Simplifying 9/2 X 15/6: A Step-by-Step Guide

by Admin 45 views
Simplifying 9/2 x 15/6: A Step-by-Step Guide

Hey guys! Let's break down how to simplify the fraction (9/2) x (15/6). It might look a little intimidating at first, but don't worry, it's totally manageable once we go through it step by step. We'll cover everything from the initial multiplication to reducing the fraction to its simplest form. So, grab your calculators (or your trusty mental math skills) and let's get started!

Understanding the Basics of Fraction Multiplication

Before we dive into this specific problem, let's quickly refresh the basics of multiplying fractions. When you multiply fractions, you simply multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. That's it! So, if you have (a/b) x (c/d), the result is (ac) / (bd). This principle is super important, and it's the foundation for everything else we're going to do. Make sure you're comfortable with this before moving on. Think of it like building blocks; you need a solid base to build something amazing! Once you understand this simple rule, you'll find that multiplying fractions is actually pretty straightforward. Remember, numerator times numerator, denominator times denominator. Keep that in mind, and you'll be golden.

Now, why is this important? Well, understanding the basic principle allows you to tackle any fraction multiplication problem with confidence. You're not just memorizing steps; you're actually understanding why you're doing what you're doing. This understanding will help you in more complex math problems down the road. Plus, it's kind of cool to know the underlying logic, right? So, let's keep this fundamental rule in mind as we move forward and simplify our fraction (9/2) x (15/6). You've got this!

Step-by-Step Multiplication of 9/2 x 15/6

Okay, let's get into the nitty-gritty of multiplying 9/2 by 15/6. The first thing we need to do is multiply the numerators together. That means we're multiplying 9 by 15. If you do that calculation, you'll find that 9 multiplied by 15 is 135. So, the numerator of our new fraction is going to be 135. Easy peasy, right? Next up, we need to multiply the denominators. That means we're multiplying 2 by 6. And 2 times 6 is, of course, 12. So, the denominator of our new fraction is 12. So, after multiplying the numerators and denominators, we get the fraction 135/12. This is the result of the multiplication, but we're not quite done yet. We need to simplify this fraction to its simplest form.

To recap, we started with (9/2) x (15/6). We multiplied the numerators (9 x 15) to get 135, and we multiplied the denominators (2 x 6) to get 12. This gave us the fraction 135/12. Remember to double-check your multiplication to make sure you didn't make any silly mistakes. It's always a good idea to be thorough. Now that we have our initial fraction, 135/12, we can move on to the next step: simplifying it. Don't worry, it's not as scary as it sounds! We're just going to find a common factor and divide both the numerator and the denominator by that factor.

Finding the Greatest Common Divisor (GCD)

Now comes the fun part: simplifying! To simplify 135/12, we need to find the greatest common divisor (GCD) of 135 and 12. The GCD is the largest number that divides both 135 and 12 without leaving a remainder. There are a few ways to find the GCD. One way is to list all the factors of each number and find the largest one they have in common. Another way is to use the Euclidean algorithm, which is a bit more advanced but can be faster for larger numbers. For this example, let's list the factors.

The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 135 are 1, 3, 5, 9, 15, 27, 45, and 135. Looking at these lists, we can see that the largest number that appears in both lists is 3. Therefore, the GCD of 135 and 12 is 3. So, we're going to divide both the numerator and the denominator by 3 to simplify the fraction. Finding the GCD is a crucial step in simplifying fractions, so make sure you understand how to do it. If you're not comfortable finding the GCD, there are plenty of online resources and videos that can help you out. Once you get the hang of it, it'll become second nature!

Simplifying the Fraction by Dividing by the GCD

Now that we know the GCD of 135 and 12 is 3, we can simplify the fraction 135/12. To do this, we divide both the numerator and the denominator by 3. So, we divide 135 by 3, which gives us 45. And we divide 12 by 3, which gives us 4. Therefore, the simplified fraction is 45/4. This fraction is now in its simplest form because 45 and 4 have no common factors other than 1. That means we can't simplify it any further. So, the final answer to the problem (9/2) x (15/6) simplified is 45/4.

Always remember to double-check that your simplified fraction is indeed in its simplest form. Make sure there are no common factors between the numerator and the denominator. If there are, you need to divide them by that common factor until you can't simplify any further. Simplifying fractions is like tidying up; you want to make sure everything is in its proper place and as neat as possible. It's a satisfying feeling when you finally get to the simplest form!

Converting the Improper Fraction to a Mixed Number (Optional)

Okay, so we've simplified the fraction to 45/4. This is an improper fraction because the numerator (45) is larger than the denominator (4). Sometimes, you might want to convert an improper fraction to a mixed number. A mixed number is a whole number and a proper fraction combined. To convert 45/4 to a mixed number, we need to divide 45 by 4. 4 goes into 45 eleven times (11 x 4 = 44), with a remainder of 1. So, the whole number part of our mixed number is 11, and the remainder is 1, which becomes the numerator of our proper fraction, with the denominator staying the same (4). Therefore, 45/4 is equal to the mixed number 11 1/4.

Converting to a mixed number is totally optional, and it depends on what your teacher or the problem is asking for. Sometimes, they'll specifically want the answer as an improper fraction, and other times, they'll want it as a mixed number. Just make sure you read the instructions carefully. And hey, even if they don't ask for it, it's still a good skill to have! Knowing how to convert between improper fractions and mixed numbers will definitely come in handy in future math problems. Plus, it's just another tool in your mathematical toolbox!

Conclusion: Final Answer and Key Takeaways

Alright, guys! We made it! We successfully simplified (9/2) x (15/6) to 45/4, and we even converted it to the mixed number 11 1/4. Give yourselves a pat on the back! The key takeaways from this exercise are: First, remember to multiply the numerators and the denominators separately. Second, find the greatest common divisor (GCD) of the numerator and denominator. Third, divide both the numerator and the denominator by the GCD to simplify the fraction. And finally, don't be afraid to convert to a mixed number if needed!

Simplifying fractions might seem a bit tricky at first, but with practice, you'll become a pro in no time. The more you work with fractions, the more comfortable you'll become with them. And remember, math is like a muscle; the more you use it, the stronger it gets. So, keep practicing, keep learning, and don't be afraid to ask for help when you need it. You've got this! Now go out there and conquer those fractions!