Problem 14 - Algebra 2

Question

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

Step-by-Step Solution

Answer

\( p = \frac{I}{rt} \)

Step 1: Identify the Known Equation

The equation provided is the formula for simple interest: \(I = prt\), where \(I\) stands for interest, \(p\) represents the principal amount, \(r\) is the rate, and \(t\) is the time.

Step 2: Isolate the Specified Variable

The problem asks us to solve for \(p\), the principal. We need to rearrange the equation to make \(p\) the subject. Currently, \(p\) is multiplied by \(r\) and \(t\).

Step 3: Rearrange the Equation

To isolate \(p\), divide both sides of the equation by \(rt\). This gives us: \[ p = \frac{I}{rt} \].

Step 4: Verifying the Solution

Check that the equation is solved correctly by substituting \(p = \frac{I}{rt}\) back into the original equation. Doing this, we get \(I = prt = \left(\frac{I}{rt}\right)rt = I\), confirming our solution is correct.

Key Concepts

Simple Interest Formula

The Simple Interest Formula is a straightforward way to calculate interest over time. It's an essential concept in mathematics, especially when dealing with financial calculations. The formula is expressed as \( I = prt \), where:

• \( I \) is the interest earned.
• \( p \) is the principal amount, which is the initial sum of money.
• \( r \) is the interest rate, usually expressed as a decimal.
• \( t \) is the time the money is invested or borrowed for, typically in years.

To find any of these values when you have the others, you'll need to rearrange the formula, as taught in algebra. This formula doesn't account for the effect of compounding, making it simpler and thus easier for basic calculations.

Variable Isolation

In algebra, isolating a variable means solving the equation so that the variable you are interested in is alone on one side of the equation. In the context of our simple interest problem, we want to find "\( p \)" by itself.

To begin isolating \( p \), identify what operations are currently affecting it. Here, \( p \) is multiplied by \( r \) and \( t \). Thus, to isolate \( p \), you will need to perform the opposite operations.

• The objective is to undo what is done to \( p \), which means dividing by \( rt \) as both \( r \) and \( t \) are multiplied with \( p \).

Mastering variable isolation is crucial for solving equations, as it makes sure you understand how each term affects the variable you are trying to solve for.

Equation Rearrangement

Rearranging equations is often necessary in algebra. It involves changing the structure of an equation to make it easier to solve a specific part. In our case, we needed to change the setup of the simple interest formula to solve for the principal, \( p \).

The original equation is \( I = prt \).

• To focus on \( p \), divide every term by \( rt \).
• This rearrangement yields the formula \( p = \frac{I}{rt} \).

By rearranging, you modify the equation without changing its meaning, allowing you to solve for any part once other values are known. This flexibility highlights the power of algebra in practical problem-solving.

Substitution Method

The substitution method is a powerful tool in algebra to check the validity of solutions. Once you have rearranged an equation and isolated a variable, substitution allows you to verify your answer.

In our example, after finding \( p = \frac{I}{rt} \), substituting this back into the original formula \( I = prt \) helps ensure the algebraic manipulation was correct:

• Substitute \( p \) with \( \frac{I}{rt} \). This results in \( I = \left(\frac{I}{rt}\right)rt \).
• The \( rt \) terms cancel out, verifying that \( I = I \).

Thus, the substitution method not only checks your solution but also reinforces your understanding of how equations behave during manipulation.