Solving (-7) × (-3/14) × (-3/4) A Step-by-Step Guide
Hey guys! Let's break down this math problem together: (-7) × (-3/14) × (-3/4). Don't worry, it looks more intimidating than it actually is. We'll go through it step-by-step, so you'll be a pro in no time!
Understanding the Basics
Before we jump into the solution, let's refresh some fundamental concepts. Remember, multiplying negative numbers can be a little tricky, but once you get the hang of it, you'll be golden. The key thing to remember is that a negative times a negative equals a positive, and a negative times a positive equals a negative. This rule is super important for this problem, so keep it in the back of your mind. Additionally, when multiplying fractions, we multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. If there are whole numbers, like -7 in our case, we can treat them as fractions by placing them over 1. So, -7 becomes -7/1. This makes the multiplication process much smoother.
Now, let's talk about simplifying fractions. Simplifying fractions before multiplying can save us a lot of work. It involves finding common factors between the numerators and denominators and canceling them out. For example, if we have a fraction like 6/8, we can see that both 6 and 8 are divisible by 2. Dividing both by 2, we get 3/4, which is the simplified form. In our problem, we'll see how simplifying fractions can make the calculation much easier. We will also use the concept of multiplying three numbers together, we can choose to multiply the first two numbers first, and then multiply the result by the third number. Or, we can multiply the last two numbers first and then multiply the result by the first number. The order doesn't matter because of the associative property of multiplication. This flexibility is super helpful in making the calculations easier and more manageable. The trick is to look for opportunities to simplify and make the numbers easier to work with. That's the secret weapon for tackling these kinds of problems. Understanding these basics will not only help you solve this particular problem but will also build a strong foundation for tackling more complex math problems in the future. So, let's keep these tips in mind as we dive into solving the equation step-by-step. Are you ready? Let’s do this!
Step 1: Rewrite the Expression
The first thing we want to do is rewrite the expression to make it easier to work with. As we discussed earlier, we'll turn the whole number into a fraction. So, we rewrite -7 as -7/1. Now our expression looks like this: (-7/1) × (-3/14) × (-3/4). See? It already looks less intimidating. This simple step sets us up for smoother sailing in the calculations ahead. Rewriting whole numbers as fractions standardizes the format and helps avoid confusion when we start multiplying. It’s a small change, but it makes a big difference in how clearly we can visualize the problem. Think of it as laying the groundwork for a solid solution. By doing this, we ensure that all the numbers are in the same format (fractions), which allows us to apply the multiplication rules consistently. This is a neat little trick that simplifies the process and makes it less prone to errors. Remember, math isn't just about getting the right answer; it's also about setting up the problem in a way that makes sense to you. This rewriting step does just that—it makes sense of the problem and prepares us for the next steps. So, let’s move on to the next step, where the magic of simplification happens! We're building up to the final answer piece by piece, and each step is a victory. Keep up the great work, and remember to enjoy the process of problem-solving. Math can be a fun puzzle, and we're putting the pieces together one by one.
Step 2: Simplify Before Multiplying
This is where the magic happens! Before we start multiplying everything, let’s see if we can simplify any fractions. Look closely, guys. Do you notice anything? Yes! We can simplify -7/1 and -3/14. Both -7 and 14 share a common factor: 7. So, we can divide -7 by 7 to get -1, and we can divide 14 by 7 to get 2. This simplifies our expression to: (-1/1) × (-3/2) × (-3/4). See how much cleaner that looks? Simplifying before multiplying makes our lives so much easier because we're dealing with smaller numbers. It’s like decluttering your desk before you start a big project – it helps you focus and reduces the chances of making mistakes. The beauty of simplifying is that it doesn't change the value of the expression; it just makes it easier to handle. We’re essentially reducing the fractions to their simplest forms before performing the multiplication, which means less work and less room for error. This step is a game-changer, especially when you’re working with larger numbers. It's a smart strategy that can save you time and headaches. So always remember, before you multiply, take a moment to see if you can simplify. It’s a math superpower that you’ll use again and again. Now that we’ve simplified, the multiplication part will be a breeze. We've done the hard work upfront, and now it’s time to reap the rewards of our efforts. Let’s move on to the next step and multiply these simplified fractions together. We’re getting closer to the solution, and you're doing an amazing job following along!
