Question

List some formulas that occur in applications. Solve each formula for the indicated variable. $$ \text { Finance } A=P(1+r t) \text { for } r $$

Short answer

The solution for \(r\) is \(r = \frac{\frac{A}{P} - 1}{t}\).

Step-by-step solution

Identify the Given Formula

The formula provided is for finance: \(A = P(1 + rt)\). The goal is to solve for the variable \(r\).

Isolate the Term with the Variable

First, isolate the term containing \(r\). Divide both sides of the equation by \(P\): \(\frac{A}{P} = 1 + rt\).

Subtract 1 from Both Sides

To isolate \(rt\), subtract 1 from both sides of the equation: \(\frac{A}{P} - 1 = rt\).

Solve for the Variable \(r\)

Finally, divide both sides of the equation by \(t\) to solve for \(r\): \(r = \frac{\frac{A}{P} - 1}{t}\).

Key concepts

Financial equations

Financial equations play a critical role in understanding and managing personal and business finances. They help in calculating values like future investments, loans, and interests. One common formula is the Future Value formula expressed as: \(A = P(1 + rt)\), where:

• A is the amount of money accumulated after a certain period.
• P is the principal amount (initial investment).
• r is the rate of interest.
• t is the time over which interest is applied.

These equations are foundational in finance for planning and decision-making. Understanding them is essential for tasks such as budgeting, saving, and investing. A strong grasp of financial equations helps in achieving long-term financial goals.

Solving for a variable

Solving for a variable means isolating that variable on one side of the equation. It involves manipulation to express the variable in terms of the other given quantities. Given the equation \(A = P(1 + rt)\):

• Identify the formula and the target variable, in this case, solving for \(r\).
• Isolate the term containing the variable. Start by dividing both sides by \(P\) to get \(\frac{A}{P} = 1 + rt\).
• Subtract 1 from both sides to isolate \(rt\): \(\frac{A}{P} - 1 = rt\).
• Finally, divide both sides by \(t\) to solve for \(r\): \(r = \frac{\frac{A}{P} - 1}{t}\).

Mastering these steps is crucial for solving various types of equations, making mathematical manipulation much more intuitive.

Interest rate calculation

Interest rate calculation is key to understanding how much interest will accumulate over time on savings or loans. The interest rate, denoted as \(r\), can be found by rearranging the Future Value formula: \(A = P(1 + rt)\).
To find \(r\):

• First, isolate the term with \(r\) by dividing both sides by \(P\): \(\frac{A}{P} = 1 + rt\).
• Subtract 1 from both sides to isolate \(rt\): \(\frac{A}{P} - 1 = rt\).
• Divide by \(t\) to solve for \(r\): \(r = \frac{\frac{A}{P} - 1}{t}\).

This formula helps in understanding how different interest rates and time periods affect the future value of investments or loans. Being adept at these calculations is beneficial for making informed financial decisions.

Algebraic manipulation

Algebraic manipulation involves the use of various techniques to rearrange and solve equations. It is critical for solving financial formulas and other mathematical problems. The process includes:

• Isolating the variable you need to solve for.
• Using operations such as addition, subtraction, multiplication, and division.
• Maintaining balance in the equation by performing the same operations on both sides.

In the equation \(A = P(1 + rt)\):

• Divide both sides by \(P\) to get \(\frac{A}{P} = 1 + rt\).
• Subtract 1 from both sides: \(\frac{A}{P} - 1 = rt\).
• Divide by \(t\) to isolate \(r\): \(r = \frac{\frac{A}{P} - 1}{t}\).

Practicing algebraic manipulation with various equations makes it easier to solve complex problems encountered in finance and other fields. It enhances problem-solving skills and mathematical reasoning.