Evaluating The Inequality [1/3 + 1/7] ≠ [1/3 × 1/7] A Step-by-Step Guide
Introduction
Hey guys! Today, we are diving into a fun little mathematical exploration. We're going to evaluate and discuss the inequality $\left[\frac{1}{3}+\frac{1}{7}\right] \neq\left[\frac{1}{3} \times \frac{1}{7}\right]$. This problem involves basic arithmetic operations with fractions and a comparison to check if the two sides are indeed not equal. So, grab your thinking caps, and let's get started! We'll break this down step by step to make sure everyone's on the same page. Remember, math can be enjoyable when you take it one piece at a time!
Step-by-Step Evaluation
Let's begin by evaluating the left-hand side (LHS) of the inequality, which is the sum of two fractions: $\frac1}{3}+\frac{1}{7}$. To add these fractions, we need to find a common denominator. The least common multiple (LCM) of 3 and 7 is 21. So, we will convert both fractions to have this denominator. The process involves multiplying the numerator and the denominator of each fraction by the factor that turns the original denominator into the common denominator. For $\frac{1}{3}$, we multiply both the numerator and the denominator by 7, which gives us $\frac{1 \times 7}{3 \times 7} = \frac{7}{21}$. Similarly, for $\frac{1}{7}$, we multiply both the numerator and the denominator by 3, resulting in $\frac{1 \times 3}{7 \times 3} = \frac{3}{21}$. Now that both fractions have the same denominator, we can add them together{21} + \frac{3}{21} = \frac{7+3}{21} = \frac{10}{21}$. So, the left-hand side simplifies to $\frac{10}{21}$. This fraction is less than 1, but greater than 0, which is an important observation for our final comparison. We’ve successfully navigated the addition of fractions, which is a crucial skill in many mathematical contexts. Great job, everyone!
Now, let's move on to the right-hand side (RHS) of the inequality. This involves the multiplication of two fractions: $\frac{1}{3} \times \frac{1}{7}$. Multiplying fractions is actually quite straightforward. You simply multiply the numerators together and the denominators together. So, in this case, we have $\frac{1 \times 1}{3 \times 7} = \frac{1}{21}$. This means the right-hand side simplifies to $\frac{1}{21}$. Again, this fraction is also less than 1 but clearly smaller than the result we obtained on the left-hand side. Understanding how to multiply fractions is fundamental, and you've aced it! Keep up the excellent work as we move towards the comparison step. We're making progress, and the solution is becoming clearer!
Finally, we need to compare the results we obtained for both sides of the inequality. We found that the left-hand side (LHS), $\frac1}{3}+\frac{1}{7}$, simplifies to $\frac{10}{21}$, and the right-hand side (RHS), $\frac{1}{3} \times \frac{1}{7}$, simplifies to $\frac{1}{21}$. Now, we ask ourselves{21}$ not equal to $\frac{1}{21}$? To answer this, we simply compare the two fractions. It's quite clear that $\frac{10}{21}$ is much larger than $\frac{1}{21}$. The numerators are different while the denominators are the same, making the comparison straightforward. Since 10 is not equal to 1, we can definitively say that $\frac{10}{21}$ is not equal to $\frac{1}{21}$. Therefore, the original inequality $\left[\frac{1}{3}+\frac{1}{7}\right] \neq\left[\frac{1}{3} \times \frac{1}{7}\right]$ holds true. You've successfully compared the two sides and confirmed the inequality. Fantastic job in working through this problem!
Detailed Discussion
In this section, let's dive a little deeper into the concepts we've used and why this inequality holds true. We started with the evaluation of fractions. Fractions are a fundamental part of mathematics, representing a portion of a whole. Understanding how to add, subtract, multiply, and divide fractions is crucial for various mathematical operations and real-world applications. When we added $\frac{1}{3}$ and $\frac{1}{7}$, we needed to find a common denominator. This step is essential because you can only add or subtract fractions directly if they have the same denominator. The least common multiple (LCM) of the denominators becomes the common denominator, ensuring that we are adding comparable parts. In our case, the LCM of 3 and 7 was 21, allowing us to add the fractions accurately. Mastering fraction manipulation is a cornerstone of arithmetic, guys! Think of it like ensuring all the slices of a pie are the same size before you start counting how many you have. This skill not only helps in simple arithmetic but also in more complex algebraic expressions and equations.
