Evaluating 60n(2-n) When N=3 A Step-by-Step Guide

by Felix Dubois 50 views

Hey guys! Let's dive into a fun math problem today. We're going to evaluate the expression 60n(2-n) when n is equal to 3. This might seem a bit intimidating at first, but don't worry, we'll break it down step by step and make it super easy to understand. So, grab your calculators (or your brains, if you're feeling extra sharp!), and let's get started!

Understanding the Expression

Before we jump into plugging in the value of n, let's take a moment to really understand what this expression, 60n(2-n), is all about. At its core, it's an algebraic expression. This means it's a combination of numbers, variables (in this case, n), and mathematical operations like multiplication and subtraction. Think of it like a recipe, where n is one of the ingredients, and the expression tells us how to combine everything to get a final result.

  • The Role of 'n': The letter n is our variable, which means it can represent different values. In this problem, we're specifically told that n is equal to 3. So, wherever we see n in the expression, we'll replace it with 3.
  • Order of Operations: Remember good old PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction)? This is our guiding principle for solving expressions. It tells us the order in which we need to perform the operations. In our expression, we have parentheses, multiplication, and subtraction. So, we'll tackle the parentheses first, then the multiplication.
  • Breaking it Down: Let's visualize the expression piece by piece. We have 60 multiplied by n, and then the result is multiplied by (2 - n). This parenthetical part, (2 - n), means we'll subtract the value of n from 2 before doing anything else. This is a crucial step to ensure we get the correct answer.

Understanding the expression is half the battle, guys. It's like having a map before you start a journey. Now that we know what we're dealing with, let's move on to the exciting part – substituting and simplifying!

Step-by-Step Evaluation of 60n(2-n) When n=3

Alright, let's get our hands dirty and actually solve this thing! The key here is to take it one step at a time, following the golden rule of PEMDAS. We'll substitute the value of n, simplify the expression within the parentheses, and then tackle the multiplications. Trust me, it's easier than it sounds!

1. Substitution: Replacing 'n' with 3

The very first thing we need to do is replace every instance of n in our expression, 60n(2-n), with the number 3. This gives us:

60 * 3 * (2 - 3)

See how we've simply swapped n for 3? This might seem like a small step, but it's a crucial one. We're now dealing with a purely numerical expression, which is much easier to handle.

2. Simplifying the Parentheses: (2 - 3)

Next up, let's conquer the parentheses. Remember, PEMDAS tells us to deal with parentheses first. Inside our parentheses, we have the operation 2 - 3. This is a simple subtraction:

2 - 3 = -1

So, we can replace (2 - 3) with -1 in our expression. This gives us:

60 * 3 * (-1)

We're making great progress, guys! The expression is getting simpler and simpler. Now, we just have multiplication left to tackle.

3. Multiplication: 60 * 3 * (-1)

Now comes the final showdown – multiplication. We have two multiplication operations to perform: 60 multiplied by 3, and then the result multiplied by -1. Remember, multiplication is associative, which means we can multiply in any order. Let's go from left to right:

First, let's multiply 60 by 3:

60 * 3 = 180

Now we have:

180 * (-1)

Next, we multiply 180 by -1:

180 * (-1) = -180

And there you have it! The final result of our expression is -180.

4. Putting it All Together

Let's recap the entire process to make sure we've got it down:

  1. Original Expression: 60n(2-n)
  2. Substitute n = 3: 60 * 3 * (2 - 3)
  3. Simplify Parentheses: 60 * 3 * (-1)
  4. Multiply: 180 * (-1)
  5. Final Result: -180

So, when n = 3, the expression 60n(2-n) evaluates to -180. Awesome job, guys! You've successfully navigated this algebraic problem. Pat yourselves on the back!

Common Mistakes to Avoid When Evaluating Expressions

Even though we've broken down the evaluation process step-by-step, it's still easy to make mistakes if we're not careful. Let's look at some common pitfalls that students often encounter when evaluating expressions, so we can learn how to avoid them. Identifying these potential errors beforehand can save us a lot of headaches (and incorrect answers!).

  • Ignoring Order of Operations (PEMDAS): This is probably the biggest culprit when it comes to mistakes in evaluating expressions. If you don't follow the correct order (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), you're likely to get the wrong answer. For instance, in our example, if we multiplied 60 by 3 before simplifying the parentheses (2-3), we would have ended up with a different result. So, always remember PEMDAS – it's your best friend in math!

  • Incorrectly Distributing Negatives: Negative signs can be tricky, especially when they're inside parentheses or involved in multiplication. A common mistake is to forget to distribute a negative sign across all terms within parentheses. For example, if we had an expression like -2(x + 3), we need to remember to multiply both x and 3 by -2. In our specific problem, we had a negative result within the parentheses (-1), which we needed to multiply correctly in the final step.

