Solving N/45 = 1/15 A Step-by-Step Guide

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Hey guys! Let's dive into a common type of math problem you'll often encounter: solving equations. In this article, we're going to break down the equation n/45 = 1/15 step by step. We'll explore the different methods you can use to find the value of 'n' and understand why a specific answer from the options provided is the correct one. So, buckle up, and let's get started!

Understanding the Equation n/45 = 1/15

When you first look at the equation n/45 = 1/15, it's crucial to understand what it actually means. Essentially, we're dealing with a proportion. A proportion is a statement that two ratios are equal. In our case, the ratio of 'n' to 45 is equal to the ratio of 1 to 15.

Think of it like this: If you have a fraction where the numerator is 'n' and the denominator is 45, it should be equivalent to the fraction 1/15. Our goal is to find the specific value of 'n' that makes this statement true. To achieve this, we need to isolate 'n' on one side of the equation. There are a couple of primary methods we can use to do this, which we'll delve into shortly. Understanding the concept of proportions is fundamental not just for solving this particular equation but also for a wide range of mathematical problems you'll encounter later on. This includes concepts like scaling, percentages, and even more complex algebraic equations. So, grasping this basic principle is key to your mathematical journey.

Methods to Solve for 'n'

Now that we understand the equation, let's explore the different ways we can solve for 'n'. There are two primary methods that are commonly used and super effective: the cross-multiplication method and the multiplication property of equality. Let's break down each one.

1. Cross-Multiplication Method

The cross-multiplication method is a classic technique for solving proportions. It's based on the principle that if two fractions are equal, then the product of the numerator of the first fraction and the denominator of the second fraction is equal to the product of the denominator of the first fraction and the numerator of the second fraction.

In simpler terms, if we have a/b = c/d, then a * d = b * c. Applying this to our equation, n/45 = 1/15, we cross-multiply to get: n * 15 = 1 * 45. This simplifies to 15n = 45. Now, to isolate 'n', we divide both sides of the equation by 15. This gives us n = 45 / 15, which simplifies to n = 3. So, using the cross-multiplication method, we've found that the value of 'n' that makes the equation true is 3. This method is particularly handy because it provides a direct and efficient way to solve proportions, making it a go-to technique for many students and mathematicians.

2. Multiplication Property of Equality

The multiplication property of equality is a fundamental concept in algebra that states that you can multiply both sides of an equation by the same non-zero number without changing the solution. This is a powerful tool because it allows us to manipulate equations while maintaining their balance.

In our equation, n/45 = 1/15, we want to isolate 'n'. To do this, we can multiply both sides of the equation by 45. This will cancel out the denominator on the left side, leaving us with just 'n'. So, multiplying both sides by 45, we get: (n/45) * 45 = (1/15) * 45. This simplifies to n = 45/15. Now, we just need to simplify the fraction 45/15. Dividing 45 by 15, we get 3. Therefore, n = 3. This method is particularly useful because it highlights the importance of maintaining balance in an equation. By performing the same operation on both sides, we ensure that the equation remains true and that we arrive at the correct solution.

Step-by-Step Solution Using Cross-Multiplication

Let's walk through the cross-multiplication method step-by-step to really solidify our understanding. This method, as we've discussed, is a direct way to solve proportions.

  1. Write down the equation: Start by writing down the equation we need to solve: n/45 = 1/15. This is our starting point, and it's crucial to have it clearly written out so we can follow the steps accurately.
  2. Cross-multiply: The next step is to cross-multiply. This involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa. So, we multiply 'n' by 15 and 1 by 45. This gives us: n * 15 = 1 * 45. Cross-multiplication is a powerful technique because it transforms the proportion into a linear equation, which is much easier to solve.
  3. Simplify: Now, let's simplify the equation. n * 15 simplifies to 15n, and 1 * 45 simplifies to 45. So, our equation now looks like this: 15n = 45. Simplifying the equation makes it clearer and brings us closer to isolating the variable 'n'.
  4. Isolate 'n': Our goal is to get 'n' by itself on one side of the equation. To do this, we need to undo the multiplication by 15. We can do this by dividing both sides of the equation by 15. So, we divide both 15n and 45 by 15. This gives us: 15n / 15 = 45 / 15. Dividing both sides by the same number maintains the balance of the equation and allows us to isolate 'n'.
  5. Solve for 'n': Finally, we solve for 'n'. 15n / 15 simplifies to n, and 45 / 15 simplifies to 3. Therefore, our solution is n = 3. This is the value of 'n' that makes the original equation true. By following these steps, we've successfully used cross-multiplication to find the solution to our equation.

Step-by-Step Solution Using the Multiplication Property of Equality

Now, let's tackle the same equation, n/45 = 1/15, but this time we'll use the multiplication property of equality. This method emphasizes the principle of maintaining balance in an equation.

