Problem 67 - A Graphical Approach to Precalculus with Limits

Question

Solve each formula for the specified variable. \(I=P R T\) for \(P \quad\) (Simple interest)

Step-by-Step Solution

Answer

The solution is: \(P = \frac{I}{RT}\).

Step 1: Identify the Given Formula

The given formula is the simple interest formula: \(I = PRT\), where \(I\) is the interest, \(P\) is the principal amount, \(R\) is the rate of interest, and \(T\) is the time period.

Step 2: Understand the Target Variable

We need to solve for \(P\), which is the principal amount in the given formula.

Step 3: Rearrange the Formula

To isolate \(P\), divide both sides of the equation by \(RT\): \[P = \frac{I}{RT}\].

Step 4: Confirm the Solution

Check that the formula \(P = \frac{I}{RT}\) makes logical sense by plugging back the rearranged terms to ensure it matches with the simple interest formula.

Key Concepts

Principal Amount

The principal amount is the original sum of money that is being invested or loaned. It is crucial in financial calculations because it serves as the base amount upon which interest is calculated. Understanding the principal is simple:

• It is denoted by the symbol 'P' in formulas.
• In financial terms, it represents the initial capital or money.
• The principal does not include any interest earned or paid over time.

For example, if you deposit $1,000 into a savings account, that $1,000 is your principal amount. Over time, as this money earns interest, you will have more than the principal available — but to calculate simple interest, you'll continue to use the original sum.

Interest Rate

The interest rate is a percentage that reflects how much interest will be charged or earned based on the principal amount over a specific time period. It is essential to understanding how your money will grow or how much you'll owe in a loan.

• Represented by 'R' in the simple interest formula.
• Given as a percentage, such as 5% or 10%, but for calculations, it's converted into a decimal like 0.05 or 0.10.

Knowing the interest rate allows you to determine how fast your investment will grow or your loans will cost. For instance, with a principal of \(1,000 and an interest rate of 5%, you can expect to earn \)50 in one year if using simple interest, calculated as follows: \(I = P \times R \times T = 1000 \times 0.05 \times 1\).

Time Period

The time period is a critical element in calculating simple interest. It defines how long the money is invested or borrowed without compounding, which means the interest is not added back to the principal.

• Denoted by 'T' in the formula, usually measured in years.
• It represents the duration for which the principal is utilized to calculate interest.

Understanding the time period helps in planning investments or loans. A longer time period generally results in more interest accrued since interest will be calculated over many years. For example, if you keep $1,000 in a bank for 3 years at an interest rate of 5%, the interest earned would be: \(I = P \times R \times T = 1000 \times 0.05 \times 3 = 150\).

Rearranging Formulas

Rearranging formulas is a powerful mathematical tool that allows you to solve for different variables or unknowns. It's about isolating the variable you want to find on one side of the equation. In the simple interest formula:

• Start from: \(I = P \times R \times T\)
• If you need to solve for one specific variable, manipulate the equation. For example, to find the principal amount, divide both sides by the product of R and T:
• Rearranged: \(P = \frac{I}{R \times T}\)

This method allows you to transform the foundation of your formula to suit your calculation needs. It's useful in numerous real-life situations where certain variables remain unknown, letting you extract the necessary information effectively. Always check your rearranged formula by plugging in known values to ensure its correctness.