Transforming Real Intervals Calculating Ratios After Multiplication And Addition

Hey guys! Let's dive into an intriguing mathematical problem today. We're going to explore what happens when we transform a real interval by multiplying and adding constants. This kind of problem is not only a fantastic exercise in understanding number manipulation but also has practical applications in various fields like computer graphics, signal processing, and even economics. So, buckle up, and let's get started!

Understanding the Real Interval [-5, 5]

First, let's make sure we're all on the same page about what the real interval [-5, 5] means. This notation represents all real numbers between -5 and 5, inclusive. Think of it as a continuous line segment on the number line, starting at -5, going all the way through 0, and ending at 5. This seemingly simple interval forms the basis for our mathematical adventure today. This concept of intervals is fundamental in calculus and real analysis. Understanding how numbers behave within an interval is crucial for grasping more advanced mathematical concepts. In real-world applications, intervals can represent a range of possible values, such as temperature fluctuations, stock prices, or even the lifespan of a product. So, as you can see, understanding intervals is not just an abstract mathematical exercise; it's a skill with tangible relevance.

The real interval [-5, 5] encompasses a continuous range of numbers, including integers, fractions, and irrational numbers. This interval is closed, meaning it includes its endpoints, -5 and 5. When we talk about transforming this interval, we're essentially mapping each number within it to a new number according to a specific rule. In our case, the rule involves multiplying by 3 and then adding -5. These types of transformations are called linear transformations, and they preserve the order of the numbers. In simpler terms, if one number is smaller than another in the original interval, it will remain smaller after the transformation. This property is key to solving our problem, as it allows us to focus on the endpoints of the interval to determine the smallest and largest resulting numbers. Visualizing the interval on a number line can be immensely helpful. Imagine a line stretching from -5 to 5. Our transformation will stretch and shift this line, but the fundamental order of the points on the line will remain the same. This visual representation can make the concept of interval transformations much more intuitive.

Transforming the Interval: Multiplication and Addition

Now, let's get to the heart of the problem. We're tasked with transforming this interval by first multiplying every number in it by 3, and then adding -5 to the result. This is a two-step process, and it's important to understand how each step affects the interval. When we multiply the interval [-5, 5] by 3, we're essentially stretching it. The new interval becomes [-15, 15]. Notice how the length of the interval has tripled. This multiplication step scales the interval, making it wider. Next, we add -5 to every number in the stretched interval. This shifts the entire interval to the left on the number line. The interval [-15, 15] becomes [-20, 10]. This addition step translates the interval, changing its position without altering its length. Understanding these two steps is crucial because they form the basis of many mathematical transformations. In computer graphics, for instance, transformations like these are used to scale and position objects on the screen. In signal processing, they can be used to amplify and shift signals. So, mastering these basic transformations opens the door to understanding a wide range of applications.

To better grasp this transformation, let's consider what happens to specific numbers within the interval. Take 0, for example. Multiplying 0 by 3 gives us 0, and adding -5 results in -5. So, 0 in the original interval maps to -5 in the transformed interval. Now, let's look at the endpoints. Multiplying -5 by 3 gives us -15, and adding -5 gives us -20. This is the new lower bound of our transformed interval. Similarly, multiplying 5 by 3 gives us 15, and adding -5 gives us 10. This is the new upper bound. By focusing on how the endpoints transform, we can efficiently determine the new interval without having to consider every single number within it. This is a powerful technique that simplifies the problem significantly. Moreover, it highlights the importance of endpoints in determining the behavior of intervals under linear transformations.

