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 interest rate, \(r\), is 14.5\%.

Step 1: Understand the Problem

We need to rearrange the formula \(i = Prt\) to solve for the interest rate, \(r\). Then we'll substitute the given values for \(i\), \(P\), and \(t\), and calculate \(r\). Finally, express \(r\) as a percentage.

Step 2: Rearrange the Formula

To solve for \(r\), divide both sides of the equation \(i = Prt\) by \(Pt\):\[ r = \frac{i}{Pt}. \]

Step 3: Substitute Given Values

Substitute \(i = 159.50\), \(P = 2200\), and \(t = 0.5\) into the equation: \[ r = \frac{159.50}{2200 \, \times \, 0.5}. \]

Step 4: Calculate \(r\)

Perform the multiplication and division: \(2200 \, \times \, 0.5 = 1100\), then \( r = \frac{159.50}{1100} \approx 0.145\).

Step 5: Convert to Percentage

Convert \(r = 0.145\) to a percentage by multiplying by 100: \(0.145 \times 100 = 14.5\%\).

Key Concepts

Interest Rate Calculation

Understanding how to calculate the interest rate from the simple interest formula is a key skill in managing finances. We use the formula for simple interest, which is: \[i = Prt\]Where:- \(i\) is the interest earned,- \(P\) is the principal amount,- \(r\) is the interest rate,- \(t\) is the time period the money is invested or borrowed for.To find the interest rate \(r\), we need to rearrange this formula. Dividing both sides of the equation by \(Pt\) gives us the formula to calculate \(r\): \[r = \frac{i}{Pt}\]In the given problem, we are provided with a specific interest amount, principal, and time period, and by inserting these values into the rearranged formula, we can solve for \(r\). This simple rearrangement allows us to quantify how much return we can get on our investment over a given time, knowing how much interest we've earned.

Percentage Conversion

Once we have determined the interest rate as a decimal, the next step is to convert it into a percentage. Conversion between decimals and percentages is essential in various fields, such as finance and statistics. Here's how you perform this conversion:- When working with decimals, multiplying by 100 converts the decimal to a percentage.- Move the decimal point two places to the right to achieve the same effect.For example, in the original exercise, once the rate \(r\) is found as \(0.145\), converting it to a percentage involves this simple step: \(0.145 \times 100 = 14.5\%\)Expressing the rate as a percentage provides an easily understandable way to communicate the interest and compare it easily to other rates. Remember, the percentage value is more intuitive and preferable for interpretation in most practical situations.

Algebraic Rearrangement

Rearranging algebraic equations is a fundamental skill that simplifies solving for unknowns. This involves performing operations to isolate the variable of interest. Here are the main steps taken in the original exercise:- Begin with the equation: \(i = Prt\)- If you're solving for the variable \(r\), identify what needs to be done to isolate \(r\).- Divide both sides of the equation by the product \(Pt\) to achieve this:\[r = \frac{i}{Pt}\]This transformation allows us to plug in known values and simplifies the process of finding the unknown variable. Such rearrangements are applicable in many problem-solving scenarios in mathematics and science alike, where clarity and simplicity are necessary to find solutions efficiently.