Solving Fractions: A Step-by-Step Guide

Hey there, math enthusiasts! Today, we're diving into the world of fractions and figuring out how to solve a problem that might seem a little daunting at first: calculating -5/9 - 4/15 : 3/20. Don't worry, we'll break it down into easy-to-understand steps, so you'll be acing this in no time. Think of it like a fun puzzle – each step brings you closer to the solution! We'll go through the process, explaining each part, so you can confidently tackle similar problems in the future. So, grab your pencils, your favorite snacks, and let's get started!

Step 1: Understanding the Order of Operations

Before we jump into the numbers, let's chat about the order of operations, sometimes remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This rule tells us the sequence in which we should solve a math problem. In our case, we've got subtraction and division. According to PEMDAS, division comes before subtraction. So, first things first, we'll tackle the division part of our problem: 4/15 : 3/20. This is super important because doing things in the wrong order can lead to a completely different (and wrong!) answer. Remember, the right order is key to unlocking the correct solution. Knowing this rule helps us stay organized and accurate, ensuring our math journey is smooth and successful. Let's make sure we've got this order down pat – it's the foundation of our entire calculation!

To make sure we're all on the same page, let's reiterate the importance of PEMDAS. This isn't just a random set of rules; it's the language of math. Understanding it allows us to read and solve complex equations with confidence. Without knowing the order of operations, we’d be like trying to read a book with the pages out of order – it simply wouldn’t make sense! So, keep PEMDAS in mind as we move forward. It’s our guide, our compass, leading us to the correct answer. Now, let’s get down to the fun part: solving the fractions!

Step 2: Dividing Fractions

Alright, folks, time to divide some fractions! Dividing fractions isn't as scary as it sounds. The golden rule here is to flip and multiply. What does that mean? Well, when you divide by a fraction, you actually multiply by its reciprocal. The reciprocal is just the fraction flipped upside down. So, instead of dividing 4/15 by 3/20, we'll multiply 4/15 by 20/3. This little trick is the key to simplifying the problem and making it easier to solve. Trust me; it's much easier to multiply fractions than to divide them directly. This step transforms our equation from a division problem into a multiplication one, which we're much more comfortable with.

So, let’s do it! We're changing 4/15 : 3/20 into 4/15 * 20/3. When we multiply fractions, we multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. So, 4 multiplied by 20 gives us 80, and 15 multiplied by 3 gives us 45. This means 4/15 * 20/3 = 80/45. But, hold on a sec – we can simplify this fraction! Both 80 and 45 are divisible by 5. Dividing both the numerator and the denominator by 5, we get 16/9.

Now, let's take a moment to really grasp what we've just done. We've transformed a division problem into a multiplication problem, then simplified our answer to its lowest terms. This ability to manipulate fractions is a super valuable skill, helping us solve everything from simple equations to more complex problems. With each fraction we solve, we become more confident and capable. Remember, practice makes perfect. The more you do it, the easier it becomes. Now we've got a much simpler fraction to work with: 16/9. Keep this fraction in mind – we will use it in our next step. It’s all coming together!

Step 3: Subtracting Fractions with Different Denominators

Now that we've got our division sorted out, we can get to the subtraction part. Remember that original problem? -5/9 - 4/15 : 3/20? We've worked out the division bit, so we're left with -5/9 - 16/9. When subtracting fractions, we need to have a common denominator. Lucky for us, they already have a common denominator. In this case, both of the fractions we are dealing with now have the same denominator, so, we can simply subtract the numerators and keep the same denominator. It’s pretty straightforward! The beauty of this process is in its simplicity. Let's see how this all plays out.

So, what does this actually look like in practice? We are subtracting 16/9 from -5/9. Since both fractions now have the same denominator, we can directly subtract the numerators. So, we do -5 - 16, which gives us -21. We keep the denominator as 9. Thus, our new fraction is -21/9. But, can we simplify this fraction? Yes, we can! Both -21 and 9 are divisible by 3. Dividing both the numerator and denominator by 3, we get -7/3. Here we've done it! We've subtracted our fractions and simplified the answer. Now we have -7/3. This is our final result. But wait, we can also convert this to a mixed number!

Step 4: Converting to a Mixed Number (Optional)

Great job, everyone! We've solved the problem and arrived at the answer -7/3. But, we can take it one step further and convert this improper fraction (where the numerator is greater than the denominator) into a mixed number. A mixed number is a whole number combined with a fraction. It’s like saying, "I have a few full pizzas and a slice left over." Converting to a mixed number makes the answer easier to understand, especially in real-world scenarios. We're going to transform -7/3 into something that's a bit more intuitive to visualize. This step is about making the answer more digestible, giving a more complete picture of the quantity. Let's make that conversion!

To convert -7/3 to a mixed number, we divide the numerator (-7) by the denominator (3). -7 divided by 3 is -2 with a remainder of -1. This means we have -2 whole numbers, and a fraction of -1/3. So, -7/3 as a mixed number is -2 1/3. This answer tells us we have negative two full units and a negative third of another unit. So there you have it – the final answer to our problem, expressed both as an improper fraction and a mixed number. This conversion is a great way to showcase our understanding and make our answer even more accessible and easy to understand. We’ve not only solved the math problem but also learned how to express the answer in different ways.

Conclusion: We did it!

Congratulations, guys! We've successfully calculated -5/9 - 4/15 : 3/20, and we’ve done it step-by-step. We started with the order of operations, tackled dividing fractions by flipping and multiplying, and then confidently subtracted fractions to arrive at the solution. We even converted our answer into a mixed number. We started with what seemed like a complex problem, and now we've broken it down to manageable steps.

Remember, mastering fractions takes practice. The more you work with them, the more comfortable and confident you'll become. Each problem you solve is a victory, a testament to your hard work and persistence. Keep practicing, keep exploring, and keep the math adventure going. You've got this! And remember, math is everywhere. Now, you’ve not only solved the problem, but you’ve also expanded your mathematical toolkit. So, pat yourselves on the back, and get ready for the next challenge! You’re all math wizards now.