Step-by-Step Guide Solving 2/5 Of 3/4 Of Half Of 180

Hey guys! Ever stumbled upon a math problem that looks like a jumbled mess of fractions and numbers? Don't worry, we've all been there! Today, we're going to break down a classic problem: solving 2/5 of 3/4 of half of 180. It might seem intimidating at first, but with a step-by-step approach, you'll see it's totally manageable. So, grab your pencils, and let's dive in!

Understanding the Problem: What Does 'of' Mean in Math?

Before we start crunching numbers, let's clarify what the word "of" means in mathematical terms. In this context, "of" simply means multiplication. So, when we say "2/5 of 3/4," we're actually saying "2/5 multiplied by 3/4." This is a crucial understanding that will help us unravel the problem. Think of it like this: you're taking a fraction of another number. The “of” signifies that we are finding a part of a part, which directly translates to multiplication in mathematical operations. Understanding this core concept is the first step toward mastering not just this problem, but also a wide range of similar mathematical challenges. It's like having a secret key that unlocks a whole new level of mathematical confidence. The beauty of math often lies in its simplicity, and recognizing that "of" means multiplication is one such simple yet powerful insight.

Now, let's delve deeper into why "of" translates to multiplication. Imagine you have a pizza cut into four slices, and you want to eat half of it. What are you doing? You're taking a fraction (1/2) of the whole (the pizza). Mathematically, this means you're multiplying 1/2 by the total number of slices (4), which gives you 2 slices. This real-world example clearly demonstrates the multiplicative relationship implied by "of." Furthermore, this concept extends beyond just fractions and applies to percentages as well. For instance, saying "20% of 100" is the same as calculating 0.20 multiplied by 100, which equals 20. So, whether it's fractions, percentages, or decimals, the principle remains the same: "of" signifies multiplication. By grasping this fundamental idea, you'll be able to approach a variety of mathematical problems with greater clarity and ease. It's like having a universal translator for mathematical language, allowing you to decipher complex expressions into simpler, more manageable forms. So, the next time you encounter "of" in a math problem, remember this explanation, and you'll be well on your way to solving it!

Now that we've nailed down the meaning of "of," let's move on to the next crucial element of our problem: fractions. Fractions, those seemingly simple yet sometimes perplexing numbers, play a starring role in our equation. They represent parts of a whole, and understanding how they work is essential for conquering mathematical challenges. But before we dive into the specific fractions in our problem (2/5 and 3/4), let's take a moment to refresh our understanding of what fractions are and how they operate. A fraction consists of two key components: the numerator and the denominator. The numerator (the top number) tells us how many parts we have, while the denominator (the bottom number) tells us the total number of parts the whole is divided into. For example, in the fraction 2/5, the numerator 2 indicates that we have two parts, and the denominator 5 indicates that the whole is divided into five parts. So, 2/5 represents two out of five equal parts of something. Similarly, in the fraction 3/4, the numerator 3 tells us we have three parts, and the denominator 4 tells us the whole is divided into four parts. So, 3/4 represents three out of four equal parts of something. Visualizing fractions can be incredibly helpful. Imagine a pizza cut into four slices. If you eat three of those slices, you've eaten 3/4 of the pizza. Or imagine a pie cut into five slices. If you take two of those slices, you've taken 2/5 of the pie. This visual representation makes it easier to understand the proportion that each fraction represents.

Step 1: Finding Half of 180

The first part of our problem is to find half of 180. This is pretty straightforward. To find half of a number, we simply divide it by 2.

180 / 2 = 90

So, half of 180 is 90. We've conquered the first hurdle! This initial step sets the stage for the rest of the problem. Think of it as laying the foundation for a building. If the foundation is solid, the rest of the structure will be stable. Similarly, getting the first step right ensures that the subsequent calculations will be accurate. In this case, finding half of 180 is a fundamental arithmetic operation that paves the way for the more complex fraction multiplications that follow. It's a simple division, but it's a critical piece of the puzzle. Many mathematical problems involve breaking down complex tasks into smaller, more manageable steps. This approach not only makes the problem less intimidating but also reduces the chances of errors. By focusing on one step at a time, we can maintain clarity and ensure that each calculation is performed correctly. So, in the spirit of this strategy, we've successfully tackled the first step, and now we're ready to move on to the next. It's like climbing a staircase; each step brings us closer to the top. And with each step we take, our confidence grows, and the problem becomes less daunting. So, let's keep climbing, step by step, until we reach the final solution!

Step 2: Finding 3/4 of 90

Now that we know half of 180 is 90, we need to find 3/4 of 90. Remember, "of" means multiplication, so we need to multiply 3/4 by 90.

