Problem 12 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The average is \(A = \frac{a + b + c}{3}\).

Step 1: Understand the Concept of Average

The average of a set of numbers is calculated by summing all the numbers in the set and then dividing the sum by the count of numbers. In this problem, we're considering three numbers: \(a, b,\) and \(c\).

Step 2: Sum the Numbers

To find the sum of the three numbers given, simply add them together. This is written mathematically as: \(a + b + c\).

Step 3: Divide by the Number of Elements

Since there are three numbers (\(a, b,\) and \(c\)), divide the sum obtained in Step 2 by 3. The formula for the average \(A\) is therefore: \[ A = \frac{a + b + c}{3} \]

Step 4: Write the Final Formula

Combine the results from the previous steps to express the final formula for the average of the three numbers: \(A = \frac{a + b + c}{3}\). This is the algebraic formula that represents the average of the three numbers \(a, b,\) and \(c\).

Key Concepts

Average Calculation

The concept of average in mathematics refers to finding a central value of a set of numbers. To calculate the average of a group of numbers, you first need to sum all the numbers. Then, you divide the total sum by the number of items in the group.
For example, given three numbers such as \(a\), \(b\), and \(c\), the process involves these steps:

• Calculate the total sum of the numbers. So, add \(a\), \(b\), and \(c\) to find the combined value.
• Count how many numbers you have. In this case, it's three numbers.
• Divide the total sum by the count of numbers. This gives you the average value of these three numbers.

Using the formula for average, you express it as: \[ A = \frac{a + b + c}{3} \] This formula is crucial for determining the central value when three numbers are involved. It's a fundamental concept in algebra that simplifies understanding and interpreting data.

Sum of Numbers

The sum of numbers is one of the most straightforward mathematical operations. To sum three numbers, all you need to do is add them together. This operation forms the foundation for many other calculations including finding an average or total.

• To find the sum of \(a\), \(b\), and \(c\), simply write it as \(a + b + c\).
• This addition consolidates multiple values into one single number, making it easier to work with in later calculations.

The operation of summing numbers forms the first step in various algebraic processes, such as calculating the average or even finding a total value over time or items.
Understanding sum as a concept is crucial since it appears in diverse areas of algebra, simplifying many problems and equations. It’s essential for mastering algebra and mathematics in general.

Division in Algebra

Division plays a critical role in algebra, especially when calculating averages, proportions, or rates. In the context of finding an average, division is used to spread the total sum across the number of items.
For three numbers \(a\), \(b\), and \(c\), division helps you convert the total sum into a more interpretable measure—the average. Here’s how division fits into the algebraic process:

• First, complete the sum of all numbers involved. In our case, this is \(a + b + c\).
• Next, identify how many numbers are summed. Here, there are three.
• Then, divide the sum by this number of elements. The expression is then \(\frac{a + b + c}{3}\).

This division operation means you’re spreading the sum equally among the counted numbers to find a mean value.
Division in algebra isn't just limited to finding averages. It helps solve many equations and distribute values evenly, making it a versatile tool in mathematics.