Problem 66 - Elementary Algebra

Question

For the following problems, solve each literal equation for the designated letter. \(A=P(1+r t)\) for \(r\).

Step-by-Step Solution

Answer

Question: Solve the given compound interest formula, \(A = P(1 + rt)\), for the variable \(r\). Answer: \(r = \frac{A - P}{Pt}\)

Step 1: Write down the given equation

The given equation is \(A=P(1+rt)\), where we need to solve for \(r\).

Step 2: Divide both sides by P

To isolate the term containing \(r\), we will first divide both sides of the equation by \(P\): \(\frac{A}{P} = 1 + rt\)

Step 3: Subtract 1 from both sides

Now, we want to isolate the term containing \(r\). To achieve this, we will subtract 1 from both sides of the equation: \(\frac{A}{P} - 1 = rt\)

Step 4: Divide both sides by t

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

Step 5: Simplify the expression

Now, we can simplify the expression for \(r\) by combining the fractions: \(r = \frac{A - P}{Pt}\) The literal equation is now solved for \(r\): \(r = \frac{A - P}{Pt}\)

Key Concepts

Algebra

Algebra is a fundamental branch of mathematics focused on solving equations and understanding how various elements can relate to one another. In algebra, letters and symbols are used to represent numbers and quantities in formulas and equations.
This method allows us to formulate generalized rules and principles that can be applied universally. The power of algebra stems from its ability to model real-world situations and solve for unknown values.

Key skills in algebra include:

• Recognizing patterns and relationships between numbers and variables.
• Understanding how to manipulate mathematical expressions to simplify or solve them.
• Knowledge of solving various types of equations, including literal equations, which express relationships between variables with known quantities.

Mastering these skills can significantly enhance your problem-solving abilities and mathematical literacy.

Solving Equations

Solving equations is at the heart of algebra, and it involves finding the value of unknown variables that make the equation true. To solve equations effectively, one must isolate the variable of interest. This process often requires multiple steps and different operations.
In the given exercise, the goal was to solve the literal equation for the variable \( r \). The steps involved simplifying the equation to isolate \( r \):

• Starting with the given equation \( A = P(1 + rt) \), the first step was to divide both sides by \( P \) to eliminate it from the right side.
• Next, subtracting 1 helped further isolate the \( rt \) term.
• Finally, dividing by \( t \) provided the solution for \( r \).

Each of these operations takes us closer to revealing the value of the unknown, which in this case was \( r \). The importance of solving equations lies in its application to real-world problems, such as financial calculations or scientific measurements.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operation symbols that represent a value. In algebra, these expressions can often be simplified or manipulated to solve equations. Understanding and working with expressions is key to mastering literal equations.

Let's break down the expression in the exercise:

• We started with \( A = P(1 + rt) \), an expression involving several operations and terms.
• The process of solving for \( r \) involved simplifying this expression by strategically removing terms and isolating the desired variable.
• Each step, like dividing or subtracting, was aimed at transforming the expression into a more manageable form.

Easier mathematical manipulation and interpretation of expressions lead to clearer understanding and solutions. It's an essential skill as it helps decode complex relationships in numerous real-life situations.