  1. Write down the equation: Just like before, we start by writing down the equation: n/45 = 1/15. Having the equation clearly written is the first step to solving it accurately.
  2. Multiply both sides by 45: The key to using the multiplication property of equality is to multiply both sides of the equation by the same number. In this case, we want to eliminate the denominator 45 on the left side. So, we multiply both sides of the equation by 45. This gives us: (n/45) * 45 = (1/15) * 45. Multiplying both sides by 45 is a strategic move to isolate 'n'.
  3. Simplify: Now, let's simplify both sides of the equation. On the left side, (n/45) * 45 simplifies to just 'n', because the 45 in the numerator and the 45 in the denominator cancel each other out. On the right side, (1/15) * 45 simplifies to 45/15. So, our equation now looks like this: n = 45/15. Simplifying the equation makes it easier to see the next step.
  4. Solve for 'n': The final step is to simplify the fraction 45/15. Dividing 45 by 15, we get 3. Therefore, our solution is n = 3. This is the same answer we got using the cross-multiplication method, which reinforces the fact that there can be different paths to the same correct solution. By following these steps, we've successfully used the multiplication property of equality to find the value of 'n'.

Identifying the Correct Answer Choice

Alright, guys, we've done the hard work of solving the equation n/45 = 1/15. We used both cross-multiplication and the multiplication property of equality, and in both cases, we arrived at the same solution: n = 3. Now, let's look at the answer choices provided and pinpoint the correct one.

We were given the following options:

  • a. n = 1/3
  • c. n = 675
    1. n = 4
  • d. n = 3

By carefully comparing our solution (n = 3) with the answer choices, we can clearly see that option d, n = 3, is the correct answer. The other options (n = 1/3, n = 675, and n = 4) are incorrect because they do not satisfy the original equation. If we were to substitute any of these values for 'n' in the equation n/45 = 1/15, the equation would not hold true. This step of identifying the correct answer choice is crucial because it ensures that we've not only solved the equation correctly but also understood how to apply our solution to the context of the problem. It's a final check to make sure we're confident in our answer.

Why Other Options are Incorrect

It's always a good idea to understand why the incorrect options are wrong. This helps to reinforce your understanding of the problem and the solution process. Let's take a look at why the other options provided for n/45 = 1/15 are incorrect.

  • a. n = 1/3: If we substitute n = 1/3 into the equation, we get (1/3) / 45 = 1/15. This simplifies to 1/135 = 1/15, which is not true. Therefore, n = 1/3 is not the correct solution. The mistake here might be a misunderstanding of how to divide a fraction by a whole number.
  • c. n = 675: Substituting n = 675 into the equation gives us 675/45 = 1/15. Simplifying 675/45, we get 15 = 1/15, which is clearly false. This option is way off, suggesting a possible error in the initial setup or a misunderstanding of the scale of the numbers involved.
  • 5. n = 4: If we plug in n = 4 into the equation, we get 4/45 = 1/15. To check if this is true, we can cross-multiply: 4 * 15 = 1 * 45, which simplifies to 60 = 45. This is not true, so n = 4 is not the correct solution. This option is closer to the correct answer than the others, but it still doesn't satisfy the equation, highlighting the importance of precise calculations.

Understanding why these options are incorrect helps to solidify your understanding of the correct solution and the process of solving the equation. It also helps you identify common mistakes and avoid them in the future.

Real-World Applications of Proportions

Understanding proportions isn't just about solving equations in a textbook; it's a fundamental skill that has wide-ranging applications in the real world. Proportions are used in various fields, from cooking and baking to construction and engineering. Let's explore some of these practical applications.

  • Cooking and Baking: In the kitchen, proportions are essential for scaling recipes. If you want to double a recipe, you need to double all the ingredients while maintaining the correct ratios. For example, if a recipe calls for 1 cup of flour and 1/2 cup of sugar, doubling the recipe would require 2 cups of flour and 1 cup of sugar. Understanding proportions ensures that your dishes turn out just right, with the flavors and textures perfectly balanced.
  • Construction and Engineering: In construction, proportions are crucial for creating accurate blueprints and models. Architects and engineers use proportions to scale down large structures to manageable sizes for planning and design purposes. They also use proportions to calculate the amounts of materials needed for a project, ensuring that the structure is safe and stable. Whether it's scaling a blueprint or mixing concrete, proportions play a vital role in construction and engineering.
  • Map Reading: Maps are scaled-down representations of real-world locations, and understanding the scale is essential for navigation. Map scales are expressed as ratios, such as 1:10,000, which means that 1 unit on the map represents 10,000 units in the real world. By understanding these proportions, you can accurately estimate distances and plan your route effectively. Proportions are the key to making sense of maps and finding your way around.
  • Business and Finance: Proportions are also used in business and finance for various calculations. For example, when calculating profit margins, interest rates, or percentage changes, proportions are essential. Understanding how to work with proportions allows businesses to make informed decisions about pricing, investments, and financial planning. From calculating sales tax to projecting revenue, proportions are a valuable tool in the world of business and finance.

These are just a few examples of how proportions are used in the real world. By mastering the concept of proportions, you're not just learning a math skill; you're gaining a valuable tool that will help you in many aspects of life.

Conclusion: Mastering Proportions

Alright, guys, we've reached the end of our journey in solving the equation n/45 = 1/15. We've explored the equation, learned two powerful methods for solving it – cross-multiplication and the multiplication property of equality – and walked through step-by-step solutions. We've also identified the correct answer choice (n = 3) and understood why the other options were incorrect. But more importantly, we've seen how proportions are a fundamental concept with real-world applications in various fields.

Mastering proportions is a valuable skill that will benefit you not only in mathematics but also in everyday life. Whether you're scaling a recipe, planning a trip using a map, or making financial decisions, understanding proportions will help you make accurate calculations and informed choices. So, keep practicing, keep exploring, and keep applying your knowledge of proportions to the world around you. You've got this!