(3/4) * 90 = (3 * 90) / 4 = 270 / 4

Now, we simplify the fraction 270/4. Both 270 and 4 are divisible by 2:

270 / 2 = 135 4 / 2 = 2

So, 270/4 simplifies to 135/2. We can leave it as an improper fraction, or convert it to a mixed number:

135 / 2 = 67 1/2

So, 3/4 of 90 is 67 1/2. This step involves a little more arithmetic than the previous one, but it's still well within our grasp. We're essentially finding a fraction of a number, which is a common operation in many real-world scenarios. Imagine, for example, you have a recipe that calls for 90 grams of flour, but you only want to make 3/4 of the recipe. This step is exactly what you would do to calculate how much flour you need. The key to mastering fraction multiplication is to break it down into smaller, more manageable steps. First, we multiply the numerator of the fraction (3) by the whole number (90). This gives us 270. Then, we divide the result (270) by the denominator of the fraction (4). This gives us 270/4, which is an improper fraction. An improper fraction is simply a fraction where the numerator is greater than the denominator. While improper fractions are perfectly valid, they can sometimes be a bit difficult to visualize. That's why we often convert them to mixed numbers. A mixed number consists of a whole number and a proper fraction (where the numerator is less than the denominator). To convert an improper fraction to a mixed number, we divide the numerator by the denominator. The quotient becomes the whole number part of the mixed number, and the remainder becomes the numerator of the fractional part. The denominator stays the same. So, when we divide 270 by 4, we get a quotient of 67 and a remainder of 2. This means that 270/4 is equal to 67 2/4. We can further simplify the fractional part by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gives us 67 1/2. So, 3/4 of 90 is 67 1/2. We've successfully navigated this step, and we're one step closer to the final answer!

Step 3: Finding 2/5 of 67 1/2

Now for the final step! We need to find 2/5 of 67 1/2. To make this easier, let's first convert the mixed number 67 1/2 into an improper fraction.

67 1/2 = (67 * 2 + 1) / 2 = 135 / 2

Now we multiply 2/5 by 135/2:

(2/5) * (135/2) = (2 * 135) / (5 * 2) = 270 / 10

Simplify the fraction:

270 / 10 = 27

So, 2/5 of 67 1/2 is 27. We've reached the finish line! This final step brings together all the concepts we've discussed so far: understanding "of" as multiplication, working with fractions, and converting between mixed numbers and improper fractions. It's a culmination of our efforts, and the satisfaction of reaching the final answer is well-deserved. Converting the mixed number to an improper fraction is a crucial step in this process. It allows us to multiply the fractions more easily. Remember, multiplying fractions involves multiplying the numerators together and the denominators together. So, by converting 67 1/2 to 135/2, we can directly multiply it by 2/5. The multiplication process itself is straightforward. We multiply the numerators (2 and 135) to get 270, and we multiply the denominators (5 and 2) to get 10. This gives us the improper fraction 270/10. The final step is to simplify the fraction. We can do this by dividing both the numerator and the denominator by their greatest common divisor, which is 10. This gives us 270/10 = 27. So, we've successfully found that 2/5 of 67 1/2 is 27. This result represents the solution to our original problem: 2/5 of 3/4 of half of 180. We've broken down a seemingly complex problem into a series of manageable steps, and we've conquered it! This demonstrates the power of a systematic approach to problem-solving. By breaking down a large problem into smaller steps, we can tackle it with greater confidence and accuracy. So, the next time you encounter a challenging mathematical problem, remember this step-by-step approach, and you'll be well-equipped to find the solution.

Final Answer

Therefore, 2/5 of 3/4 of half of 180 is 27. You did it! You successfully navigated through the fractions and found the solution. Give yourself a pat on the back! This whole process highlights the importance of breaking down complex problems into smaller, more manageable steps. By tackling each step individually, the entire problem becomes less daunting and easier to solve. It's like eating an elephant one bite at a time – a seemingly impossible task becomes achievable when approached strategically. Moreover, this step-by-step approach not only simplifies the problem-solving process but also reduces the chances of making errors. When we try to solve a complex problem all at once, it's easy to get lost in the details and make mistakes. However, by focusing on one step at a time, we can maintain clarity and ensure that each calculation is performed correctly. This is particularly important in mathematics, where even a small error can lead to a wrong answer. So, the next time you encounter a challenging problem, remember this valuable lesson: break it down into smaller steps, and you'll be well on your way to finding the solution. And don't forget to celebrate your successes along the way! Each step you complete is a victory, and acknowledging these victories can boost your confidence and motivation. So, congratulations on solving this problem! You've demonstrated your ability to work with fractions, understand mathematical operations, and apply a systematic approach to problem-solving. These are valuable skills that will serve you well in many areas of life. Keep practicing, keep learning, and keep challenging yourself. The world of mathematics is full of exciting challenges, and you're well-equipped to explore them!

Practice Makes Perfect

Now that you've mastered this problem, try tackling similar ones! The more you practice, the more comfortable you'll become with fractions and multi-step problems. Remember, math is like a muscle – the more you use it, the stronger it gets! You can find plenty of practice problems online or in textbooks. Look for problems that involve fractions, the word "of," and multiple steps. Start with simpler problems and gradually work your way up to more challenging ones. Don't be afraid to make mistakes – mistakes are a natural part of the learning process. When you make a mistake, take the time to understand why you made it and how you can avoid it in the future. This is how you learn and grow. Another great way to improve your math skills is to work with others. Collaborate with classmates, friends, or family members. Discuss different approaches to solving problems and learn from each other. Teaching someone else is also a fantastic way to solidify your own understanding of the material. When you have to explain a concept to someone else, you're forced to think about it in a deeper and more organized way. And most importantly, don't give up! Math can be challenging at times, but it's also incredibly rewarding. The feeling of finally understanding a difficult concept is a truly satisfying one. So, keep practicing, keep learning, and keep challenging yourself. You've got this!