Chapter 11: Problem 6
If the average of \(6,3,-2,5,11,\) and \(x\) is \(5,\) what is the value of \(x ?\)
Short Answer
Expert verified
x = 7
Step by step solution
01
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 \[ \frac{a_1 + a_2 + ... + a_n}{n} \]
02
Set Up the Equation
Given the numbers are \(6, 3, -2, 5, 11, \) and \(x\), and the average is \(5\). Therefore, the equation becomes \[ \frac{6 + 3 - 2 + 5 + 11 + x}{6} = 5 \]
03
Sum the Known Numbers
First, sum the known numbers: \(6 + 3 - 2 + 5 + 11 = 23\). Now the equation is \[ \frac{23 + x}{6} = 5 \]
04
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, \[ x = 7 \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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 \(x\)
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 \(x\)
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
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
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.