Solving Proportions Finding The Value Of N

Hey there, math enthusiasts! Today, we're diving into a fun little problem that involves proportions and finding the value of an unknown variable. Specifically, we're going to tackle the equation:

$$\frac{n \text { pounds }}{28 \text { ounces }} = \frac{3 \text { pounds }}{48 \text { ounces }}$$

Our mission? To find the value of n and express it in its simplest form, especially if it turns out to be a fraction. So, grab your thinking caps, and let's get started!

Understanding Proportions: The Foundation of Our Solution

Before we jump into solving for n, let's quickly recap what proportions are all about. Proportions are essentially statements that two ratios or fractions are equal. They're a fundamental concept in mathematics and pop up in various real-world scenarios, from scaling recipes to calculating distances on maps. In our case, we have a proportion that relates pounds to ounces. The key to solving proportions lies in understanding how to manipulate them to isolate the variable we're interested in.

In this particular problem, we're dealing with a proportion that compares the ratio of pounds to ounces on both sides of the equation. The left side has n pounds over 28 ounces, while the right side has 3 pounds over 48 ounces. Our goal is to figure out what value of n makes these two ratios equivalent. Think of it like this: we're trying to find a weight in pounds (n) that, when compared to 28 ounces, maintains the same relationship as 3 pounds compared to 48 ounces. This kind of problem is super common in everyday situations, like when you're adjusting a recipe or figuring out how much of an ingredient you need based on a different serving size. The beauty of proportions is that they provide a straightforward way to solve these kinds of problems.

To really nail this, remember that proportions are all about maintaining balance. Whatever you do to one side of the equation, you have to do to the other to keep things equal. This principle will guide us as we work through the steps to solve for n. So, with this understanding in place, we're ready to roll up our sleeves and get into the nitty-gritty of solving the equation. Let's move on to the next section where we'll explore the different methods we can use to isolate n and find its value.

Methods to Solve for n: Cross-Multiplication and More

Alright, guys, let's get down to the nitty-gritty of solving for n. There are a couple of cool ways we can tackle this proportion, but the most common and efficient method is cross-multiplication. This technique is a real workhorse when dealing with proportions, and it's super handy to have in your math toolkit.

Cross-Multiplication: Our Go-To Technique

Cross-multiplication is a neat trick that simplifies proportions into linear equations, which are much easier to solve. The basic idea is to multiply the numerator of one fraction by the denominator of the other fraction, and then set the two products equal to each other. In our equation:

$$\frac{n \text { pounds }}{28 \text { ounces }} = \frac{3 \text { pounds }}{48 \text { ounces }}$$

We would cross-multiply like this:

• n (numerator of the left fraction) multiplied by 48 (denominator of the right fraction)
• 3 (numerator of the right fraction) multiplied by 28 (denominator of the left fraction)

This gives us the equation:

$$48n = 3 \times 28$$

See how we've transformed a proportion into a simple linear equation? Now, all we need to do is solve for n. This involves performing the multiplication on the right side and then isolating n by dividing both sides of the equation by 48. We'll walk through those steps in detail in the next section.

An Alternative Approach: Scaling the Fractions

While cross-multiplication is often the quickest route, there's another way to think about solving proportions that can be helpful in certain situations. This method involves scaling one or both fractions so that they have the same denominator. Once the denominators are the same, you can simply equate the numerators. For example, in our problem, we could try to manipulate the fraction on the left side so that its denominator is 48 ounces, just like the fraction on the right side. To do this, we'd need to figure out what number we can multiply 28 by to get 48. This approach might involve a bit more arithmetic, especially if the numbers aren't as clean, but it reinforces the fundamental concept of equivalent fractions and can sometimes lead to a more intuitive understanding of the problem.

However, for this particular problem, cross-multiplication is definitely the most straightforward method. It minimizes the number of steps and reduces the chances of making a calculation error. So, let's stick with cross-multiplication and move on to the next step: solving the resulting equation.

Solving the Equation: Isolating n and Finding the Value

Okay, let's roll up our sleeves and get our hands dirty with the actual solving part. We've already used cross-multiplication to transform our proportion into a linear equation. Remember, we ended up with:

$$48n = 3 \times 28$$

The first step here is to simplify the right side of the equation. We need to multiply 3 by 28. If you're comfortable doing this in your head, awesome! If not, no worries, grab a piece of paper or a calculator. 3 times 28 equals 84. So, our equation now looks like this:

$$48n = 84$$

Now comes the crucial part: isolating n. Our goal is to get n all by itself on one side of the equation. Currently, n is being multiplied by 48. To undo this multiplication, we need to perform the opposite operation, which is division. We're going to divide both sides of the equation by 48. This is super important: whatever we do to one side of the equation, we have to do to the other to keep things balanced. So, we divide both sides by 48 like this:

$$\frac{48n}{48} = \frac{84}{48}$$

On the left side, the 48s cancel each other out, leaving us with just n. On the right side, we have the fraction 84/48. This is the value of n, but we're not quite done yet. Remember, the problem asked us to express our answer in its simplest form, which means we need to reduce this fraction to its lowest terms.

Simplifying the Fraction: Expressing the Answer in Simplest Form

Alright, we've arrived at the final step: simplifying the fraction 84/48. This is a crucial step because we want to present our answer in the most concise and clear way possible. To simplify a fraction, we need to find the greatest common divisor (GCD) of the numerator (84) and the denominator (48) and then divide both the numerator and the denominator by that GCD.

There are a couple of ways to find the GCD. One method is to list the factors of both numbers and identify the largest factor they have in common. Another method, which can be particularly useful for larger numbers, is the Euclidean algorithm. But for 84 and 48, listing the factors works well. Let's do that:

• Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
• Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

Looking at these lists, we can see that the greatest common factor of 84 and 48 is 12. So, we're going to divide both the numerator and the denominator of our fraction by 12:

$$\frac{84 \div 12}{48 \div 12} = \frac{7}{4}$$

And there we have it! The simplified fraction is 7/4. This is our final answer for the value of n. We've successfully navigated the proportion, isolated n, and expressed our answer in its simplest form. Pat yourselves on the back, guys! You've tackled a math problem like a pro.

Final Answer: Putting It All Together

So, after all that brainpower and number crunching, we've arrived at our final answer. The value of n that satisfies the proportion is 7/4. This means that:

$$n = \frac{7}{4}$$

To recap, we started with the proportion:

$$\frac{n \text { pounds }}{28 \text { ounces }} = \frac{3 \text { pounds }}{48 \text { ounces }}$$

We then used cross-multiplication to transform it into a linear equation:

$$48n = 84$$

Next, we isolated n by dividing both sides of the equation by 48:

$$n = \frac{84}{48}$$

Finally, we simplified the fraction 84/48 by dividing both the numerator and the denominator by their greatest common divisor, which was 12, resulting in:

$$n = \frac{7}{4}$$

Therefore, the value of n is 7/4. We've successfully solved the problem and expressed the answer as a fraction in its simplest form, just as the problem instructed. This entire process highlights the power of proportions in solving real-world problems and reinforces the importance of simplifying fractions to present answers clearly and concisely. Great job, everyone! You've conquered this math challenge, and hopefully, you feel a little more confident in your problem-solving abilities now. Keep practicing, and you'll become a math whiz in no time!