Question

For the real integers \(x\) and \(y\), what must \(\frac{2 x+2 y}{4} \quad\) equal? (A) The mode of \(x\) and \(y\) (B) \(x^{2}+y^{2}\) (C) The arithmetic mean of \(x\) and \(y\) (D) The median of \(2 x\) and \(2 y\)

Short answer

The expression equals the arithmetic mean of \(x\) and \(y\) (C).

Step-by-step solution

Simplify the Expression

We start with the expression \(\frac{2x + 2y}{4}\). We can simplify it by factoring out a 2 from the numerator: \(\frac{2(x + y)}{4}\). Simplify further: \(\frac{x + y}{2}\). This expression \(\frac{x + y}{2}\) represents the average of \(x\) and \(y\).

Identify the Matching Option

The options are: (A) The mode of \(x\) and \(y\), (B) \(x^2 + y^2\), (C) The arithmetic mean of \(x\) and \(y\), (D) The median of \(2x\) and \(2y\). From Step 1, we determined that \(\frac{x+y}{2}\) is the arithmetic mean of \(x\) and \(y\). Therefore, the correct choice is option (C).

Key concepts

Real Integers

In mathematics, real integers are a set of numbers that include all positive whole numbers, negative whole numbers, and zero. They do not include fractions, decimals, or imaginary numbers. The set of real integers can be represented as \(\{-\infty, ..., -3, -2, -1, 0, 1, 2, 3, ... ,\infty\}\u007f\). These numbers are foundational as they are used to build more complex number systems. They are also called whole numbers or simply integers.
Real integers are crucial because:

• They provide a way to express quantities without any fractional components.
• They are used in everyday counting and ordering.
• They serve as the basis for more advanced mathematical concepts, such as rational and complex numbers.

Understanding real integers helps pave the way for comprehending algebraic structures and solving a wide variety of mathematical problems.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This reduces complexity and makes calculations easier to perform and understand. Simplification usually involves combining like terms and factoring common values to streamline expressions.
In our exercise, we started with the expression \[ \frac{2x + 2y}{4} \]To simplify, we performed the following steps:

• Factor out the common factor (2) from the terms in the numerator to get \[ \frac{2(x + y)}{4} \]
• Divide the numerator and the denominator by 2, resulting in \[ \frac{x + y}{2} \]

This simplified form is not only easier to understand but also reveals the underlying structure and relationships between the variables involved. Mastery of simplifying expressions is essential for solving algebraic equations and making mathematical calculations more efficient.

Mathematical Averages

The arithmetic mean, often referred to simply as the average, is a measure of central tendency. It represents the sum of values divided by the number of values. This gives a central value around which the data tends to cluster.
In mathematical terms, for two variables \(x\) and \(y\), the arithmetic mean is calculated as:
\[ \text{Arithmetic Mean} = \frac{x + y}{2} \]
Finding the arithmetic mean helps in understanding the dataset, which is especially useful in analysis and decision-making processes.

• It provides a quick snapshot of the central position of the data.
• It is useful to compare different datasets.
• It is applicable in various fields beyond mathematics, such as economics, psychology, and everyday decision making.

Understanding how to calculate and interpret the arithmetic mean is fundamental in statistics and valuable in various real-world applications.