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Chapter 1: Problem 74

Write each word statement as an equation. Use \(x\) as the variable. Find all solutions from the set \(\\{2,4,6,8,10\\} .\) See Example 5 Twelve divided by a number equals \(\frac{1}{3}\) times that number.

Short Answer

Expert verified
The solution is \(x = 6\).

Step by step solution

01

Understand the Problem

Identify the key information in the word problem. Here, the problem states that twelve divided by a number equals one-third of that number. Let the number be represented by the variable, which will be denoted as \(x\).
02

Write the Word Statement as an Equation

Translate the word statement into a mathematical equation. The problem states 'Twelve divided by a number', which can be written as \( \frac{12}{x} \). It also states, 'equals one-third of that number', which is written as \( \frac{1}{3} x \). Therefore, the equation is: \[ \frac{12}{x} = \frac{1}{3} x \]
03

Solve the Equation for \(x\)

To find \(x\), cross-multiply the equation: \[ 12 = \frac{1}{3} x^2 \]. Multiply both sides by 3 to clear the fraction: \[ 36 = x^2 \]. Take the square root of both sides to solve for \(x\): \[ x = \pm 6 \]. However, since \(x\) must be selected from the set \{2,4,6,8,10\}, choose \(x = 6\).
04

Verify Solution

Verify that the solution \(x = 6\) is indeed a solution to the original problem. Substitute \(x = 6\) back into the equation: \[ \frac{12}{6} = \frac{1}{3} \times 6 \]. This simplifies to: \[ 2 = 2 \], confirming that \(x = 6\) is a correct solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Translating Word Statements
Word problems often seem complicated, but breaking them down into smaller parts can make them easier. The trick is to understand what mathematical operations the words are describing.
For example, let's look at the phrase 'Twelve divided by a number equals one-third of that number.' The key parts here are:
  • 'Twelve divided by a number' which translates to the mathematical expression: \(\frac{12}{x}\).
  • 'Equals one-third of that number' which translates to: \(\frac{1}{3} x\).
Putting it all together, we can write the equation: \(\frac{12}{x} = \frac{1}{3} x\). Learning to translate word statements is an essential skill that makes solving these problems much easier. It turns words into an equation, which you can then solve using algebra.
Solving Algebraic Equations
Once you have your equation, the next step is to solve it. Solving algebraic equations involves finding the value of the unknown variable that makes the equation true.
In our example, the equation is: \(\frac{12}{x} = \frac{1}{3} x\). Let's solve it step-by-step:
  • First, clear the fraction by cross-multiplying: 12 = \(\frac{1}{3} x^2\).
  • Next, get rid of the fraction by multiplying both sides by 3: 36 = \x^2\.
  • Finally, solve for x by taking the square root of both sides: x = \ \(\text{卤}6\).
However, we need to choose the solution that fits the given set \ \(\text{\textbackslash\text{\textbackslash\textbraceleft}}2,4,6,8,10\text{\textbackslash\textbackslash textbraceright}\). Therefore, the valid solution is x = 6.
Verifying Solutions
After solving the equation, it's important to verify that your solution is correct. This ensures you didn't make any mistakes during your calculations.
To verify, substitute the solution back into the original equation and check if both sides are equal.
For our problem, substitute x = 6 back into the equation: \ \(\frac{12}{6} = \frac{1}{3} \times 6\) which simplifies to 2 = 2.
This confirms that our solution x = 6 is correct.
Verification is a critical step because it confirms the accuracy of your solution and helps you understand your work better.

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