  • Simple Arithmetic Errors: Sometimes, the mistake isn't in the algebra, but in the basic arithmetic. A simple addition, subtraction, multiplication, or division error can throw off the entire solution. This is why it's always a good idea to double-check your calculations, especially when you're dealing with larger numbers or multiple operations.

  • Misinterpreting the Expression: Before you even start substituting values and simplifying, make sure you fully understand what the expression is telling you to do. Read the expression carefully, identify the variables, operations, and any special groupings (like parentheses). If you misinterpret the expression from the beginning, you're likely to go down the wrong path.

  • Forgetting to Substitute Correctly: When substituting the value of a variable (like n in our case), make sure you replace every instance of the variable with its value. It's easy to overlook one or two instances, especially in longer expressions. A good practice is to carefully cross-check your substituted expression with the original to ensure you haven't missed anything.

By being aware of these common mistakes, we can be more mindful and careful in our approach to evaluating expressions. Math is all about precision, guys, so let's strive for accuracy in every step!

Real-World Applications of Evaluating Expressions

Okay, we've conquered the math problem, but you might be wondering, “Why is this even important? Where will I ever use this in real life?” That's a valid question! Evaluating expressions isn't just an abstract math concept; it's a fundamental skill that pops up in all sorts of real-world scenarios. Let's explore some practical applications to see how this skill can be useful in your daily life.

  • Calculating Costs and Budgets: Imagine you're planning a party. You need to figure out how much it will cost to buy decorations, food, and drinks. You might have a formula that calculates the total cost based on the number of guests. This formula is an expression, and you'll need to evaluate it by plugging in the number of guests to find the total cost. Similarly, when you're creating a budget, you're essentially evaluating expressions to see how much money you have coming in versus how much is going out.

  • Cooking and Baking: Recipes are essentially mathematical expressions! They tell you the quantities of each ingredient you need. If you want to double or triple a recipe, you're evaluating expressions to adjust the amounts of each ingredient. For example, if a recipe calls for 1/2 cup of flour and you want to double it, you're evaluating the expression 2 * (1/2) to find the new amount of flour needed.

  • Physics and Engineering: Many concepts in physics and engineering are described using mathematical expressions. For example, the distance an object travels can be calculated using an expression involving its speed and time. Engineers use these expressions to design structures, calculate forces, and predict the behavior of systems. Evaluating these expressions accurately is crucial for ensuring safety and efficiency.

  • Computer Programming: At its core, computer programming is all about evaluating expressions. When you write code, you're creating instructions that the computer follows to perform calculations and manipulate data. These instructions often involve evaluating mathematical and logical expressions. So, mastering the skill of evaluating expressions is a huge asset for anyone interested in programming.

  • Financial Planning: When you're planning for your financial future, you might use expressions to calculate things like investment returns, loan payments, and retirement savings. These expressions can help you make informed decisions about your money and achieve your financial goals. For example, you might use an expression to calculate how much your investment will grow over time based on the interest rate and the amount you invest.

These are just a few examples, but the truth is, guys, evaluating expressions is a skill that underlies countless aspects of our lives. It's not just about solving math problems in a textbook; it's about developing a way of thinking that helps us make sense of the world around us. So, the next time you're faced with an expression, remember that you're not just doing math – you're building a valuable life skill!

Conclusion: Mastering the Art of Evaluating Expressions

Well, guys, we've reached the end of our journey into the world of evaluating expressions! We started with a seemingly simple problem, 60n(2-n) when n=3, and we've not only solved it step-by-step but also explored the broader importance of this skill. Hopefully, you're feeling confident and empowered to tackle any expression that comes your way.

We've learned that evaluating expressions is more than just plugging in numbers and performing calculations. It's about understanding the structure of the expression, following the order of operations, and being mindful of potential pitfalls. It's about precision, attention to detail, and a systematic approach to problem-solving.

But perhaps even more importantly, we've seen that evaluating expressions is a skill with real-world applications. From calculating costs and budgets to cooking and baking, from physics and engineering to computer programming and financial planning, the ability to evaluate expressions is a valuable asset in countless areas of life. It empowers us to make informed decisions, solve problems effectively, and navigate the complexities of the world around us.

So, as you continue your math journey, remember that the skills you're learning are not just for the classroom. They're tools that you can use to build a brighter future for yourself. Keep practicing, keep exploring, and keep challenging yourselves. You've got this!

And remember, guys, math can be fun! It's a way to unlock the secrets of the universe and develop your critical thinking skills. So, embrace the challenge, ask questions, and never stop learning. The world of math is vast and fascinating, and there's always something new to discover.

Keep up the great work, and I'll see you in the next math adventure!