Problem 11 - Algebra and Trigonometry

Question

Write an algebraic formula for the given quantity.. The average \(A\) of two numbers \(a\) and \(b\)

Step-by-Step Solution

Answer

The formula is \(A = \frac{a + b}{2}\).

Step 1: Understanding the Average

The average of two numbers is found by adding the numbers together and dividing by two. Here, we need to find the formula for the average of the numbers \(a\) and \(b\).

Step 2: Add the Numbers

First, add the two numbers \(a\) and \(b\) together. The sum can be written as \(a + b\).

Step 3: Divide the Sum by Two

Take the sum \(a + b\) and divide it by two to find the average. This is represented as \(\frac{a + b}{2}\).

Step 4: Write the Formula

The algebraic formula for the average \(A\) of the two numbers \(a\) and \(b\) is \(A = \frac{a + b}{2}\).

Key Concepts

Algebraic Expressions

Algebraic expressions are a way to represent numbers and operations using variables and constants. They allow us to generalize a problem so it can be applied to any specific situation. For example, in the calculation of an average, we often use the expression \( \frac{a + b}{2} \) to find the average of any two numbers \(a\) and \(b\).

This expression consists of:

• Variables like \(a\) and \(b\) that stand for numbers.
• Operations such as addition \((+)\) and division \((\div)\), which show how numbers are processed.

To write an effective algebraic expression, understanding the elements involved is crucial as it can represent countless possible values and scenarios.

Step-by-Step Problem Solving

The key to problem-solving in mathematics is to break down tasks into manageable steps. This method provides clarity and direction, ensuring that each part of the problem is understood and tackled systematically.

For instance, in the given problem of finding the average of two numbers \(a\) and \(b\), the process involves:

• Understanding what average means: It's the central value of two numbers, calculated by summing the numbers and dividing by their count.
• Adding the two numbers: First, calculate \(a + b\).
• Dividing the sum by two: This gives us the average \(\frac{a + b}{2}\).

Each step builds on the last, ensuring a logical progression from understanding the context to coming up with the solution. This approach can be applied to various mathematical problems.

Mathematical Formula Derivation

Deriving mathematical formulas allows us to create a general method for solving similar problems with different inputs. The process ensures that the formula is accurate by logically following the mathematical principles at work.

For the average of two numbers, the derivation follows these principles:

• Identify the operation: The goal is to find the average.
• Establish operations: Begin with summing the numbers \(a\) and \(b\), resulting in \(a + b\).
• Divide by the count: Since there are two numbers, dividing by 2 gives the average.

This logical derivation results in the formula \(A = \frac{a + b}{2}\), exemplifying how mathematical reasoning leads to a reliable and reusable formula. Understanding how each step connects helps in grasping why the formula is structured as it is.