Problem 5 - Algebra for College Students

Question

5\. Solve \(i=P r t\) for \(r\), given that \(P=\$ 600, t=2 \frac{1}{2}\) years, and \(i=\$ 90\). Express \(r\) as a percent.

Step-by-Step Solution

Answer

The interest rate \(r\) is 6\%.

Step 1: Understand the Formula

The formula given is for simple interest: \(i = P \cdot r \cdot t\). Here, \(i\) is the interest earned, \(P\) is the principal amount, \(r\) is the rate of interest, and \(t\) is the time in years. We want to solve for \(r\).

Step 2: Substitute the Given Values

We are given \(P = 600\), \(t = 2.5\), and \(i = 90\). Substitute these values into the formula: \[ 90 = 600 \cdot r \cdot 2.5 \].

Step 3: Simplify the Equation

Simplify the equation by performing the multiplication on the right side: \[ 90 = 1500 \cdot r \].

Step 4: Solve for r

To find \(r\), divide both sides of the equation by 1500: \[ r = \frac{90}{1500} \].

Step 5: Calculate r

Perform the division: \[ r = 0.06 \].

Step 6: Convert to Percentage

Convert \(r\) from a decimal to a percentage by multiplying by 100: \[ r = 0.06 \times 100 = 6\% \].

Key Concepts

Interest Rate

Interest rate is a key concept in finance that determines how much extra money you pay or earn over a period of time on borrowed or invested principal amounts. It is expressed as a percentage. In the context of the simple interest formula, the rate is what you are looking to solve for, when you know how much interest has been earned or paid.

• The higher the interest rate, the more you will earn or pay over time.
• Expressed as a decimal before calculating; as a percentage afterwards.

For example, if you borrow money with an interest rate of 5%, it means you will pay an additional 5% of the principal amount each year.

Principal Amount

The principal amount is the initial sum of money borrowed or invested, before earning or paying additional interest. It is the value of the actual loan or investment, not taking into account any additional interest that accumulates over time.

• Original sum of money in finance calculations.
• Does not include interest or charges.
• Essential for calculating total costs or returns over time.

In our example, the principal amount is $600. This is the base upon which interest is calculated, meaning our calculations start here to determine any earned or owed interest.

Time in Years

Time in years is the period over which the simple interest is calculated. It reflects how long you will be borrowing the money or holding an investment. This is crucial because even a small interest rate can accumulate to a significant amount over a longer period.

• Expressed in years, potentially as fractions (like 2.5).
• Longer periods usually mean more interest earned or owed.

In this example, the time is 2.5 years. It's essential to include exact timeframes, including fractions, to accurately calculate total interest using the formula.