How To Simplify 6/9 - 1/8 A Step-by-Step Guide

by Scholario Team 47 views

Hey guys! Today, we're diving deep into the fascinating world of fractions. Specifically, we're going to tackle a common problem that many students (and even adults!) face: how to combine fractions with different denominators. Our main goal? To simplify the expression 6/9 - 1/8. Sounds intimidating? Don't worry! We'll break it down step-by-step, making it super easy to understand. So, grab your pencils and let's get started!

Understanding the Basics of Fractions

Before we jump into the nitty-gritty of combining 6/9 and 1/8, let's make sure we're all on the same page with the basic concepts of fractions. Think of a fraction as a part of a whole. It's represented by two numbers separated by a line: the numerator (the top number) and the denominator (the bottom number).

The denominator tells us how many equal parts the whole is divided into. For example, in the fraction 1/4, the denominator 4 means the whole is divided into four equal parts. The numerator, on the other hand, tells us how many of these parts we're talking about. So, in 1/4, the numerator 1 means we're considering one of those four parts.

Fractions can represent anything from slices of a pizza to portions of a cake. They're a fundamental concept in mathematics and are used in everyday life, from cooking and baking to measuring and building. Understanding fractions is crucial for tackling more complex math problems, so let's make sure we've got a solid foundation.

Now, what happens when we want to add or subtract fractions? It's not as simple as just adding or subtracting the numerators and denominators separately. We need to make sure the fractions have a common denominator – a shared bottom number. This allows us to compare and combine the fractions accurately. Imagine trying to add apples and oranges – you can't do it directly! You need a common unit, like "fruit," to add them together. The same principle applies to fractions.

So, how do we find this common denominator? That's where the concept of the Least Common Multiple (LCM) comes into play. The LCM is the smallest number that is a multiple of both denominators. Once we find the LCM, we can rewrite the fractions with this common denominator and then perform the addition or subtraction. We'll explore this in more detail in the next section.

In the context of our problem, 6/9 - 1/8, we have two fractions with different denominators: 9 and 8. This means we can't directly subtract them yet. We need to find the LCM of 9 and 8, and then rewrite both fractions with that common denominator. This is the key to solving the problem, and we'll tackle it step-by-step.

Remember, fractions are all about understanding parts of a whole. By grasping this fundamental concept and the idea of a common denominator, we can confidently tackle any fraction problem that comes our way. So, let's move on and find the LCM of 9 and 8 – the next crucial step in simplifying 6/9 - 1/8!

Finding the Least Common Multiple (LCM)

The Least Common Multiple (LCM) is a crucial concept when it comes to adding or subtracting fractions with different denominators. Think of it as the magic key that unlocks the solution! The LCM is the smallest number that is a multiple of both denominators. In our case, we need to find the LCM of 9 and 8. Why? Because that will be our common denominator, allowing us to combine 6/9 and 1/8 effectively.

There are a couple of ways to find the LCM, and we'll explore two common methods: listing multiples and prime factorization. Let's start with the listing multiples method. This involves listing out the multiples of each number until we find a common one. Multiples are simply the numbers you get when you multiply a number by an integer (1, 2, 3, and so on).

So, let's list the multiples of 9:

  • 9 x 1 = 9
  • 9 x 2 = 18
  • 9 x 3 = 27
  • 9 x 4 = 36
  • 9 x 5 = 45
  • 9 x 6 = 54
  • 9 x 7 = 63
  • 9 x 8 = 72
  • 9 x 9 = 81
  • ...

And now, let's list the multiples of 8:

  • 8 x 1 = 8
  • 8 x 2 = 16
  • 8 x 3 = 24
  • 8 x 4 = 32
  • 8 x 5 = 40
  • 8 x 6 = 48
  • 8 x 7 = 56
  • 8 x 8 = 64
  • 8 x 9 = 72
  • ...

Looking at the lists, we can see that the smallest multiple that both 9 and 8 share is 72. Therefore, the LCM of 9 and 8 is 72. This means we'll be rewriting our fractions with a denominator of 72.

Now, let's explore the second method: prime factorization. This method involves breaking down each number into its prime factors – prime numbers that multiply together to give the original number. Prime numbers are numbers greater than 1 that are only divisible by 1 and themselves (e.g., 2, 3, 5, 7, 11).

Let's find the prime factorization of 9:

  • 9 = 3 x 3 = 3²

And now, the prime factorization of 8:

  • 8 = 2 x 2 x 2 = 2³

To find the LCM using prime factorization, we take the highest power of each prime factor that appears in either factorization and multiply them together. In this case, we have 3² (from 9) and 2³ (from 8).

So, the LCM is 3² x 2³ = 9 x 8 = 72. Again, we arrive at the same answer: the LCM of 9 and 8 is 72.

Whether you choose the listing multiples method or the prime factorization method, the goal is the same: to find the LCM. In our case, we've determined that the LCM of 9 and 8 is 72. This is a crucial step because it allows us to rewrite the fractions 6/9 and 1/8 with a common denominator, making them ready for subtraction. In the next section, we'll do just that – rewrite the fractions and prepare them for the final step!

Rewriting Fractions with a Common Denominator

Alright, guys, now that we've conquered the LCM and found that the magic number for 9 and 8 is 72, it's time to put that knowledge into action! We need to rewrite the fractions 6/9 and 1/8 so that they both have a denominator of 72. This is a crucial step because it allows us to finally subtract the fractions. Remember, we can only add or subtract fractions when they have the same denominator – it's like comparing apples to apples, not apples to oranges!

