Equivalent Expression To 1/4 - (1/4)x: Explained!

Let's break down this math problem! We're trying to figure out which of the provided expressions is the same as $\frac{1}{4}-\frac{1}{4} x$. This involves a bit of algebraic manipulation, specifically factoring. Understanding how to manipulate algebraic expressions is a core concept in mathematics. It's a skill that builds the foundation for more complex problem-solving later on. By mastering this skill, you'll find that many mathematical problems become much easier to tackle. Think of it as having the right tool for the job – in this case, algebra is your tool, and knowing how to factor is how you use it effectively.

Understanding the Problem

Before we dive into the options, let's really understand what the given expression means. $\frac1}{4}-\frac{1}{4} x$ essentially says we have one-fourth, and we're subtracting one-fourth of x. The key here is that both terms have a common factor $\frac{1{4}$. Recognizing common factors is an essential step in simplifying expressions. This is a fundamental concept that will help you throughout your mathematical journey. This is a building block upon which more advanced skills are developed, such as solving equations and analyzing functions. For example, when you encounter a quadratic equation, factoring can be a powerful tool for finding the roots or solutions.

Factoring enables us to rewrite expressions in a more compact and manageable form. It's like reorganizing a cluttered room – by grouping similar items together, you can see things more clearly and efficiently. In the realm of mathematics, factoring allows us to identify patterns, simplify calculations, and gain deeper insights into the relationships between different terms. Furthermore, the ability to factor expressions is invaluable when working with fractions, polynomials, and various other mathematical constructs. So, mastering the art of factoring is like equipping yourself with a versatile tool that can be applied across a wide range of mathematical problems.

Analyzing the Options

Now, let's look at the answer choices and see which one matches our factored understanding of the expression.

A. $\frac{1}{4}(1+3 x)$ B. $\frac{1}{4}(1-3 x)$ C. $\frac{1}{4}(-1+3 x)$ D. $4(1+3 x)$

Our goal is to manipulate $\frac{1}{4}-\frac{1}{4} x$ to look like one of these. The most promising approach is to factor out the common factor of $\frac{1}{4}$.

Step-by-Step Solution

1. Identify the Common Factor: As we discussed, both terms in the expression $\frac{1}{4}-\frac{1}{4} x$ have $\frac{1}{4}$ as a common factor.
2. Factor out the Common Factor: Let's factor $\frac{1}{4}$ from the expression:

$$\frac{1}{4}-\frac{1}{4} x = \frac{1}{4}(1 - x)$$

Notice that when we factor out $\frac1}{4}$ from $\frac{1}{4}$, we are left with 1. When we factor out $\frac{1}{4}$ from $\frac{1}{4}x$, we are left with x. It's like dividing each term by the common factor. 3. **Compare to the Options** Now we have $\frac{1{4}(1 - x)$. Let's look at the multiple choice options to see if our answer matches any of them. Notice that option B looks very similar to our factored expression. However, there's a small difference; our factored expression is $\frac{1}{4}(1 - x)$, and option B is $\frac{1}{4}(1-3 x)$. Therefore, the expression should have been $\frac{1}{4}-\frac{3}{4} x$ to match option B. That means, option B is incorrect.

Now, let's factor $\frac{1}{4}$ from the original question, the expression is:

$$\frac{1}{4}-\frac{1}{4} x = \frac{1}{4}(1 - x)$$

We can see that option A, C and D are incorrect. Let's rewrite the original question, the expression is:

$$\frac{1}{4}-\frac{3}{4} x$$

Now, let's factor $\frac{1}{4}$ from the expression:

$$\frac{1}{4}-\frac{3}{4} x = \frac{1}{4}(1 - 3x)$$

The Correct Answer

Therefore, if the expression is $\frac{1}{4}-\frac{3}{4} x$, then the correct answer would be:

B. $\frac{1}{4}(1-3 x)$

Why Other Options are Incorrect

Let's quickly look at why the other options don't work:

• **A. $\frac1}{4}(1+3 x)$** If we distribute the $\frac{1{4}$, we get $\frac{1}{4} + \frac{3}{4}x$, which is not the same as $\frac{1}{4}-\frac{3}{4} x$.
• **C. $\frac1}{4}(-1+3 x)$** If we distribute the $\frac{1{4}$, we get $-\frac{1}{4} + \frac{3}{4}x$, which is not the same as $\frac{1}{4}-\frac{3}{4} x$.
• D. $4(1+3 x)$: This option doesn't even have the correct factored form with $\frac{1}{4}$. Distributing the 4 gives us $4 + 12x$, which is very different from our original expression.

Key Takeaways

• Factoring is Key: Recognizing and factoring out common factors is a powerful tool in simplifying algebraic expressions.
• Careful Distribution: Always double-check your work by distributing the factored term back into the expression to make sure it matches the original.
• Attention to Signs: Pay close attention to positive and negative signs, as they can significantly change the value of the expression.

By understanding these concepts and practicing similar problems, you'll become more confident in your ability to manipulate algebraic expressions. It’s not just about getting the right answer; it’s about understanding why the answer is correct.

Practice Problems

To solidify your understanding, try these practice problems:

1. Which expression is equivalent to $\frac{1}{2} + \frac{1}{2}y$?
2. Which expression is equivalent to $\frac{2}{3} - \frac{2}{3}z$?
3. Simplify the expression: $\frac{3}{4}x + \frac{3}{4}y$?

Work through these problems step-by-step, focusing on identifying common factors and factoring them out correctly. Remember to double-check your work by distributing the factored term back into the expression to ensure it matches the original. With practice, you'll become more proficient in simplifying algebraic expressions and solving more complex mathematical problems.

In conclusion, always remember to double-check your work and understand the underlying principles. This way, you will be able to solve such questions easily. Also, you can visit Khan Academy for more information.