert the decimal rate to a percentage by multiplying by 100: \\[ r = 0.145 imes 100 = 14.5\% \]

Step 6: Verify the Solution

To ensure the calculation is correct, we should substitute \( r = 14.5\% \) back into the original equation to verify that the interest matches \( i = 159.50 \). Substitute \( r \) as a decimal (0.145): \\[ 2200 \times 0.145 \times 0.5 = 159.50 \] \Since this is true, the solution is verified.

Key Concepts

Solving Equations

When dealing with simple interest problems, solving for the unknown variable often requires rearranging equations. In our problem, we have the formula for simple interest:

• \(i = Prt\)
• Where:
  • \(i\) is the interest
  • \(P\) is the principal amount
  • \(r\) is the rate of interest
  • \(t\) is the time in years

Given:

• \(i = 159.50\) (interest earned)
• \(P = 2200\) (principal)
• \(t = 0.5\) year

We need to solve for the interest rate, \(r\). Initially, substitute the known values into the equation:\[159.50 = 2200 \times r \times 0.5\]Rearrange to isolate \(r\): divide both sides by \(2200 \times 0.5\). This manipulation gives:\[r = \frac{159.50}{1100}\]Once arranged in this form, you can easily solve for \(r\) by performing the division.

Interest Rate Calculation

Calculating the interest rate, especially with simple interest, is straightforward. Once the equation to solve for \(r\) is set:

• Substitute the known values (we had: \(i = 159.50\), \(P = 2200\), \(t = 0.5\))
• Rearrange to obtain:\[r = \frac{159.50}{2200 \times 0.5}\]
• Which simplifies to:\[r = \frac{159.50}{1100}\]

Perform the division:

• Result: \(r = 0.145\)

This is the decimal form of our interest rate. You need this division to determine how much interest grows per unit of the principal over the given time. Make sure to solve carefully to ensure the figures add up correctly.

Percent Conversion

Once the calculation for the interest rate \(r\) is complete in its decimal form, converting it to a percentage makes it more intuitive. Percentages are easier to understand and communicate. Here’s how you do it:The decimal value we found was \(r = 0.145\). To convert it to a percentage:

• Multiply the decimal by 100.\[r = 0.145 \times 100 = 14.5\%\]

This means the interest problem’s effective rate is 14.5%. Generally, converting decimals to percentages is always done by multiplying by 100.
This step is crucial because it presents the rate in a way that's commonly used in financial discussions. It allows for a clear and concise display of how interest impacts the principal over the time period given.