So, how do we rewrite these fractions? The key is to multiply both the numerator and the denominator of each fraction by a specific number that will result in the desired denominator of 72. Let's start with the fraction 6/9. We need to figure out what number we can multiply 9 by to get 72. We know from our LCM exploration that 9 x 8 = 72. So, we'll multiply both the numerator and the denominator of 6/9 by 8.

Here's how it looks:

(6/9) x (8/8) = (6 x 8) / (9 x 8) = 48/72

Notice that we're multiplying by 8/8, which is essentially multiplying by 1. This means we're not changing the value of the fraction; we're just changing the way it looks. 48/72 is equivalent to 6/9, but now it has our desired denominator of 72.

Now, let's tackle the fraction 1/8. We need to figure out what number we can multiply 8 by to get 72. Again, our LCM knowledge comes in handy: 8 x 9 = 72. So, we'll multiply both the numerator and the denominator of 1/8 by 9.

Here's the calculation:

(1/8) x (9/9) = (1 x 9) / (8 x 9) = 9/72

Just like before, we're multiplying by 9/9, which is equal to 1. We're not changing the value of the fraction; we're simply expressing it with a different denominator. 9/72 is equivalent to 1/8, and it now has our common denominator of 72.

Now, take a moment to appreciate what we've accomplished! We started with two fractions, 6/9 and 1/8, that had different denominators. We found the LCM of 9 and 8, which is 72. And then, we rewrote both fractions with a denominator of 72. We now have 48/72 and 9/72 – fractions that are ready to be subtracted!

This step is super important because it sets us up for the final stage of solving the problem. We've successfully transformed the fractions into a form where we can easily perform the subtraction. In the next section, we'll do just that: subtract the fractions and simplify the result. We're almost there, guys! Let's keep going!

Subtracting the Fractions and Simplifying

Okay, folks, the moment we've been waiting for! We've successfully navigated the tricky parts – finding the LCM and rewriting the fractions with a common denominator. Now, we're at the home stretch: subtracting the fractions and simplifying the result. This is where all our hard work pays off!

We have the fractions 48/72 and 9/72. Since they now have the same denominator, we can directly subtract the numerators. The denominator stays the same. So, here's the calculation:

48/72 - 9/72 = (48 - 9) / 72 = 39/72

We've done it! We've subtracted the fractions and arrived at the answer 39/72. But hold on a second… we're not quite finished yet. Remember, the goal is to simplify the answer as much as possible. This means we need to see if we can reduce the fraction to its lowest terms. In other words, can we find a number that divides both the numerator (39) and the denominator (72) without leaving a remainder?

To simplify a fraction, we look for the Greatest Common Factor (GCF), also known as the Greatest Common Divisor (GCD), of the numerator and the denominator. The GCF is the largest number that divides both numbers evenly. There are a few ways to find the GCF, but one common method is to list the factors of each number and identify the largest one they share.

Let's list the factors of 39:

  • 1, 3, 13, 39

And now, the factors of 72:

  • 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

Looking at the lists, we can see that the largest factor that 39 and 72 share is 3. Therefore, the GCF of 39 and 72 is 3.

To simplify the fraction, we divide both the numerator and the denominator by the GCF:

(39/3) / (72/3) = 13/24

And there we have it! We've simplified the fraction 39/72 to 13/24. This is the final answer, and it's in its simplest form because 13 and 24 have no common factors other than 1.

So, to recap, we started with the problem 6/9 - 1/8. We found the LCM of 9 and 8, which is 72. We rewrote the fractions with a common denominator: 48/72 and 9/72. We subtracted the fractions to get 39/72. And finally, we simplified the result to 13/24. Phew! That was quite a journey, but we made it!

Conclusion: Mastering Fraction Simplification

Fantastic job, everyone! We've successfully navigated the world of fraction subtraction and simplification. We started with a seemingly complex problem, 6/9 - 1/8, and broke it down into manageable steps. We learned about the importance of understanding fractions, finding the Least Common Multiple (LCM), rewriting fractions with a common denominator, subtracting the fractions, and simplifying the result using the Greatest Common Factor (GCF).

This process might seem like a lot of steps, but with practice, it becomes second nature. The key is to understand the underlying concepts and to approach each step methodically. Remember, fractions are a fundamental part of mathematics, and mastering them opens the door to more advanced topics.

So, what are the key takeaways from this journey?

  • Understanding Fractions: Grasp the basic concept of a fraction as a part of a whole, represented by a numerator and a denominator.
  • Finding the LCM: The LCM is crucial for adding or subtracting fractions with different denominators. We explored two methods: listing multiples and prime factorization.
  • Rewriting Fractions: This step allows us to combine fractions by expressing them with a common denominator, without changing their value.
  • Subtracting Fractions: Once we have a common denominator, we can simply subtract the numerators, keeping the denominator the same.
  • Simplifying Fractions: Always simplify your answer to its lowest terms by finding the GCF and dividing both the numerator and denominator by it.

By following these steps, you can confidently tackle any fraction problem that comes your way. And remember, practice makes perfect! The more you work with fractions, the more comfortable you'll become.

So, go out there and conquer the world of fractions! You've got the tools and the knowledge to succeed. Keep practicing, keep learning, and most importantly, have fun with math!

If you ever encounter a fraction problem that seems daunting, remember this journey we took together. Break it down into smaller steps, apply the concepts we've learned, and you'll be simplifying fractions like a pro in no time!