Problem 5 - Algebra for College Students

Question

Solve \(i=\) Prt 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 the simple interest formula: \(i = P \cdot r \cdot t\), where \(i\) is the interest, \(P\) is the principal amount, \(r\) is the rate of interest (as a decimal), and \(t\) is the time in years.

Step 2: Substitute the Given Values

Substitute the values into the formula: \(P = 600\), \(t = 2.5\) years, and \(i = 90\). This gives us the equation: \(90 = 600 \cdot r \cdot 2.5\).

Step 3: Solve for \(r\)

Rearrange the equation to solve for \(r\). Divide both sides by \(600 \cdot 2.5\) to isolate \(r\): \(r = \frac{90}{600 \cdot 2.5}\).

Step 4: Calculate \(r\)

Calculate \(r\): \(r = \frac{90}{1500} = 0.06\).

Step 5: Convert \(r\) to a Percentage

Multiply \(r\) by 100 to convert it from a decimal to a percentage: \(r = 0.06 \times 100 = 6\%\).

Key Concepts

Interest Rate Calculation

When dealing with simple interest, calculating the interest rate is an important step, especially if it's unknown. Simple interest is calculated using the formula: \( i = P \cdot r \cdot t \).

• \( i \): Interest, the extra amount charged or earned.
• \( P \): Principal amount, the initial sum of money.
• \( r \): Interest rate, expressed as a decimal.
• \( t \): Time in years.

To find the interest rate \( r \), you rearrange the formula:1. Place all known values on one side of the equation.2. Isolate \( r \) by dividing by the product of \( P \) and \( t \).For example, if you know: \( i = 90 \), \( P = 600 \), \( t = 2.5 \), calculate \( r \) as follows: \[ r = \frac{i}{P \cdot t} = \frac{90}{600 \cdot 2.5} = 0.06 \]Finally, converting this decimal to a percentage gives \( 6\% \). Always remember that percentages give a clearer picture of rates and are easier to understand.

Principal Amount

The principal amount \( P \) is the initial sum of money that earns or incurs interest over a period. It forms the base over which interest calculations are made. When calculating simple interest, knowing the principal is crucial, as it helps you understand how much money is either being invested or charged interest on. Important points about the principal amount:- It's your starting financial figure.- Interest accrued over time builds upon this initial amount.- Helps determine total interest when combined with rate and time.When solving for interest rate or interest payment, you substitute this value in the equation for \( P \). For example, in our exercise, \( P = 600 \), gives us a clear starting point for our calculations.

Time in Years

Time \( t \), expressed in years, is another key component of the simple interest formula. It's the duration over which the principal amount is either earning or being charged interest. Why is time in years important?- It directly affects the amount of interest: longer time means more interest, assuming rate and principal remain constant.- It modifies the effect of the interest rate: the same rate will yield different interest sums if the time periods differ.Time must be correctly converted into years if not already in that form, while fractions of a year should be dealt with accordingly, such as converting months into a decimal. For instance, 2 years and 6 months becomes \( t = 2.5 \) years in numerical terms.In our example problem, time is given as \( 2.5 \) years, directly influencing the calculated interest by its length.