Problem 4 - Algebra for College Students

Question

4\. Solve \(i=P r t\) for \(t\), given that \(P=\$ 250, r=12 \%\), and \(i=\$ 120\).

Step-by-Step Solution

Answer

The value of \( t \) is 4.

Step 1: Identify the Formula

The formula we are using is the simple interest formula given by \( i = P \cdot r \cdot t \), where \( i \) is the interest, \( P \) is the principal, \( r \) is the rate, and \( t \) is the time.

Step 2: Convert Percentage Rate to Decimal

Convert the percentage rate of 12% into a decimal for our calculations. This is done by dividing by 100: \( r = \frac{12}{100} = 0.12 \).

Step 3: Substitute Known Values into the Equation

Substitute the known values into the formula: \( 120 = 250 \cdot 0.12 \cdot t \).

Step 4: Simplify the Equation

Multiply \( 250 \cdot 0.12 \) to simplify the equation: \( 120 = 30 \cdot t \).

Step 5: Solve for t

Solve for \( t \) by dividing both sides of the equation by 30: \( t = \frac{120}{30} = 4 \).

Key Concepts

Understanding Percentage Rate

Percentage rate is a way of expressing a number as a fraction of 100. For instance, if you have a percentage rate of 12%, it is important to note that this is another way of writing 12 out of 100.
To use it in formulas like the simple interest formula, we need to convert it to a decimal.
This conversion is done by dividing the percentage by 100. For example, 12% becomes 0.12 when expressed as a decimal.

• This conversion is crucial because it allows us to perform mathematical operations correctly in equations.
• Always double-check your conversion to avoid errors in your computations.

Understanding how to adjust percentage rates into decimals is essential for accurately solving many mathematical problems involving interests and percentages.

Solving Equations Efficiently

Solving equations involves finding the value of the unknown variable. In simple interest problems, this often means rearranging the formula to solve for the desired variable, such as time, rate, or principal.
For the equation given: \(i = P \cdot r \cdot t\), you may need to isolate the variable \(t\) to find the time.

• Start by identifying your known values and substituting them into the equation.
• After substitution, simplify the equation by performing any necessary arithmetic operations like division or multiplication.
• Finally, isolate the variable by performing inverse operations like division to find its value.

Pay close attention to correctly carry out each step and verify your solution by substituting the found value back into the original equation.

The Process of Interest Calculation

Interest calculation using the simple interest formula is a straightforward approach to find how much interest a sum of money will earn over a period of time.
The key here is to correctly apply the formula \(i = P \cdot r \cdot t\), where each term accurately represents your specific scenario.

• Identify your principal (\(P\)), which is the initial amount of money.
• Determine the rate of interest (\(r\)), remembering to convert it to a decimal.
• Understand the time period (\(t\)), which often needs to be expressed in years.

This calculation helps predict financial growth or loans, making it an important skill for personal and professional finance management. Properly understanding each component ensures accurate and meaningful results.