Problem 67 - Algebra 2

Question

Solve each equation or formula for the specified variable. $$ I=p r t, \text { for } t $$

Step-by-Step Solution

Answer

The formula for \( t \) is \( t = \frac{I}{pr} \).

Step 1: Understanding the Equation

The equation given is \( I = prt \). This is a formula for computing simple interest, where \( I \) is the interest, \( p \) is the principal amount, \( r \) is the rate, and \( t \) is the time. Our task is to solve this equation for the variable \( t \).

Step 2: Isolating the Variable

To solve for \( t \), we need to isolate \( t \) on one side of the equation. We start by dividing both sides of the equation \( I = prt \) by \( pr \) to get \( t \) alone. This gives us: \( \frac{I}{pr} = t \).

Step 3: Rewriting the Equation

After isolating \( t \), we rewrite the equation to clearly state \( t \) in terms of \( I \), \( p \), and \( r \). Therefore, \( t = \frac{I}{pr} \). This is the rearranged formula for calculating time \( t \).

Key Concepts

Simple Interest Formula

The simple interest formula is a fundamental concept in finance and mathematics. It helps you understand how interest works for loans and investments. The formula is: \[ I = prt \] Here's what each variable represents:

• I = Simple Interest: This is the amount of interest earned or paid over the specified period.
• p = Principal: The initial sum of money placed or borrowed.
• r = Rate: The interest rate, expressed as a decimal.
• t = Time: The period over which interest is calculated, often in years.

Simple interest calculations are much easier compared to compound interest, making them useful for understanding basic interest concepts. This formula is used in various real-life scenarios like calculating car loan interest or savings account interest. By understanding this formula, you can make better financial decisions.

Isolating Variables

In mathematics, isolating a variable means rearranging an equation so that one specific variable stands alone on one side. This is a crucial step in solving equations, allowing for better understanding and application of formulas. To isolate a variable like \( t \) in the simple interest formula \( I = prt \), you perform algebraic manipulations to express \( t \) in terms of the other variables \( I, p, \text{and } r \). Here’s how you can do it:

• The goal is to move all terms involving \( t \) to one side of the equation.
• In our equation \( I = prt \), you want to divide both sides by \( pr \) to isolate \( t \).
• This results in the equation \( t = \frac{I}{pr} \).

By successfully isolating the variable, you can solve the equation for different scenarios, such as finding the time required to earn a certain amount of interest, given the principal and rate.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations through various operations. This includes adding, subtracting, multiplying, dividing, and factoring within equations to solve for unknowns like a pro. Here are some basic steps in algebraic manipulation, using the simple interest formula example:

• Identifying operations: You decide what operations to use in isolating your desired variable. In \( I = prt \), the operation needed is division.
• Applying operations: You apply the operations across the equation to keep it balanced. So, you divide both sides of \( I = prt \) by \( pr \) to get \( \frac{I}{pr} = t \).
• Simplifying the equation: You rewrite the equation cleanly, ensuring the isolated variable is clear: \( t = \frac{I}{pr} \).

Good algebraic manipulation skills are key to success in mathematics. They help you derive solutions efficiently and work through real-world problem scenarios. Whether it’s isolating one variable or simplifying complex expressions, these skills aid in understanding and solving equations.