Question

If the average of \(6,3,-2,5,11,\) and \(x\) is \(5,\) what is the value of \(x ?\)

Short answer

x = 7

Step-by-step solution

Understand the Average Formula

The average of a set of numbers is the sum of the numbers divided by the count of numbers. For a set of numbers \{a_1, a_2, ..., a_n\}, the average is given by \[ \frac{a_1 + a_2 + ... + a_n}{n} \]

Set Up the Equation

Given the numbers are \(6, 3, -2, 5, 11, \) and \(x\), and the average is \(5\). Therefore, the equation becomes \[ \frac{6 + 3 - 2 + 5 + 11 + x}{6} = 5 \]

Sum the Known Numbers

First, sum the known numbers: \(6 + 3 - 2 + 5 + 11 = 23\). Now the equation is \[ \frac{23 + x}{6} = 5 \]

Solve for \(x\)

Multiply both sides of the equation by \(6\) to isolate \(x\): \[ 23 + x = 30 \]. Next, subtract \(23\) from both sides: \[ x = 30 - 23 \]. Hence, \[ x = 7 \]

Key concepts

solving algebraic equations

Algebraic equations are mathematical statements that use variables and constants along with arithmetic operations to show equality. Solving them often involves finding the value of the unknown variable. For example, in the given problem, we had to find the value of \(x\) such that the average of the numbers was \(5\). We used the equation \[ \frac{23 + x}{6} = 5 \] and applied algebraic techniques to isolate \(x\). Here are the steps we followed:

• Summed the known numbers
• Formulated the equation using the average formula
• Multiplied both sides by the number of terms
• Simplified to isolate and solve for \(x\)

The key skills applied were addition, multiplication, and understanding of equality in equations.

arithmetic mean

The arithmetic mean is another term for the average. It provides a central value for a set of numbers. To find the arithmetic mean, you sum all the elements and divide by the count of elements, as shown in this problem. For the numbers \[ 6, 3, -2, 5, 11, \] and \( x \), the mean is given to be \(5\). Applying the arithmetic mean formula, we derive an equation:

• Sum the known numbers: \(23\)
• Set up the average equation: \[ \frac{23 + x}{6} = 5 \]
• Solve for the unknown \(x\)

This way, we ensure that all values synchronize to provide the given mean, making the arithmetic mean a useful measure of central tendency.

GRE math problems

When tackling GRE math problems, mastering key algebra concepts and arithmetic operations is essential. The problem here is a typical example, challenging students to apply the average formula and solve for an unknown variable. Key takeaways for GRE prep include:

• Understanding fundamental formulas like the arithmetic mean
• Using basic algebraic manipulations
• Practicing step-by-step problem-solving

This approach not only strengthens mathematical understanding but also boosts problem-solving speed and accuracy, crucial for success on standardized tests like the GRE.

algebra basics

Basics of algebra revolve around manipulating variables and constants to establish relationships and solve equations. This problem involved several algebra basics:

• Understanding variables and constants
• Summing a set of numbers
• Setting up an equation based on the average formula
• Isolating the variable through multiplication and subtraction

These foundational skills are critical not only for algebra courses but also for advanced mathematical applications. Practicing problems like the given exercise helps solidify these concepts.

step-by-step math solutions

Breaking down complex problems into manageable steps is a hallmark of effective math instruction. Here’s how we approached the given problem in a step-by-step manner:

• Identify the known values and what needs to be found
• Sum the known numbers naturally
• Formulate the target equation


• Perform operations to isolate the variable and solve

By structuring our solution this way, we ensure clarity and logical flow, aiding in better understanding and retention of the method. This approach can be applied universally to various math problems, enhancing problem-solving efficiency.