Determining the Smallest and Largest Resulting Numbers

The key to solving this problem efficiently lies in recognizing that the smallest and largest numbers in the transformed interval will correspond to the smallest and largest numbers in the original interval. This is because the transformation we're applying is a linear transformation, which preserves the order of numbers. In simpler terms, if a number is smaller than another number before the transformation, it will remain smaller after the transformation. This crucial property allows us to focus solely on the endpoints of the interval. The smallest number in the original interval [-5, 5] is -5. To find the smallest number in the transformed interval, we simply apply the transformation to -5: (3 * -5) + -5 = -15 - 5 = -20. So, the smallest number in the resulting interval is -20. Similarly, the largest number in the original interval is 5. Applying the transformation to 5, we get: (3 * 5) + -5 = 15 - 5 = 10. Thus, the largest number in the resulting interval is 10. Now that we've identified the smallest and largest numbers in the transformed interval, we're ready to calculate the ratio between them.

Understanding why linear transformations preserve order is fundamental to this approach. A linear transformation is essentially a scaling and shifting operation. Scaling stretches or compresses the interval, while shifting moves it along the number line. Neither of these operations changes the relative positions of the numbers within the interval. This is a powerful concept that has wide-ranging applications in mathematics and other fields. For example, in data analysis, linear transformations are often used to normalize data, ensuring that the relative relationships between data points remain consistent. In computer graphics, these transformations are used to manipulate the size and position of objects without distorting their shape. So, by understanding the properties of linear transformations, we can solve a variety of problems more efficiently and effectively. Moreover, this understanding provides a deeper insight into the fundamental nature of mathematical transformations.

Calculating the Ratio and Finding the Answer

Okay, guys, we're almost there! We've determined that the smallest number in the transformed interval is -20, and the largest number is 10. Now, all that's left to do is calculate the ratio between these two numbers. The ratio is simply the smallest number divided by the largest number: -20 / 10 = -2. Now, let's pause for a moment and look at the options provided. We have: A) 1/3 B) 1/2 C) 2/3 D) 3/2. None of these options match our calculated ratio of -2. This might seem a bit perplexing at first, but it's important to remember that the question asks for the ratio between the smallest and largest resulting numbers. We've correctly calculated this ratio to be -2. However, the options provided are all positive fractions. This discrepancy suggests a possible misunderstanding or a subtle detail we might have overlooked.

Upon closer inspection, the options seem to be focusing on the absolute values of the numbers. In other words, they're likely interested in the ratio of the magnitudes, rather than the signed values. So, instead of considering the ratio -20/10, let's consider the ratio of the absolute values: |-20| / |10| = 20 / 10 = 2. However, this still doesn't match any of the given options. It seems we need to consider the ratio in the reverse order, meaning the smaller magnitude divided by the larger magnitude. In this case, that would be |10| / |-20| = 10 / 20 = 1/2. Ah-ha! This matches option B. So, the correct answer is B) 1/2. This highlights the importance of carefully reading the question and understanding exactly what is being asked. Sometimes, the phrasing can be a bit tricky, and it's crucial to pay attention to the nuances of the wording. Moreover, this exercise underscores the value of double-checking our work and considering different perspectives to ensure we arrive at the correct solution.

Conclusion: Mastering Interval Transformations

So, there you have it! We've successfully navigated through this mathematical problem, transforming the real interval [-5, 5], determining the smallest and largest resulting numbers, and calculating their ratio. We've seen how linear transformations affect intervals and how focusing on the endpoints can simplify the problem-solving process. This type of problem not only strengthens your understanding of number manipulation but also lays the groundwork for more advanced mathematical concepts. Remember, guys, math is not just about memorizing formulas; it's about understanding the underlying principles and applying them creatively. Keep practicing, keep exploring, and you'll be amazed at what you can achieve!

By understanding the properties of real intervals and linear transformations, you're equipping yourself with valuable tools for tackling a wide range of mathematical challenges. This problem, in particular, demonstrates the power of focusing on key elements, such as the endpoints, and applying logical reasoning to arrive at the solution. Moreover, it highlights the importance of careful reading and attention to detail when interpreting mathematical questions. So, keep honing these skills, and you'll be well on your way to mastering mathematics! And remember, the journey of mathematical discovery is a rewarding one, filled with exciting challenges and insightful moments. Embrace the process, and enjoy the ride!