Solving Proportions: Find 'v' In 9/v = 13/7
Hey guys! Today, we're diving into a common math problem: solving proportions. Proportions are all about finding equivalent ratios, and in this case, we're tackling the equation 9/v = 13/7. Our mission? To find the value of 'v' that makes this equation true, and then round our answer to the nearest tenth. Sounds like fun, right? Let's get started!
Understanding Proportions
Before we jump into solving, let's make sure we're all on the same page about what a proportion actually is. A proportion is simply a statement that two ratios are equal. A ratio, in turn, is a comparison of two quantities, often expressed as a fraction. So, when we see 9/v = 13/7, we're saying that the ratio of 9 to 'v' is the same as the ratio of 13 to 7.
Why are proportions important? Well, they pop up all over the place in real life! From scaling recipes to calculating distances on a map, proportions help us understand relationships between different quantities. Mastering proportions is a fundamental skill in math, and it's super useful in many everyday situations. Think about it: if you're doubling a recipe, you're using proportions to figure out how much of each ingredient you need. Or, if you're looking at a map, you're using the scale (a proportion!) to understand the actual distances between places.
In our specific problem, 9/v = 13/7, we have an unknown value, 'v'. Our goal is to isolate 'v' and figure out what number it represents. There are a couple of ways we can do this, but one of the most common and reliable methods is cross-multiplication. So, let's get into that!
Cross-Multiplication: The Key to Unlocking 'v'
Okay, let's talk about cross-multiplication. This technique is a neat trick for solving proportions, and it's based on the idea that if two ratios are equal, their cross-products are also equal. What does that mean in plain English? Well, if we have a proportion like a/b = c/d, cross-multiplication tells us that a * d = b * c. We're essentially multiplying the numerator of one fraction by the denominator of the other, and vice versa.
So, how does this apply to our problem, 9/v = 13/7? Simple! We multiply 9 by 7 and set it equal to 13 multiplied by 'v'. This gives us the equation: 9 * 7 = 13 * v. See how we've transformed our proportion into a linear equation? This is a crucial step because linear equations are much easier to solve for a single variable.
Let's break it down further: 9 * 7 equals 63, so we have 63 = 13v. Now, we're just one step away from isolating 'v'. We have 'v' multiplied by 13, so to undo that multiplication, we need to perform the inverse operation: division. We'll divide both sides of the equation by 13, which will leave 'v' all by itself on one side. Are you ready to see the next step?
Isolating 'v': Division to the Rescue
Alright, we've reached the point where we need to isolate 'v'. We've got the equation 63 = 13v, and as we discussed, we need to get rid of that 13 that's hanging out with 'v'. To do that, we'll divide both sides of the equation by 13. Remember, whatever we do to one side of an equation, we must do to the other side to keep things balanced. It's like a mathematical seesaw – we need to maintain equilibrium!
So, we divide both sides by 13: 63 / 13 = (13v) / 13. On the right side, the 13s cancel each other out, leaving us with just 'v'. On the left side, we have 63 / 13. Now, this isn't a whole number, so we're going to end up with a decimal. That's perfectly okay! In fact, the problem specifically asks us to round our answer to the nearest tenth, so we know we're heading in the right direction.
Using a calculator (or long division, if you're feeling old-school!), we find that 63 / 13 is approximately 4.846. But wait! We're not quite done yet. We need to round this to the nearest tenth. Let's talk about how to do that.
Rounding to the Nearest Tenth: Getting the Final Answer
Okay, we've arrived at the final stage: rounding our answer to the nearest tenth. We've got v ≈ 4.846, and we want to express this value with only one digit after the decimal point. So, how do we do it? The key is to look at the digit in the hundredths place (the second digit after the decimal), which in this case is 4.
Here's the rule: If the digit in the hundredths place is 5 or greater, we round the digit in the tenths place up. If it's less than 5, we leave the digit in the tenths place as it is. Since our digit in the hundredths place is 4, which is less than 5, we don't round up. We simply drop the 4 and the 6.
Therefore, when we round 4.846 to the nearest tenth, we get 4.8. And that, my friends, is our final answer! We've successfully solved the proportion for 'v' and rounded it as requested. Pat yourselves on the back – you've earned it!
So, to recap, we started with the proportion 9/v = 13/7, used cross-multiplication to transform it into a linear equation, isolated 'v' by dividing both sides by 13, and then rounded our answer to the nearest tenth. And there you have it: v ≈ 4.8.
Final Answer
Therefore, the solution to the proportion 9/v = 13/7, rounded to the nearest tenth, is:
v = 4.8
Great job, everyone! Solving proportions is a valuable skill, and you've just added another tool to your mathematical toolbox. Keep practicing, and you'll become a proportion pro in no time!