Solving Proportions What Is The Missing Number In 7/8 = ?/48

Solving Proportions: A Step-by-Step Guide

Hey guys! Ever get those math problems that look like a puzzle? Today, we're tackling one of those: solving proportions. Our question is: $7 / 8=? / 48$. It looks intimidating, but don't worry, we'll break it down together.

So, what are proportions? Simply put, a proportion is just a statement that two ratios are equal. A ratio, in turn, is a comparison of two quantities, often written as a fraction. Think of it like this: if you're baking a cake, the ratio of flour to sugar needs to be just right for the cake to taste delicious. Change that ratio, and you might end up with a culinary disaster! In our problem, $7 / 8$ is one ratio, and $? / 48$ is the other. Our mission is to find the missing number that makes these two ratios equivalent. The key concept here is equivalent fractions. Remember those? Equivalent fractions are fractions that represent the same value, even though they have different numerators and denominators. For example, $1 / 2$ and $2 / 4$ are equivalent fractions. They both represent the same half of something. Our goal is to find the fraction with a denominator of 48 that's equivalent to $7 / 8$.

There are a couple of ways we can tackle this. One method, which is super helpful for understanding the concept, is to think about what we need to multiply the denominator (8) by to get the new denominator (48). In this case, 8 multiplied by what equals 48? If you're thinking 6, you're spot on! Now, here's the crucial part: to keep the fractions equivalent, we need to multiply the numerator (7) by the same number (6). So, 7 multiplied by 6 is 42. This means our missing number is 42, and the equivalent fraction is $42 / 48$. Another way to solve proportions is using cross-multiplication. This method might seem a bit like a magic trick at first, but it's based on solid mathematical principles. Here's how it works: In our proportion, $7 / 8=? / 48$, we cross-multiply. This means we multiply the numerator of the first fraction (7) by the denominator of the second fraction (48), and we multiply the denominator of the first fraction (8) by the numerator of the second fraction (which is our unknown, let's call it 'x'). This gives us two products: 7 * 48 and 8 * x. The fundamental rule of proportions is that these two products must be equal. So, we have the equation 7 * 48 = 8 * x. Now it's just a matter of solving for x. First, we calculate 7 * 48, which equals 336. So our equation becomes 336 = 8 * x. To isolate x, we need to divide both sides of the equation by 8. 336 divided by 8 is 42. Therefore, x = 42, which is the same answer we got using the first method. See? No magic, just math!

So, whether you prefer thinking about equivalent fractions or using cross-multiplication, both methods lead us to the same answer: the missing number is 42. This means the proportion $7 / 8=? / 48$ is solved when the question mark is replaced with 42. And that's it! You've successfully navigated the world of proportions. Remember, proportions are all about finding equivalent ratios, and there are different ways to get there. The more you practice, the more comfortable you'll become with these techniques. Now, let's look at the answer choices provided and confidently select the correct one.

Identifying the Correct Answer

Alright, now that we've cracked the code and found our missing number, let's zero in on the correct answer from the options given. Remember, we determined that the missing number in the proportion $7 / 8=? / 48$ is 42. This means that the fraction $42 / 48$ is equivalent to $7 / 8$. We arrived at this conclusion by understanding that to get from a denominator of 8 to a denominator of 48, we need to multiply by 6. Consequently, we also multiplied the numerator 7 by 6, resulting in 42. Alternatively, we employed the method of cross-multiplication, where we multiplied 7 by 48 and 8 by our unknown (x), setting the products equal to each other (7 * 48 = 8 * x). Solving this equation led us to the same answer: x = 42.

Now, let's take a look at the answer choices we have:

• A) 13
• B) 6
• C) 42
• D) 1

Scanning through these options, it's clear that option C) 42 perfectly matches our calculated answer. The other options – 13, 6, and 1 – are incorrect and would not maintain the proportionality between the two ratios. Choosing the correct answer in a multiple-choice question is not just about arriving at the right solution; it's also about confidently identifying it among the distractors. Distractors are the incorrect answer choices that are designed to mislead you if you haven't fully understood the problem or made a calculation error. In this case, the distractors might have been chosen based on common mistakes students make when working with proportions, such as multiplying only the denominator or forgetting to multiply both the numerator and denominator by the same factor. Therefore, it's essential to double-check your work and make sure your answer makes logical sense in the context of the problem. In our situation, we can quickly verify that $42 / 48$ simplifies to $7 / 8$ by dividing both the numerator and denominator by their greatest common divisor, which is 6. This confirms that 42 is indeed the correct missing number. So, without a doubt, we can confidently select C) 42 as the solution to our proportion problem.

Selecting the correct answer involves a combination of accurate problem-solving skills and careful attention to detail. By understanding the underlying concepts and practicing different methods, you can confidently navigate proportion problems and choose the right answer every time. Now, let's move on to discussing the broader category of mathematics that this problem falls under and why understanding these concepts is so important.

The Importance of Understanding Proportions in Mathematics

So, we've successfully solved our proportion problem, but you might be wondering,