Multiply Fractions: 3/4 By -6/7

Hey guys! Today we're diving into a cool math problem that's all about multiplying fractions, and not just any fractions, but one positive and one negative. So, if you're wondering, what is the product of 3/4 and -6/7? We're going to break it down step-by-step, making sure you get it. This is a fundamental skill in mathematics, and understanding how to multiply fractions, especially with negative numbers involved, will set you up for success in more complex problems down the line. So, grab your notebooks, and let's get this math party started! We'll cover the basic rules of fraction multiplication, how to handle those pesky negative signs, and how to simplify your answer to its lowest terms. By the end of this, you'll be a fraction-multiplying pro, ready to tackle any similar problem that comes your way. We'll also touch on why this concept is important and where you might see it used in the real world, though maybe not directly multiplying 3/4 by -6/7, but the principles are universal. Remember, math is everywhere, and understanding these building blocks makes it all the more accessible and, dare I say, fun!

Understanding Fraction Multiplication

Alright, let's get down to business with what is the product of 3/4 and -6/7? First things first, when you're multiplying fractions, the process is pretty straightforward. You simply multiply the numerators (the top numbers) together to get the new numerator, and then multiply the denominators (the bottom numbers) together to get the new denominator. It's like a direct line multiplication: top times top, bottom times bottom. So, for our problem, we're looking at $\frac{3}{4} \times -\frac{6}{7}$. The numerators are 3 and -6, and the denominators are 4 and 7. The rule for multiplying fractions is: $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$. Easy enough, right? Now, let's not forget about the negative sign. When you multiply a positive number by a negative number, the result is always negative. This is a super important rule in arithmetic that you need to keep in mind. So, our final answer is definitely going to be a negative number. Before we jump straight into multiplying, it's also a good habit to see if you can simplify any of the fractions before you multiply. This is called cross-simplifying. You look diagonally to see if the numerator of one fraction shares any common factors with the denominator of the other. In our case, we have $\frac{3}{4}$ and $-\frac{6}{7}$. Can we simplify diagonally? Let's check. Between 3 and 7? Nope, no common factors other than 1. How about between 4 and -6? Yes! Both 4 and 6 are even numbers, so they share a common factor of 2. We can divide both 4 and 6 by 2. This step isn't strictly necessary to get the correct answer, but it makes the final multiplication much easier and reduces the chance of errors when simplifying later. It’s like taking a shortcut that leads you to the same, correct destination, but with less effort. So, let's keep this simplifying trick in our back pocket as we move forward.

Performing the Multiplication

Now that we've got the rules down, let's actually do the multiplication for what is the product of 3/4 and -6/7? We have our fractions: $\frac{3}{4}$ and $-\frac{6}{7}$. As we discussed, we multiply the numerators and the denominators. So, the new numerator will be $3 \times -6$. And the new denominator will be $4 \times 7$. Let's calculate those: $3 \times -6 = -18$. And $4 \times 7 = 28$. So, our product is $-\frac{18}{28}$. See? It wasn't that bad! We multiplied the top numbers ($3 \times -6 = -18$) and the bottom numbers ($4 \times 7 = 28$), and we remembered that a positive times a negative gives us a negative result. So, we have $-\frac{18}{28}$. Now, here's the crucial part: mathematicians always want answers in their simplest form. This means we need to reduce the fraction $-\frac{18}{28}$ to its lowest terms. To do this, we find the greatest common divisor (GCD) of the numerator (18) and the denominator (28). What's the biggest number that divides evenly into both 18 and 28? Let's list the factors of 18: 1, 2, 3, 6, 9, 18. And the factors of 28: 1, 2, 4, 7, 14, 28. Looking at both lists, the greatest common factor is 2. So, we divide both the numerator and the denominator by 2. $-18 \div 2 = -9$. And $28 \div 2 = 14$. Therefore, our simplified product is $-\frac{9}{14}$. This is our final answer, guys! It’s in its simplest form because 9 and 14 don't share any common factors other than 1. This step of simplification is just as important as the multiplication itself, so don't skip it!

The Importance of Simplification and Negative Numbers

So, we've figured out what is the product of 3/4 and -6/7? and arrived at $-\frac{9}{14}$. Let's take a moment to appreciate why two steps – simplification before multiplication and simplification after multiplication – are so vital, and why handling negative numbers correctly is non-negotiable. The rule of multiplying a positive by a negative yielding a negative is a cornerstone of algebra and arithmetic. Getting this wrong can throw off entire equations. Think of it like this: if you're dealing with temperatures, a positive number might mean above zero, and a negative number below. If you multiply a temperature change by a number of days, you need the sign to be correct to understand if the temperature is rising or falling overall. For our fraction problem, $-\frac{18}{28}$ is technically correct, but it's like presenting a messy solution. Simplifying it to $-\frac{9}{14}$ makes it clear, concise, and easier to compare with other fractions. It's the standard mathematical etiquette. We saw that before we even multiplied, we could have simplified the 4 and the 6 by dividing by 2. Let's quickly revisit that. If we had $\frac{3}{4} \times -\frac{6}{7}$, we can see that 4 and 6 share a factor of 2. So, we could rewrite it as $\frac{3}{2 \times 2} \times -\frac{2 \times 3}{7}$. Then, we cancel out one of the 2s from the denominator of the first fraction with the 2 from the numerator of the second fraction. This leaves us with $\frac{3}{2} \times -\frac{3}{7}$. Now, multiply: $3 \times -3 = -9$ and $2 \times 7 = 14$. This gives us $-\frac{9}{14}$ directly! See how much easier the numbers are? This pre-simplification is a game-changer, especially with larger numbers. It prevents you from getting huge numerators and denominators that are a nightmare to simplify later. So, always look for those opportunities to simplify diagonally before you multiply. And, of course, always double-check your signs. Positive times negative is negative. Negative times negative is positive. Positive times positive is positive. These sign rules are your best friends when working with fractions and all sorts of numbers. Mastering these basics ensures that your mathematical journey is smoother and your results are accurate and elegant.

Conclusion: The Answer to What is the Product of 3/4 and -6/7?

So, after all that breakdown, the answer to the question what is the product of 3/4 and -6/7? is definitively $-\frac{9}{14}$. We learned the fundamental rule of multiplying fractions: multiply the numerators together and the denominators together. We also reinforced the critical rule that a positive number multiplied by a negative number always results in a negative number. This gave us an intermediate answer of $-\frac{18}{28}$. Then, we emphasized the importance of simplifying fractions to their lowest terms. By finding the greatest common divisor of 18 and 28, which is 2, we divided both the numerator and the denominator by 2. This led us to our final, simplified answer: $-\frac{9}{14}$. We also explored the power of simplifying before multiplying, which can make the calculations much more manageable, especially with larger numbers. This strategy, combined with a firm grasp of multiplication rules and sign conventions, is what makes solving these problems efficient and accurate. Remember, guys, whether you're dealing with simple fraction multiplication or more complex algebraic expressions, these core principles are what hold everything together. Keep practicing, keep questioning, and don't be afraid to simplify. It’s these small, consistent efforts that build a strong foundation in mathematics. So next time you see a fraction multiplication problem, especially one involving negatives, you'll know exactly what to do! Keep up the great work, and happy calculating!

The options provided were: A. $-\frac{7}{8}$ B. $-\frac{9}{14}$ C. $\frac{9}{14}$ D. $\frac{7}{8}$

Based on our calculations, the correct answer is B. $-\frac{9}{14}$.