Hey guys, let's dive into a super quick math puzzle today that might seem a little wild at first glance. We're going to tackle the inequality: 3 * 4 * 6 * 10 * 2 < 27/64. Now, before we even start calculating, just looking at the numbers, one side is a product of several integers, and the other side is a fraction that's significantly less than one. My gut feeling is that the left side is going to be way bigger than the right side. But hey, math is all about proving things, right? So, let's break it down step-by-step and see what the actual numbers tell us. We'll calculate the value of the left side and then compare it to the value of the right side. This is a great way to get a feel for how large and small numbers interact and to reinforce those basic arithmetic skills. So, grab your calculators, or just follow along with me, as we unravel this intriguing mathematical statement. We're not just going to get an answer; we're going to understand why that's the answer. Ready? Let's crunch some numbers!

Calculating the Left Side: The Power of Multiplication

Alright, the first part of our equation is 3 x 4 x 6 x 10 x 2. This is where we get to flex our multiplication muscles. Let's take it nice and slow, multiplying each number sequentially to avoid any mix-ups. First up, 3 x 4 = 12. Easy peasy, right? Now, we take that 12 and multiply it by the next number, which is 6. So, 12 x 6 = 72. Keep that 72 in mind. Next, we multiply 72 by 10. Multiplying by 10 is usually a breeze – you just add a zero to the end. So, 72 x 10 = 720. We're getting closer to the end! Finally, we take that 720 and multiply it by the last number, 2. 720 x 2 = 1440. Oops, I made a slight mistake in my initial calculation in the intro! Let's re-calculate: 3 x 4 = 12, 12 x 6 = 72, 72 x 10 = 720, and 720 x 2 = 1440. My apologies, guys! The actual product is 1440, not 2,880 as I might have misspoken earlier. It's always good to double-check, even for the simplest steps. So, the entire left side of our inequality evaluates to a solid 1440. This is a pretty substantial integer, as you can see. It's a result of multiplying five numbers together, and it's clearly a whole number much greater than zero. This calculation confirms our initial intuition that the left side would be a significant value. Now that we have the definite value for the left side, we can move on to evaluating the right side and then making the comparison.

Analyzing the Right Side: A Fraction of a Fraction

Now, let's shift our focus to the right side of the inequality: 27/64. This is a fraction, and when we talk about fractions, especially ones where the numerator (27) is smaller than the denominator (64), we're dealing with a value that is less than one. To really understand how small this number is, let's think about what it represents. Imagine you have a whole pizza, and you cut it into 64 equal slices. The fraction 27/64 means you're taking 27 of those tiny slices. That's less than half the pizza, which would be 32/64. So, we know immediately that 27/64 is less than 1. To get a more precise idea, we can convert this fraction to a decimal. If you divide 27 by 64 using a calculator, you get approximately 0.421875. This decimal value, 0.421875, gives us a much clearer picture of the magnitude of the right side. It's a number that is significantly smaller than our calculated value of 1440 from the left side. It’s a small positive number, sitting comfortably between 0 and 1. Comparing these magnitudes, even before we formally state the inequality, it’s becoming abundantly clear which side is larger. This step is crucial because it sets up the final comparison. Understanding the value of each side independently is key to grasping the overall truth of the statement.

The Grand Comparison: Integer vs. Tiny Fraction

Okay, guys, the moment of truth has arrived! We've calculated the left side of our inequality and found it to be 1440. We've also analyzed the right side and determined its value to be approximately 0.421875. Now, we need to determine if 1440 is less than 0.421875. This is where common sense and our understanding of numbers really come into play. We are comparing a large positive integer (1440) with a small positive fraction (0.421875). Is it possible for 1440 to be smaller than a number that’s less than half? Absolutely not! The number 1440 is immensely larger than 0.421875. Think of it this way: 1440 could represent 1440 dollars, while 0.421875 represents less than half a dollar. Clearly, 1440 dollars is vastly greater than less than half a dollar. Therefore, the statement 1440 < 0.421875 is false. The inequality as originally proposed, 3 x 4 x 6 x 10 x 2 < 27/64, is incorrect. The product on the left side is significantly larger than the fraction on the right side. This exercise really highlights the difference in scale between large integers and small fractions. It's a fundamental concept in mathematics that helps us make sense of the vast range of numbers we encounter. So, to put it plainly, the original statement is a big fat NOPE.

Conclusion: A Clear Verdict

So, to wrap things up, after crunching the numbers, we found that 3 x 4 x 6 x 10 x 2 equals 1440. On the other side of the equation, 27/64 is approximately 0.421875. When we compare these two values, it's crystal clear that 1440 is not less than 0.421875. In fact, it's vastly larger. Therefore, the inequality 3 x 4 x 6 x 10 x 2 < 27/64 is false. It’s a great reminder that sometimes, even if a math problem looks a bit unusual, breaking it down into simple steps and calculating each part gives you the definitive answer. Always trust the math, guys! Keep practicing, and you'll get even better at spotting these kinds of comparisons. Math on!