Question

If \(\frac{x}{4}=\frac{1}{2}\), then \(\frac{4(x-3)}{(-12)}\) equals which of the following? (A) \(\frac{1}{16}\) (B) \(\frac{1}{16}\) (C) \(\frac{7}{9}\) (D) \(\frac{1}{2}\)

Short answer

None of the options match the answer \( \frac{1}{3} \); there may be an error.

Step-by-step solution

Solve for x

Start by solving the equation \( \frac{x}{4} = \frac{1}{2} \) to find the value of \( x \). To do this, multiply both sides of the equation by 4 to isolate \( x \). This gives you:\[ x = 4 \times \frac{1}{2} \]Calculate the product:\[ x = 2 \]

Substitute x into the expression

Now that we know \( x = 2 \), substitute it into the expression \( \frac{4(x-3)}{(-12)} \). This means substituting \( 2 \) for \( x \):\[ \frac{4(2-3)}{(-12)} \]

Simplify the expression

First, calculate the expression inside the parentheses:\[ 2 - 3 = -1 \]Substitute this result into the expression:\[ \frac{4 \times (-1)}{(-12)} \]Simplify the multiplication in the numerator:\[ \frac{-4}{-12} \]Now, simplify the fraction. Since \( -4 / -12 = 1/3 \), the result of the expression is:\[ \frac{1}{3} \]

Compare with the choices

None of the given choices match \( \frac{1}{3} \). There might be a mistake in the problem statement, or there may be a missing option.

Key concepts

Solving Equations

Solving equations is a fundamental part of algebra that involves finding the value of unknown variables. In this exercise, we begin with the simple equation \( \frac{x}{4} = \frac{1}{2} \). The goal is to determine what number \( x \) should be to make both sides equal. To isolate \( x \), we need to eliminate the fraction. We do this by multiplying both sides of the equation by 4. By performing this operation, we get: \[ x = 4 \times \frac{1}{2} \] This simplifies to \( x = 2 \). Here are some tips to successfully solve equations:

• Always aim to isolate the variable on one side.
• Perform the same mathematical operation on both sides of the equation to keep it balanced.
• Check your solution by substituting the value back into the original equation.

Fractions

Fractions represent parts of a whole and can often show up in equations like the one provided. Understanding fractions involves:

• The numerator, which is the top number representing how many parts we have.
• The denominator, which is the bottom number indicating into how many parts the whole is divided.

To manage fractions, it often involves operations such as:

• Addition and subtraction: You need a common denominator to add or subtract fractions.
• Multiplication: Multiply the numerators together and then the denominators.
• Division: Flip the second fraction and multiply.

In this exercise, simplifying and working with fractions accurately is key to getting the correct answer.

Substitution

Substitution is a useful technique in algebra where you replace a variable with a given or known value. After finding \( x = 2 \) in the initial equation, we use substitution to continue solving the expression. We start with \[ \frac{4(x-3)}{(-12)} \] By replacing \( x \) with 2, the expression becomes: \[ \frac{4(2-3)}{(-12)} \] This method helps make complex expressions simpler and more manageable.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. Once we have the substituted value \( \frac{4(2-3)}{(-12)} \), this process allows us to tackle this: First, evaluate the expression inside the parentheses: \[ 2 - 3 = -1 \] Then, substitute back into the main expression: \[ \frac{4 \times (-1)}{(-12)} \] The next step is to multiply, which results in: \[ \frac{-4}{-12} \] Finally, simplifying the fraction: divide the numerator and the denominator by their greatest common divisor. Here, both -4 and -12 can be divided by 4, and the negative signs cancel each other out, so the fraction simplifies to \( \frac{1}{3} \). This simplification process is essential because it often reveals the answer in a cleaner form.