Dividing Fractions Step-by-Step Solution 1 1/5 By 2 1/4

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Hey guys! Ever get tripped up by fraction division? It can seem a bit daunting at first, but trust me, once you understand the steps, it’s actually quite straightforward. Let's break down how to tackle the problem: 1 1/5 divided by 2 1/4. We'll go through each step methodically, so you'll be a fraction-division whiz in no time! We will also understand the fundamental concepts, ensuring that you grasp not just the how but also the why behind each step. This understanding is crucial for tackling more complex problems and building a solid foundation in mathematics. So, let’s dive in and make fraction division a breeze!

Step 1: Converting Mixed Numbers to Improper Fractions

Before we can even think about dividing, we need to deal with those pesky mixed numbers. Mixed numbers, like 1 1/5 and 2 1/4, combine a whole number and a fraction. To make our lives easier, we're going to convert them into improper fractions. An improper fraction is simply a fraction where the numerator (the top number) is larger than or equal to the denominator (the bottom number).

So, how do we do this conversion? It's a simple process: We'll start by focusing on 1 1/5. To convert this, follow these steps:

  1. Multiply the whole number (1) by the denominator (5): 1 * 5 = 5
  2. Add the numerator (1) to the result: 5 + 1 = 6
  3. Keep the same denominator (5).

Therefore, 1 1/5 becomes 6/5. Easy peasy, right?

Now, let's tackle 2 1/4. We'll use the same method:

  1. Multiply the whole number (2) by the denominator (4): 2 * 4 = 8
  2. Add the numerator (1) to the result: 8 + 1 = 9
  3. Keep the same denominator (4).

So, 2 1/4 transforms into 9/4. We've now successfully converted our mixed numbers into improper fractions! This is a crucial step because it sets us up for the next part of the division process. Trying to divide mixed numbers directly is messy and confusing, so converting to improper fractions is the way to go.

Why does this work? Think about it this way: 1 1/5 means one whole plus one-fifth. The one whole can be thought of as 5/5 (since 5 divided by 5 equals 1). Adding that to the existing 1/5 gives us 6/5. The same logic applies to 2 1/4. Understanding the why behind the conversion helps you remember the steps and apply them confidently.

Step 2: The Key to Division: Multiplying by the Reciprocal

Okay, we've got our improper fractions – 6/5 and 9/4. Now comes the magic trick of fraction division: We don't actually divide fractions. Instead, we multiply by the reciprocal of the second fraction. What's a reciprocal? Great question! The reciprocal of a fraction is simply that fraction flipped over. The numerator becomes the denominator, and the denominator becomes the numerator.

Let's look at our second fraction, 9/4. To find its reciprocal, we just flip it: The reciprocal of 9/4 is 4/9. That's it! Now, instead of dividing 6/5 by 9/4, we're going to multiply 6/5 by 4/9. This is a fundamental rule of fraction division, and it's the key to solving these problems correctly.

Why does this work? This might seem like a strange trick, but there's a solid mathematical reason behind it. Dividing by a number is the same as multiplying by its inverse. The reciprocal of a fraction is its multiplicative inverse, which means that when you multiply a fraction by its reciprocal, you get 1. For example, (9/4) * (4/9) = 1.

So, by multiplying by the reciprocal, we're essentially undoing the division and turning it into a multiplication problem, which is much easier to handle. This concept is really important for understanding more advanced math topics later on, so make sure you grasp the idea of reciprocals and their role in division.

Now that we know this crucial step, we can rewrite our problem. The original problem was 1 1/5 divided by 2 1/4, which we converted to 6/5 divided by 9/4. Now, we're changing that to 6/5 multiplied by 4/9. See how we've transformed the division into multiplication? That's the power of the reciprocal!

Step 3: Multiplying the Fractions

We've transformed our division problem into a multiplication problem: 6/5 multiplied by 4/9. Multiplying fractions is actually quite simple. Here's the rule: Multiply the numerators (the top numbers) together, and then multiply the denominators (the bottom numbers) together.

So, let's apply this to our problem:

  • Numerator: 6 * 4 = 24
  • Denominator: 5 * 9 = 45

This gives us the fraction 24/45. We've successfully multiplied the fractions! We are now one step closer to the final answer. However, before we declare victory, there’s one more thing we need to consider: simplifying our fraction. This is where the concept of finding the greatest common factor comes into play.

When multiplying fractions, remember to double-check your work. It's easy to make a small mistake in multiplication, especially when dealing with larger numbers. A quick review can save you from an incorrect final answer. Additionally, understanding the concept of multiplying fractions visually can be helpful. Imagine you have 6 slices of a pie, each representing 1/5 of the whole pie. You're taking 4/9 of each of those slices. Multiplying 6/5 by 4/9 tells you how much of the whole pie you have in total.

Step 4: Simplifying the Fraction to Its Lowest Terms

We've arrived at 24/45, which is a perfectly valid answer. However, mathematicians (and math teachers!) generally prefer fractions to be in their simplest form, also known as lowest terms. This means reducing the fraction until the numerator and denominator have no common factors other than 1. In other words, we want to divide both the numerator and the denominator by their greatest common factor (GCF).

So, how do we find the GCF of 24 and 45? One way is to list the factors of each number:

  • Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
  • Factors of 45: 1, 3, 5, 9, 15, 45

Looking at these lists, we can see that the largest number that appears in both lists is 3. Therefore, the GCF of 24 and 45 is 3.

Now, we divide both the numerator and the denominator by 3:

  • 24 ÷ 3 = 8
  • 45 ÷ 3 = 15

This gives us the simplified fraction 8/15. This fraction is in its lowest terms because 8 and 15 have no common factors other than 1. We have successfully simplified our answer! This step is crucial because it ensures that your final answer is in the most concise and understandable form. Simplifying fractions not only makes the answer look cleaner but also makes it easier to work with in future calculations.

There are other methods for finding the GCF, such as using prime factorization, but listing the factors works well for smaller numbers. The important thing is to find the greatest common factor to simplify the fraction in one step. If you divide by a common factor that's not the greatest, you'll just need to simplify again.

Final Answer: 8/15

We've done it! We've successfully divided 1 1/5 by 2 1/4. We converted the mixed numbers to improper fractions, multiplied by the reciprocal, simplified the resulting fraction, and arrived at our final answer: 8/15. Yay us!

So, the next time you encounter a fraction division problem, remember these steps: Convert to improper fractions, multiply by the reciprocal, multiply the numerators and denominators, and simplify. With a little practice, you'll be dividing fractions like a pro! Remember, math is all about practice. The more you practice, the more confident you'll become. Try working through similar problems on your own, and don't be afraid to ask for help if you get stuck. There are plenty of resources available online and in textbooks to help you master fraction division. Keep up the great work, and you'll be amazed at what you can achieve!

And remember, guys, understanding the why behind the math is just as important as knowing the how. When you understand the underlying concepts, you're not just memorizing steps; you're building a solid foundation for future learning. Keep asking questions, keep exploring, and keep having fun with math!