Step 3: Multiply the Fractions
Alright, now for the fun part – multiplying! We have (-1/1) × (-3/2) × (-3/4). Remember, when multiplying fractions, we multiply the numerators together and the denominators together. So, let's multiply the numerators: -1 × -3 × -3. A negative times a negative is a positive, so -1 × -3 = 3. Then, 3 × -3 = -9. So, our numerator is -9. Now, let's multiply the denominators: 1 × 2 × 4 = 8. So, our denominator is 8. Putting it all together, we get -9/8. We’re almost there! Multiplying fractions might seem like a simple process, but it’s crucial to pay attention to the signs and keep track of the numerators and denominators. By breaking it down step-by-step, we ensure that we don’t miss anything and that we get the correct result. Think of it as building a house – each step is important, and if one step is off, the whole structure might be unstable. In this case, each multiplication contributes to the final answer, so we want to make sure we’re doing it right. And we are! We’ve successfully multiplied the fractions and arrived at -9/8. But, we’re not quite done yet. There’s one more step to make our answer look its best. We need to check if we can simplify our final fraction or convert it to a mixed number. This final touch is what separates a good answer from a great answer. It shows that we’re not just about getting the numbers right, but also about presenting our solution in the clearest and most understandable way. So, let’s head to the final step and put the finishing touches on our masterpiece!
Step 4: Simplify the Result (If Possible)
Okay, we've got -9/8. Can we simplify this fraction further? Nope, 9 and 8 don't share any common factors other than 1. But, we can convert this improper fraction (where the numerator is larger than the denominator) into a mixed number. To do this, we divide 9 by 8. 8 goes into 9 one time with a remainder of 1. So, -9/8 is equal to -1 1/8. And that's our final answer! Woohoo! Great job, guys! Simplifying the result is the final polish that makes our answer shine. It's like adding the perfect accessory to an outfit – it completes the look and makes everything come together. Converting an improper fraction to a mixed number makes the answer more understandable and relatable. It gives us a clearer sense of the value of the fraction. In our case, -9/8 is a perfectly correct answer, but -1 1/8 gives us a better picture of where this number falls on the number line. It tells us that it’s a little bit more than -1. This is why simplifying and converting are such important steps in problem-solving. They’re not just about following the rules; they’re about making the answer as clear and useful as possible. So, always take that extra step to simplify and convert, if possible. It’s a hallmark of a thoughtful and thorough mathematician. And with that, we’ve reached the end of our journey. We’ve successfully solved the problem and arrived at the final answer: -1 1/8. Give yourselves a pat on the back – you’ve earned it!
Conclusion
So, there you have it! We've successfully solved (-7) × (-3/14) × (-3/4). Remember, the key is to break down the problem into smaller, manageable steps. First, we rewrote the expression, then we simplified before multiplying, then we multiplied, and finally, we simplified the result. By following these steps, you can tackle similar problems with confidence. Math might seem daunting at times, but with a little practice and a step-by-step approach, you can conquer any challenge. And remember, it’s not just about getting the right answer; it’s about understanding the process and building your problem-solving skills. These skills will serve you well in all areas of life, not just in math class. So keep practicing, keep exploring, and keep having fun with math. You’ve got this! Remember, each math problem is a puzzle waiting to be solved, and every time you solve one, you become a little bit smarter and a little bit more confident. So, keep challenging yourself, keep learning, and never be afraid to ask for help when you need it. The world of math is vast and fascinating, and there’s always something new to discover. So, embrace the journey and enjoy the ride. And until next time, happy calculating!
