Evaluating [6 + (4/a) + (b/3)] When A = 4 And B = 3 A Step-by-Step Guide

Hey guys! Today, we are going to dive into a fun little math problem. We're tasked with evaluating the expression ${6 + \frac{4}{a} + \frac{b}{3}}$ when a equals 4 and b equals 3. Sounds like a piece of cake, right? Let's break it down step by step to make sure we get it perfect. This is the kind of problem you might see on national exams, so paying close attention is super important. Math can sometimes seem daunting, but trust me, with a clear approach, it becomes much easier and even enjoyable. Think of it like solving a puzzle – each piece fits perfectly to reveal the final answer. So, grab your pencils, and let's get started! The key to success here is to be methodical and pay attention to detail. We'll substitute the given values into the expression and then simplify it. It's like following a recipe; each step is crucial for the final outcome. Remember, practice makes perfect, so the more problems like this we tackle, the more confident we become. So, let's put on our thinking caps and get ready to solve this mathematical puzzle together!

Step-by-Step Solution

1. Substitute the Values

The very first thing we need to do is replace the variables in our expression with the given values. We know that a is 4 and b is 3. So, let's plug those in:

${6 + \frac{4}{a} + \frac{b}{3} = 6 + \frac{4}{4} + \frac{3}{3}}$

See? That wasn't so hard! All we did was swap the letters a and b with their numerical values. It’s like replacing ingredients in a recipe – simple substitutions that lead to a delicious result. This is a crucial step because it sets the stage for the rest of our calculation. If we don't substitute correctly, the entire answer will be off. So always double-check this step to make sure everything is in its right place. Think of it as the foundation of a building; if the foundation isn't solid, the rest of the structure won't stand properly. In the same way, a correct substitution is the foundation of our mathematical solution. Now that we've got our expression set up, we can move on to the next step: simplifying those fractions.

2. Simplify the Fractions

Now we have some fractions to deal with, but don't worry, they're super easy to simplify. We have ${\frac{4}{4}}$ and ${\frac{3}{3}}$. What happens when the numerator (the top number) and the denominator (the bottom number) are the same? They simply equal 1! So, we can rewrite our expression as:

${6 + 1 + 1}$

Isn't that much simpler? Fractions can sometimes look intimidating, but in this case, they turned out to be quite friendly. Simplifying fractions is a fundamental skill in math, and mastering it will make your life so much easier. It's like knowing a shortcut on a familiar route – you get to your destination faster and with less effort. When we simplify fractions, we reduce them to their simplest form, which makes further calculations much easier to handle. In our case, ${\frac{4}{4}}$ and ${\frac{3}{3}}$ both simplified to 1, turning our expression into a straightforward addition problem. This is a perfect example of how simplifying early can save us time and prevent potential errors later on.

3. Perform the Addition

We're almost there! Now we just need to add the numbers together. We have 6 + 1 + 1. This is basic addition, so we can easily calculate the result:

${6 + 1 + 1 = 8}$

And just like that, we have our answer! Addition is one of the most fundamental operations in math, and it's something we use every day. In this case, we're simply combining three numbers to find their total. Think of it like counting objects; you're putting them all together to find out how many you have in total. This step is the culmination of all our previous work. We simplified the fractions, and now we're adding the resulting numbers to arrive at our final answer. It's like the final brushstroke on a painting, bringing the entire artwork to completion. So, after all our hard work, we can confidently say that the value of the expression is 8.

Final Answer

The value of the expression ${6 + \frac{4}{a} + \frac{b}{3}}$ when a = 4 and b = 3 is 8. Great job, guys! We tackled this problem step by step, and we nailed it. Remember, the key to solving math problems is to break them down into smaller, manageable steps. And always double-check your work to make sure you haven't made any silly mistakes. This problem is a great example of how math can be simple and straightforward if you approach it methodically. Each step built upon the previous one, leading us to the final answer. So, whether you're facing a similar problem on an exam or just want to flex your math muscles, remember the steps we followed today: substitute, simplify, and add. With these tools in your mathematical toolkit, you'll be well-prepared to tackle any challenge that comes your way.

Key Takeaways for Exam Success

Mastering Substitution

Substitution is the bread and butter of algebra. It's a technique we use to replace variables with their corresponding values. In our problem, we substituted a with 4 and b with 3. But the importance of substitution extends far beyond this single problem. In various mathematical contexts, substitution allows us to evaluate expressions, solve equations, and simplify complex problems. Think of it as replacing a placeholder with a specific value to unlock the next stage of the problem-solving process. It’s like having a secret code that opens a door to a new level of understanding. The more proficient you become at substitution, the more confident you'll feel when tackling algebraic problems. This skill is crucial not only for exams but also for real-world applications where you often need to plug in values to formulas and equations. So, practice substitution regularly to make it second nature, and you'll be well-equipped to handle any mathematical challenge that comes your way.

The Art of Simplification

Simplification is like decluttering your mind – it makes complex problems easier to handle. In our example, we simplified the fractions ${\frac{4}{4}}$ and ${\frac{3}{3}}$ to 1. But simplification is more than just dealing with fractions. It involves reducing expressions to their simplest forms, combining like terms, and eliminating unnecessary complexity. This not only makes calculations easier but also helps in understanding the underlying structure of a problem. Think of simplification as the art of making something complex beautiful in its simplicity. It’s about distilling the essence of a problem down to its core components. This skill is invaluable in various areas of math and science, where complex formulas and equations often need to be simplified to extract meaningful information. Mastering simplification techniques will not only improve your exam scores but also enhance your problem-solving abilities in general.

Accuracy in Arithmetic

Arithmetic might seem basic, but it's the backbone of mathematics. In our problem, we performed simple addition: 6 + 1 + 1 = 8. But even simple arithmetic can be tricky if we're not careful. Accuracy is paramount, and one small error can throw off the entire solution. So, it's essential to double-check your calculations and pay close attention to detail. Think of arithmetic as the foundation of a building; if the foundation is weak, the entire structure is at risk. Similarly, if your arithmetic is flawed, your mathematical reasoning will crumble. Developing good arithmetic skills involves practicing regularly, using mental math techniques, and knowing when to use a calculator. It also means being meticulous and verifying your results. Accuracy in arithmetic is not just about getting the right answers; it's about building a solid foundation for more advanced mathematical concepts.

Practice Makes Perfect

The more you practice problems like this, the more comfortable you'll become with the concepts. Try varying the values of a and b to see how the final result changes. This will not only reinforce your understanding but also help you develop a deeper intuition for how the expression works. Remember, math isn't just about memorizing formulas; it's about understanding the underlying principles and applying them in different contexts. So, keep practicing, and you'll be amazed at how much your math skills improve. Practice is the key to mastery, and with each problem you solve, you're building your confidence and competence. It’s like training for a marathon; the more miles you run, the stronger and more prepared you become. So, embrace the challenge, put in the effort, and watch your math abilities soar.