Solve Proportions: Find X In 7/15 = X/75

by Omar Yusuf 41 views

Hey there, math enthusiasts! Today, we're diving into the world of proportions to solve for an unknown value. Proportions are a fundamental concept in mathematics and have wide-ranging applications in everyday life, from scaling recipes to calculating distances on a map. So, let's break down this problem step by step and find the value of 'x' in the proportion $ rac{7}{15} = rac{x}{75}$.

Understanding Proportions

Before we jump into the solution, let's quickly recap what proportions are all about. A proportion is simply a statement that two ratios are equal. A ratio, in turn, compares two quantities. Think of it as a way to express how much of one thing there is compared to another. For example, if you have 7 apples and 15 oranges, the ratio of apples to oranges is 7:15, which can also be written as the fraction $ rac{7}{15}$. When we say two ratios are proportional, it means they represent the same relationship, just with different numbers. Our given proportion, $ rac{7}{15} = rac{x}{75}$, tells us that the ratio 7 to 15 is equivalent to the ratio x to 75. This means that if we multiply 15 by a certain number to get 75, we should be able to multiply 7 by the same number to get x.

Why are proportions so important? They allow us to solve problems involving scaling and equivalent ratios. Imagine you're baking a cake, and the recipe calls for a certain amount of flour for a specific number of servings. If you want to make a bigger cake, you can use proportions to figure out how much flour you'll need. Or, if you're looking at a map, the scale tells you the proportion between distances on the map and actual distances on the ground. Proportions are also crucial in fields like physics, chemistry, and engineering, where scaling and ratios are constantly used.

So, with a solid understanding of what proportions are and why they matter, let's get our hands dirty and solve for that unknown 'x'!

Solving for 'x' Using Cross-Multiplication

Alright, guys, let's get down to the nitty-gritty of solving this proportion. The most common and efficient method for solving proportions is called cross-multiplication. This technique is based on the fundamental property of proportions, which states that if $ rac{a}{b} = rac{c}{d}$, then $ad = bc$. In simpler terms, we can multiply the numerator of the first fraction by the denominator of the second fraction, and set it equal to the product of the denominator of the first fraction and the numerator of the second fraction.

Let's apply this to our proportion: $ rac{7}{15} = rac{x}{75}$. To cross-multiply, we multiply 7 by 75 and 15 by x. This gives us the equation:

7imes75=15imesx7 imes 75 = 15 imes x

Now, let's simplify this equation. First, we calculate 7 multiplied by 75:

7imes75=5257 imes 75 = 525

So, our equation now looks like this:

525=15x525 = 15x

Our next step is to isolate 'x'. To do this, we need to get 'x' by itself on one side of the equation. Since 'x' is being multiplied by 15, we can do the opposite operation, which is division. We'll divide both sides of the equation by 15:

rac{525}{15} = rac{15x}{15}

This simplifies to:

35=x35 = x

Therefore, the value of x is 35. We've successfully solved for the unknown value in the proportion using cross-multiplication. This method is a powerful tool for tackling proportion problems, and it's a technique you'll use time and time again in mathematics.

Alternative Method: Scaling the Ratio

While cross-multiplication is a fantastic and widely used method, it's always good to have other tools in your mathematical toolkit. Let's explore an alternative approach to solving this proportion: scaling the ratio. This method involves looking at the relationship between the denominators (or numerators) and using that to find the missing value. In our proportion, $ rac{7}{15} = rac{x}{75}$, we can observe the relationship between the denominators, 15 and 75. We need to figure out what we need to multiply 15 by to get 75.

If you know your multiplication facts, you might immediately recognize that $15 imes 5 = 75$. If not, you can always divide 75 by 15 to find the scaling factor. The result is 5. So, we've determined that the denominator 15 was multiplied by 5 to get the denominator 75. Now, here's the key: to maintain the proportion, we must multiply the numerator by the same factor. This is because we're essentially creating an equivalent fraction. If we multiply only the denominator, we change the value of the fraction.

Therefore, we multiply the numerator 7 by 5 as well:

7imes5=357 imes 5 = 35

This tells us that the numerator 'x' in the second fraction must be 35. So, we have:

rac{7}{15} = rac{35}{75}

Just like with cross-multiplication, we've found that x = 35. This scaling method can be particularly useful when the relationship between the numbers is easy to spot. It provides a more intuitive way to think about proportions, as you're directly scaling the ratio to find the equivalent fraction. It also reinforces the concept that equivalent fractions represent the same value, even though they have different numerators and denominators.

Checking Our Solution

Before we celebrate our victory, it's always a super good idea to check our solution. This is a crucial step in any mathematical problem-solving process. It ensures that we haven't made any silly mistakes along the way and that our answer makes sense. In the context of proportions, checking our solution involves substituting the value we found for 'x' back into the original proportion and verifying that the two ratios are indeed equal.

We found that x = 35, so let's substitute that into our proportion: $ rac{7}{15} = rac{35}{75}$. To verify if these two ratios are equal, we can use a couple of different methods. One way is to simplify the second fraction, $ rac{35}{75}$. Both 35 and 75 are divisible by 5. Dividing both the numerator and denominator by 5, we get:

rac{35 eq 5}{75 eq 5} = rac{7}{15}

Lo and behold, the simplified fraction is exactly the same as the first fraction in our proportion! This confirms that our solution, x = 35, is correct. Another way to check our solution is to use cross-multiplication again. If the cross-products are equal, then the proportion is valid. Let's multiply 7 by 75 and 15 by 35:

7imes75=5257 imes 75 = 525

15imes35=52515 imes 35 = 525

The cross-products are equal (525 = 525), which further validates our solution. By checking our solution, we've gained confidence that we've solved the problem correctly. This practice of verification is essential for building strong mathematical skills and avoiding errors. So, always take a few moments to double-check your work – your future self will thank you!

