Solving Proportions 7/x = 19/114 A Step-by-Step Guide

Solving proportions might seem daunting at first, but trust me, guys, it's like pie! Once you get the hang of it, you'll be breezing through them. Proportions are simply statements that two ratios or fractions are equal. They show a relationship between two sets of numbers, and figuring them out is super useful in everyday life, from scaling recipes to understanding maps. In this article, we'll break down how to solve proportions, using the example ${\frac{7}{x} = \frac{19}{114}}$ as our guide. We’ll go through each step in detail, so you can confidently tackle any proportion problem that comes your way.

Understanding Proportions

Before we dive into solving our specific proportion, let's make sure we're all on the same page about what a proportion actually is. A proportion is essentially an equation stating that two ratios are equal. A ratio, in turn, is a comparison of two numbers by division. Think of it like this: if you have a recipe that calls for 2 cups of flour for every 1 cup of sugar, the ratio of flour to sugar is 2:1. Now, if you want to double the recipe, you'd need 4 cups of flour for 2 cups of sugar, maintaining the same ratio. This is where proportions come in handy. They allow us to find an unknown value while keeping the relationship between the other values constant.

In mathematical terms, a proportion looks like this: ${\frac{a}{b} = \frac{c}{d}}$, where a, b, c, and d are numbers. The key to solving proportions is understanding that the cross-products are equal. This means that ${a \times d = b \times c}$. This property is derived from the fundamental principles of fractions and equations. When two fractions are equal, their numerators and denominators are related in a specific way that makes their cross-products equal. This cross-product property is our main tool for solving for unknown variables in proportions. So, with this understanding, let's jump into solving our example problem. We’ll see how this cross-product thing works in practice and you'll be solving these like a pro in no time!

Step-by-Step Solution for ${\frac{7}{x} = \frac{19}{114}}$

Alright, let’s get down to business and solve the proportion ${\frac{7}{x} = \frac{19}{114}}$. We're going to break it down step-by-step so it's super clear. The main goal here is to find the value of x that makes this equation true. Remember, a proportion is just saying that two fractions are equal, so we need to figure out what x should be to keep that balance.

Step 1: Cross-Multiplication

The first thing we need to do is use that handy cross-multiplication trick we talked about earlier. This involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa. So, in our proportion, ${\frac{7}{x} = \frac{19}{114}}$, we'll multiply 7 by 114 and x by 19. This gives us the equation: ${7 \times 114 = 19 \times x}$. Cross-multiplication is a neat way to get rid of the fractions and turn our proportion into a more manageable equation. It's like transforming the problem into a format we're more used to dealing with. By cross-multiplying, we're essentially creating an equivalent equation that doesn't have any fractions, making it much easier to solve for our unknown variable.

Step 2: Simplify the Equation

Now that we've cross-multiplied, let's simplify the equation. We have ${7 \times 114 = 19 \times x}$. First, we need to calculate ${7 \times 114}$. If you do the math (or use a calculator, no judgment here!), you'll find that ${7 \times 114 = 798}$. So, our equation now looks like this: ${798 = 19x}$. Simplifying is crucial because it makes the equation less cluttered and easier to work with. It’s like decluttering your workspace before starting a big project; it helps you focus and makes the whole process smoother. In this case, multiplying the numbers on one side gives us a single value, which simplifies the equation and brings us closer to isolating x. Remember, our goal is to get x all by itself on one side of the equation, so simplifying is a key step in that process.

Step 3: Isolate the Variable

Okay, we're on the home stretch! We've got ${798 = 19x}$, and our mission is to get x all by itself. To do this, we need to isolate x by undoing the multiplication. Currently, x is being multiplied by 19, so to undo that, we need to do the opposite operation: divide. We'll divide both sides of the equation by 19. This is a crucial step because what we do to one side of an equation, we must also do to the other side to maintain balance. Think of it like a scale; if you add or remove weight from one side, you need to do the same on the other side to keep it level. So, we divide both sides by 19: ${\frac{798}{19} = \frac{19x}{19}}$. This step is all about peeling away the layers around x until it's standing alone and we can finally see its value. It's like a detective solving a mystery, carefully uncovering clues until the answer is revealed.

Step 4: Solve for x

Time to wrap things up and find the value of x. We've got ${\frac{798}{19} = \frac{19x}{19}}$. On the right side of the equation, the 19s cancel each other out, leaving us with just x. Now we need to calculate ${\frac{798}{19}}$. When you divide 798 by 19, you get 42. So, our equation simplifies to ${42 = x}$. This means that x is equal to 42. We've done it! We've successfully isolated x and found its value. This step is the culmination of all our hard work. It's the moment where the puzzle pieces fall into place and we see the solution clearly. Solving for x is like reaching the summit of a mountain after a long climb; it's a rewarding feeling knowing you've conquered the challenge.

Step 5: Check Your Answer

Before we declare victory, it’s always a good idea to check our answer. This is like proofreading a document before submitting it or taste-testing a dish before serving it. We want to make sure everything is correct and that our solution makes sense. To check our answer, we'll plug the value we found for x (which is 42) back into the original proportion: ${\frac{7}{x} = \frac{19}{114}}$. So, we replace x with 42, giving us ${\frac{7}{42} = \frac{19}{114}}$. Now, we need to see if these two fractions are indeed equal. We can simplify the fraction ${\frac{7}{42}}$ by dividing both the numerator and the denominator by 7. This gives us ${\frac{1}{6}}$. Next, let's see if ${\frac{19}{114}}$ can also be simplified to ${\frac{1}{6}}$. If we divide both 19 and 114 by 19, we get ${\frac{1}{6}}$. So, we have ${\frac{1}{6} = \frac{1}{6}}$, which is true! This confirms that our solution, x = 42, is correct. Checking our answer is a crucial step because it gives us confidence in our solution and helps us catch any mistakes we might have made along the way. It's like having a safety net that ensures we land on our feet.

Final Answer: x = 42

So, there you have it! We've successfully solved the proportion ${\frac{7}{x} = \frac{19}{114}}$. By following our step-by-step guide, we found that the value of x is 42. We started by understanding what proportions are and how they work, then we used cross-multiplication to simplify the equation. We isolated the variable x by dividing both sides of the equation, and finally, we checked our answer to make sure everything was correct. Solving proportions is a valuable skill that can be applied in many real-world situations. Whether you're calculating ingredient ratios in a recipe, determining distances on a map, or scaling up a design project, understanding proportions can help you solve problems accurately and efficiently. Remember, the key to mastering proportions is practice, so keep working at it, and you'll become a pro in no time!