Solving 4/n = 4/5 Step-by-Step Guide

Hey guys! Today, we're diving into a fun little math problem that involves fractions and solving for an unknown variable. It's like a mini-puzzle, and we're going to break it down step-by-step so you can ace similar problems in the future. Our mission, should we choose to accept it (and we do!), is to find the value of 'n' in the equation 4/n = 4/5. This might seem a bit daunting at first, but trust me, it's easier than it looks. We'll explore different approaches, making sure you not only get the answer but also understand the why behind it. So, grab your thinking caps, and let's get started!

Understanding the Basics of Equations and Fractions

Before we jump straight into solving for 'n', let's take a moment to brush up on some fundamental concepts. Think of an equation as a balanced scale. On one side, we have 4/n, and on the other, we have 4/5. The equals sign (=) is like the fulcrum, ensuring both sides remain perfectly balanced. Our goal is to manipulate the equation without disrupting this balance, until we isolate 'n' on one side. This will tell us the value of 'n' that makes the equation true.

Now, let's talk fractions. A fraction represents a part of a whole. In our equation, 4/n means we have 4 parts out of a total of 'n' parts. Similarly, 4/5 means we have 4 parts out of 5. To solve for 'n', we need to understand how the numerator (the top number) and the denominator (the bottom number) interact. Remember, the denominator tells us the total number of parts, and the numerator tells us how many of those parts we have. With these basics in mind, we're well-equipped to tackle our problem.

Cross-Multiplication: A Powerful Tool

One of the most effective techniques for solving equations involving fractions is cross-multiplication. It's like a mathematical shortcut that helps us eliminate the fractions and simplify the equation. The idea behind cross-multiplication is simple: we multiply the numerator of the first fraction by the denominator of the second fraction, and vice versa. This creates a new equation without any fractions.

In our case, we have 4/n = 4/5. To cross-multiply, we'll multiply 4 (the numerator of the left fraction) by 5 (the denominator of the right fraction), and then multiply 4 (the numerator of the right fraction) by 'n' (the denominator of the left fraction). This gives us a new equation: 4 * 5 = 4 * n. See how we've eliminated the fractions? Now, we have a much simpler equation to work with. Cross-multiplication is a powerful tool in our arsenal, and it's one you'll use frequently when dealing with fractions in equations. So, let's put it into action and see where it leads us!

Step-by-Step Solution: Finding the Value of n

Alright, let's get down to business and solve for 'n' in our equation 4/n = 4/5. We've already laid the groundwork by understanding the basics of equations and fractions, and we've learned about the handy technique of cross-multiplication. Now, it's time to put it all together and find our answer.

Applying Cross-Multiplication

As we discussed earlier, cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other. In our equation 4/n = 4/5, this means we'll multiply 4 by 5 and 4 by 'n'. This gives us the equation 4 * 5 = 4 * n. It's like we're creating a mathematical bridge between the two fractions, connecting them through multiplication.

Simplifying the Equation

Now that we've cross-multiplied, let's simplify our equation. We have 4 * 5 = 4 * n, which we can rewrite as 20 = 4n. This equation is much cleaner and easier to work with. We've reduced the fractions to a simple algebraic equation, where 'n' is the only unknown. Our next step is to isolate 'n' on one side of the equation. This will reveal its value and solve our problem.

Isolating n: The Final Step

To isolate 'n', we need to get it by itself on one side of the equation. Currently, 'n' is being multiplied by 4. To undo this multiplication, we'll perform the opposite operation: division. We'll divide both sides of the equation by 4. Remember, whatever we do to one side of the equation, we must do to the other to maintain balance.

So, we divide both sides of 20 = 4n by 4. This gives us 20 / 4 = (4n) / 4. Simplifying this, we get 5 = n. And there you have it! We've successfully solved for 'n'. The value of 'n' that makes the equation 4/n = 4/5 true is 5. It's like finding the missing piece of a puzzle, and we did it step-by-step, using our knowledge of equations, fractions, and cross-multiplication.

Verifying the Solution: Ensuring Accuracy

Before we celebrate our victory, it's always a good idea to double-check our work. In mathematics, verification is key to ensuring accuracy. It's like proofreading a document before submitting it – we want to make sure everything is correct.

Plugging the Value of n Back into the Original Equation

To verify our solution, we'll substitute the value we found for 'n' (which is 5) back into the original equation 4/n = 4/5. This means we'll replace 'n' with 5, giving us 4/5 = 4/5. Does this equation hold true? Yes, it does! The left side of the equation is equal to the right side, confirming that our solution is correct.

Why Verification Matters

Verifying our solution is not just a formality; it's a crucial step in the problem-solving process. It helps us catch any mistakes we might have made along the way, whether it's a simple arithmetic error or a misunderstanding of the concepts. By verifying, we can have confidence in our answer and know that we've solved the problem correctly. It's like having a safety net that prevents us from falling into the trap of an incorrect solution. So, always remember to verify your answers, especially in mathematics. It's a habit that will serve you well in the long run.

Analyzing the Answer Choices: Selecting the Correct Option

Now that we've confidently solved for 'n' and verified our solution, let's take a look at the answer choices provided. This is where we put our hard work to the test and select the correct option.

The answer choices were:

A. n = 5 B. n = 5/16 C. n = 3 1/5 D. n = 50

Matching Our Solution to the Options

We found that n = 5 is the solution to our equation. Looking at the answer choices, we can see that option A, n = 5, perfectly matches our solution. This confirms that we've not only solved the problem correctly but also identified the correct answer choice. It's like finding the missing piece of a puzzle and placing it perfectly in its spot.

Why Other Options Are Incorrect

It's also helpful to understand why the other answer choices are incorrect. This reinforces our understanding of the problem and the solution process.

• Option B, n = 5/16, is incorrect because if we substitute this value into the original equation, we get 4 / (5/16) = 4/5, which simplifies to 64/5 = 4/5, which is not true.
• Option C, n = 3 1/5, which is the same as 16/5, is incorrect because if we substitute this value into the original equation, we get 4 / (16/5) = 4/5, which simplifies to 5/4 = 4/5, which is also not true.
• Option D, n = 50, is incorrect because if we substitute this value into the original equation, we get 4/50 = 4/5, which simplifies to 2/25 = 4/5, which is not true either.

By understanding why the other options are incorrect, we solidify our understanding of the problem and the correct solution. It's like building a strong foundation of knowledge that will help us tackle similar problems in the future.

Conclusion: Mastering the Art of Solving Equations

Wow, guys! We've journeyed through a fun and insightful math problem together. We started with the equation 4/n = 4/5 and, through a series of steps, successfully solved for 'n'. We learned about the basics of equations and fractions, mastered the technique of cross-multiplication, and verified our solution to ensure accuracy. And guess what? The correct answer is A. n = 5.

Key Takeaways and Tips for Success

• Understand the Fundamentals: A strong grasp of basic concepts like equations and fractions is crucial for solving more complex problems.
• Master Cross-Multiplication: This technique is a powerful tool for solving equations involving fractions.
• Verify Your Solutions: Always double-check your work to ensure accuracy.
• Practice Makes Perfect: The more you practice, the more confident and proficient you'll become in solving equations.

The Joy of Problem-Solving

Solving math problems can be like cracking a code or solving a puzzle. It's a rewarding experience that challenges our minds and helps us develop critical thinking skills. So, embrace the challenge, enjoy the process, and celebrate your successes. Remember, every problem you solve is a step forward on your mathematical journey.

I hope this comprehensive guide has been helpful in understanding how to solve for 'n' in the equation 4/n = 4/5. Keep practicing, keep exploring, and keep the math magic alive! You've got this!