The Correct Answer and Why

We've journeyed through the world of proportions, tackled the problem using two different methods (cross-multiplication and scaling), and even verified our solution. Now, let's state the correct answer explicitly and discuss why it's the right choice.

As we've demonstrated, the value of 'x' in the proportion $ rac{7}{15} = rac{x}{75}$ is 35. Therefore, the correct answer is C) 35. But why is this the right answer? Well, let's revisit our understanding of proportions. A proportion states that two ratios are equal. In this case, the ratio 7:15 must be equivalent to the ratio x:75. We found that to get from 15 to 75, we multiply by 5. To maintain the equality of the ratios, we must also multiply the numerator, 7, by the same factor, 5. This gives us 7 * 5 = 35. So, the ratio 35:75 is equivalent to 7:15.

Let's also consider why the other options are incorrect:

  • A) 1.4: This answer is way off the mark. It likely comes from incorrectly dividing 7 by 5 (the scaling factor) instead of multiplying. This highlights the importance of understanding the relationship between the numbers in a proportion.
  • B) 3: This answer doesn't have a clear logical path. There's no direct mathematical operation you can perform on the numbers in the proportion to arrive at 3.
  • D) 161: This number is much larger than what we would expect for 'x'. If we substituted 161 for 'x', the ratio $ rac{161}{75}$ would be significantly larger than $ rac{7}{15}$, violating the principle of proportionality.

By understanding the concept of proportions and applying the correct methods, we can confidently arrive at the correct answer, which is C) 35.

Real-World Applications of Proportions

Okay, we've conquered this proportion problem like mathematical ninjas! But you might be thinking, "Why should I care about proportions? Where will I ever use this in the real world?" Well, guys, the truth is that proportions are everywhere! They're not just abstract concepts confined to textbooks; they pop up in a surprisingly wide range of situations.

Let's explore some real-world applications of proportions:

  • Cooking and Baking: Recipes often specify quantities of ingredients for a certain number of servings. If you want to make a larger or smaller batch, you'll need to adjust the ingredients proportionally. For example, if a recipe for 4 servings calls for 1 cup of flour, you can use a proportion to figure out how much flour you need for 8 servings (which would be 2 cups).
  • Map Reading: Maps use a scale to represent real-world distances. A scale might say that 1 inch on the map represents 10 miles in reality. If you measure the distance between two cities on the map and find it's 2.5 inches, you can use a proportion to calculate the actual distance (which would be 25 miles).
  • Unit Conversions: Converting between different units of measurement, such as inches to centimeters or pounds to kilograms, often involves proportions. For instance, if you know that 1 inch is equal to 2.54 centimeters, you can use a proportion to convert any number of inches to centimeters.
  • Scale Models: Architects and engineers use scale models to represent buildings or structures. The dimensions of the model are proportional to the dimensions of the real thing. This allows them to visualize the design and make adjustments before construction begins.
  • Photography and Graphic Design: When resizing images, it's important to maintain the proportions to avoid distortion. Software programs use proportions to ensure that the image scales correctly, preserving the aspect ratio.
  • Business and Finance: Proportions are used in calculating percentages, interest rates, and financial ratios. They help businesses make informed decisions about pricing, investments, and profitability.

These are just a few examples, guys. The list goes on and on! Proportions are a fundamental tool for problem-solving in many different fields. By mastering proportions, you're not just learning a math concept; you're developing a valuable skill that will serve you well in many aspects of life. So, keep practicing, keep exploring, and keep applying your knowledge to the world around you! You'll be amazed at how often proportions come in handy.

Conclusion

Alright, mathletes, we've reached the end of our proportional journey! We started with a simple-looking proportion, $ rac{7}{15} = rac{x}{75}$, and we've unpacked it using multiple approaches. We've seen the power of cross-multiplication, the elegance of scaling the ratio, and the importance of checking our solutions. We've also discovered that the unknown value, 'x', is indeed 35, making option C) 35 the correct answer.

But more importantly, we've gone beyond just finding the answer. We've delved into the underlying concept of proportions, understanding what they represent and why they're so useful. We've explored how proportions connect ratios and allow us to solve problems involving scaling and equivalence. And we've even taken a peek at the real-world applications of proportions, seeing how they show up in everyday situations, from cooking and baking to map reading and photography.

So, what are the key takeaways from our adventure today? Here are a few reminders to keep in your mathematical toolbox:

  • Proportions are about equivalent ratios: They express the idea that two ratios represent the same relationship.
  • Cross-multiplication is a powerful technique: It provides a systematic way to solve for unknowns in proportions.
  • Scaling the ratio offers an intuitive approach: It highlights the connection between the numbers and helps you visualize the proportion.
  • Checking your solution is crucial: It ensures accuracy and builds confidence in your problem-solving skills.
  • Proportions are everywhere! They're not just a math concept; they're a tool for understanding and navigating the world.

Remember, guys, learning mathematics isn't just about memorizing formulas and procedures. It's about developing a way of thinking, a way of approaching problems, and a way of seeing the world in terms of patterns and relationships. Proportions are a beautiful example of this. They connect seemingly different quantities and reveal the underlying order and harmony in the world around us.

So, keep practicing, keep exploring, and keep asking questions. The world of mathematics is vast and fascinating, and there's always something new to discover. And who knows, maybe the next time you're scaling a recipe or planning a road trip, you'll find yourself thinking, "Hey, this is just a proportion problem!" And you'll smile, knowing that you have the tools and the understanding to tackle it with confidence.