Problem 68 - Prealgebra

Question

Jake ran \(x\) miles on Monday, \(y\) miles on Tuesday, and \(z\) miles on Wednesday. Write an expression for the average number of miles Jake ran.

Step-by-Step Solution

Answer

\[ \frac{x + y + z}{3} \]

Step 1: Understand the Concept of Average

The average of a set of numbers is obtained by adding all the numbers together and then dividing by the count of the numbers. In this problem, we will calculate the average number of miles Jake ran over three days.

Step 2: Set Up the Expression to Sum the Miles

Write an expression that sums the miles Jake ran over the three days: On Monday, Tuesday, and Wednesday, he ran a total of: \[ S = x + y + z \]

Step 3: Divide the Sum by the Total Days

Since Jake ran on three different days, divide the total distance by 3 to find the average number of miles per day: \[ \text{Average} = \frac{S}{3} = \frac{x + y + z}{3} \]

Step 4: Finalize the Expression

The final expression for the average number of miles per day is: \[ \frac{x + y + z}{3} \]

Key Concepts

Understanding Averages

Finding the average of a set of numbers is an essential skill in both mathematics and everyday life. An average gives us a sense of what is typical or expected in a given set of data. To calculate the average, follow these simple steps:

• Add up all the values in the set. In Jake's case, this means adding up the miles he ran over three days: Monday, Tuesday, and Wednesday.
• Count how many values there are. For Jake, this is the three days of running.
• Divide the total sum of the values by the count of the values. This gives you the average.

So, if Jake ran \(x\) miles on Monday, \(y\) miles on Tuesday, and \(z\) miles on Wednesday, the average is calculated by the expression \( \frac{x + y + z}{3} \). Through this method, we can standardize multiple values into one understandable figure.

Role of Variables

Variables often serve as placeholders in mathematical expressions representing unknown or changing quantities. In algebraic expressions, letters like \(x\), \(y\), and \(z\) make it easy to generalize problems for different situations.

When Jake's running distances are described by \(x\), \(y\), and \(z\), each variable stands for the mileage of a particular day but also allows flexibility:

• They can represent any real number, which means they can accommodate actual data from Jake's week as well as any theoretical scenario.
• Using variables makes equations and expressions versatile because they are not tied to specific numbers.
• In calculations like average, variables facilitate finding a general formula applicable to various contexts with varying inputs.

Using these variables in expressions helps maintain simplicity and broad applicability, essential features in algebra.

Applying Arithmetic Operations

Arithmetic operations are the building blocks of mathematics and are crucial when working with expressions. The main arithmetic operations include addition, subtraction, multiplication, and division. In the context of finding Jake's average running distance, we primarily use addition and division:

• Addition: Combine all the miles Jake ran over the three days, resulting in the expression \(x + y + z\).
• Division: Calculate the average by dividing the total sum of miles by the number of days, which yields \( \frac{x + y + z}{3} \).

This process of using arithmetic operations is pivotal in breaking down problems into manageable parts. By consistently applying these operations, we transform raw data into valuable insights, like Jake's average running distance per day.