Dividing Fractions: Step-by-Step Solution
Hey guys! Let's dive into the world of fraction division! It might seem tricky at first, but trust me, once you get the hang of it, it's a piece of cake. We're going to break down a specific problem today: (1 + 1/45) ÷ (1/10 × 4/5). We'll go through each step slowly and carefully, so you understand exactly what's happening. Think of this as your ultimate guide to mastering fraction division!
Understanding the Problem
Okay, first things first, let's take a good look at our problem: (1 + 1/45) ÷ (1/10 × 4/5). We've got a mix of whole numbers, fractions, addition, division, and multiplication. It looks like a lot, but don't worry, we'll tackle it step by step. To solve this, we need to remember the order of operations (PEMDAS/BODMAS) and how to work with fractions. This means dealing with parentheses first, then multiplication and division (from left to right), and finally addition and subtraction. Understanding the order of operations is key to solving any mathematical problem correctly, especially when dealing with multiple operations.
Before we even start crunching numbers, let's think about what each part of the equation means. We have a mixed number (1 + 1/45), which we'll need to convert into an improper fraction. We also have multiplication of fractions (1/10 × 4/5), which we'll need to simplify. Once we've simplified these individual parts, we can move on to the division. Remember, dividing by a fraction is the same as multiplying by its reciprocal. This is a crucial concept that we'll use later on. So, let’s keep this in mind as we proceed.
Why is this important? Well, fractions are everywhere in real life, from cooking and baking to measuring and construction. Knowing how to divide fractions is a fundamental skill that will help you in many different situations. Plus, understanding the underlying concepts will make more advanced math topics easier to grasp in the future. So, let's get started and unlock the mystery of fraction division!
Step 1: Converting the Mixed Number
Our first step is to tackle the mixed number: 1 + 1/45. To make it easier to work with, we need to convert it into an improper fraction. An improper fraction is simply a fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number). To do this, we'll multiply the whole number (1) by the denominator (45) and then add the numerator (1). This will give us the new numerator, and we'll keep the same denominator.
So, let's do the math: (1 × 45) + 1 = 45 + 1 = 46. This means our new numerator is 46, and our denominator remains 45. Therefore, the improper fraction equivalent of 1 + 1/45 is 46/45. This conversion is important because it allows us to perform mathematical operations, like division, more easily. Remember, a mixed number represents a whole number plus a fraction, while an improper fraction represents the same value as a single fraction.
Why do we do this? Converting to an improper fraction makes the division process much smoother. It allows us to treat the entire quantity as a single fraction, making the subsequent calculations more straightforward. Imagine trying to divide (1 + 1/45) – it’s much easier to divide 46/45! This is a common technique in fraction arithmetic, and mastering it will significantly improve your problem-solving skills.
Step 2: Multiplying the Fractions
Next up, we need to deal with the multiplication part of our problem: 1/10 × 4/5. Multiplying fractions is actually quite straightforward. All we need to do is multiply the numerators together and multiply the denominators together. So, the new numerator will be 1 multiplied by 4, and the new denominator will be 10 multiplied by 5.
Let's calculate: 1 × 4 = 4, and 10 × 5 = 50. This gives us the fraction 4/50. But we're not quite done yet! It's always a good idea to simplify fractions to their lowest terms. Simplifying a fraction means finding the greatest common factor (GCF) of the numerator and the denominator and dividing both by it. In this case, the GCF of 4 and 50 is 2. So, we'll divide both the numerator and the denominator by 2.
Dividing 4 by 2 gives us 2, and dividing 50 by 2 gives us 25. This means our simplified fraction is 2/25. Simplifying fractions not only makes them easier to work with but also presents the answer in its most concise form. Always remember to check if your fraction can be simplified after performing any operation.
Why simplify? Simplifying fractions is like tidying up your work. It makes the fraction easier to understand and use in further calculations. Plus, it's often expected in math problems to provide the answer in its simplest form. Think of it as the final polish on your work!
Step 3: Dividing the Fractions
Now we're at the main event: dividing the fractions! We've simplified our original problem to 46/45 ÷ 2/25. Remember the golden rule of fraction division: dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is simply flipping it over, so the numerator becomes the denominator and the denominator becomes the numerator. The reciprocal of 2/25 is therefore 25/2.
So, our division problem now becomes a multiplication problem: 46/45 × 25/2. We can multiply these fractions just like we did before – multiply the numerators and multiply the denominators: 46 × 25 = 1150, and 45 × 2 = 90. This gives us the fraction 1150/90. Wow, that’s a big fraction!
Before we declare victory, let's simplify this bad boy. The GCF of 1150 and 90 is 10. Dividing both the numerator and the denominator by 10, we get 115/9. Now, this is an improper fraction, so let's convert it back to a mixed number to make it easier to understand. To do this, we'll divide the numerator (115) by the denominator (9). 115 divided by 9 is 12 with a remainder of 7. This means our mixed number is 12 7/9.
Why the reciprocal? The reciprocal rule might seem a bit like magic, but it's based on a solid mathematical principle. Dividing by a number is the same as multiplying by its inverse. For fractions, the inverse is the reciprocal. Understanding this concept will give you a deeper understanding of fraction division.
Final Answer
Alright guys, we've made it to the end! We started with a complex division problem and broke it down into manageable steps. We converted a mixed number to an improper fraction, multiplied fractions, divided fractions using the reciprocal, simplified, and finally converted back to a mixed number. Phew! Our final answer to the problem (1 + 1/45) ÷ (1/10 × 4/5) is 12 7/9.
Let's recap what we did:
- Converted the mixed number 1 + 1/45 to the improper fraction 46/45.
- Multiplied the fractions 1/10 × 4/5 to get 4/50, which we simplified to 2/25.
- Divided 46/45 by 2/25 by multiplying 46/45 by the reciprocal of 2/25, which is 25/2.
- Calculated 46/45 × 25/2 = 1150/90.
- Simplified 1150/90 to 115/9.
- Converted the improper fraction 115/9 to the mixed number 12 7/9.
Congratulations! You've successfully navigated the world of fraction division. Remember, practice makes perfect. The more you work with fractions, the more comfortable you'll become. So, keep practicing, and you'll be a fraction master in no time! If you ever get stuck, just break the problem down into smaller steps, like we did here, and remember the key concepts: converting to improper fractions, multiplying by the reciprocal, and simplifying.
Practice Problems
To really nail this down, here are a few more problems you can try on your own:
- (2 + 1/3) ÷ (3/4 × 1/2)
- (5/8 ÷ 1/4) + 2/3
- (1 1/2 × 2/5) ÷ 3/10
Try solving these problems using the steps we discussed. Remember to show your work, and don't be afraid to make mistakes – that's how we learn! Check your answers with a friend, a teacher, or an online calculator to see how you did. And remember, the most important thing is to understand the process, not just get the right answer.
If you get stuck, go back and review the steps we covered. Pay special attention to the concepts of converting mixed numbers, simplifying fractions, and multiplying by the reciprocal. These are the key building blocks of fraction division. With a little patience and practice, you'll be dividing fractions like a pro!
So go ahead, give these problems a try, and let's conquer the world of fractions together! You got this!