Then, we moved on to multiplication of fractions, which is a bit more straightforward than addition. To multiply fractions, you simply multiply the numerators together to get the new numerator and multiply the denominators together to get the new denominator. No need for a common denominator here! This simplicity makes fraction multiplication a very useful operation in many calculations. In our problem, multiplying $\frac{1}{3}$ by $\frac{1}{7}$ gave us $\frac{1}{21}$, which is significantly smaller than the sum we calculated earlier. The difference in results highlights an important point: adding fractions can often lead to a larger result than multiplying them, especially when the fractions are less than 1. This is because multiplication, in this context, is akin to taking a fraction of a fraction, which will naturally be smaller than either of the original fractions. This concept is crucial not just in math but also in understanding proportions and scaling in various scientific and practical applications. Remember, multiplying fractions is like slicing a slice of pie – you end up with a smaller piece!
Finally, we compared the two results to confirm the inequality. Comparison is a vital mathematical skill. It allows us to understand the relative sizes of different quantities, which is essential in problem-solving and decision-making. In our case, we compared $\frac{10}{21}$ and $\frac{1}{21}$. Since both fractions have the same denominator, the comparison boils down to comparing the numerators. 10 is clearly greater than 1, so $\frac{10}{21}$ is greater than $\frac{1}{21}$. This confirmed that the sum of the fractions is indeed not equal to the product of the fractions, validating the original inequality. The ability to compare quantities accurately forms the basis for many mathematical proofs and logical arguments. It's like being able to tell which piece of cake is bigger – a crucial life skill, right? This entire process showcases the importance of understanding not just the mechanics of mathematical operations but also the underlying principles that govern them. By mastering these foundational skills, you're setting yourself up for success in more advanced mathematical topics and real-world problem-solving.
Real-World Applications
Understanding fractions and inequalities isn't just about acing math tests; these concepts have numerous real-world applications that you encounter every day, sometimes without even realizing it! For example, think about cooking. Recipes often call for fractional amounts of ingredients – half a cup of flour, a quarter teaspoon of salt, and so on. Being able to add and multiply these fractions accurately is essential for following the recipe correctly and getting the delicious results you expect. If you mess up the fractions, you might end up with a cake that's too dry or a sauce that's too salty. So, mastering fractions is the key to becoming a star chef in your own kitchen! This practical application highlights how mathematical precision translates directly into real-world success. Isn't it cool how fractions help you bake a perfect cake?
Another area where fractions and inequalities come into play is in financial planning. When you're budgeting your money, you often deal with fractions of your income – saving a tenth, spending a third on rent, and so forth. Understanding how these fractions add up and comparing different financial options involves the same mathematical principles we've discussed. For instance, if you're trying to decide between two different investment opportunities, you might need to compare the fractional returns they offer to see which one is better. Similarly, when calculating discounts or sales taxes, you're essentially working with fractions. So, a solid grasp of fractions is crucial for making smart financial decisions and managing your money wisely. It’s like having a superpower for your wallet! This connection between math and money underscores the practical value of these seemingly abstract concepts.
Furthermore, fractions and inequalities are vital in various scientific and engineering fields. In physics, for example, you might need to calculate fractional changes in velocity or acceleration. In chemistry, you might work with fractional concentrations of solutions. Engineers use fractions and inequalities to design structures, calculate stresses and strains, and ensure safety and efficiency. For instance, an architect needs to ensure that the load-bearing capacity of a building is significantly greater than the expected load – an inequality in action! These applications demonstrate that the ability to work with fractions and inequalities is a foundational skill for anyone pursuing a career in STEM fields. It's the mathematical backbone behind many of the technological advancements we rely on every day. From designing bridges to formulating medications, fractions play a critical role in shaping our world. So, as you can see, the math we've discussed today isn't just an abstract exercise; it's a powerful tool that can help you in countless real-world situations!
Conclusion
In conclusion, we've successfully evaluated the inequality $\left[\frac{1}{3}+\frac{1}{7}\right] \neq\left[\frac{1}{3} \times \frac{1}{7}\right]$ and confirmed that it indeed holds true. We began by adding the fractions on the left-hand side, finding a common denominator, and simplifying the result to $\frac{10}{21}$. Then, we multiplied the fractions on the right-hand side, which gave us $\frac{1}{21}$. Finally, we compared the two results and clearly saw that $\frac{10}{21}$ is not equal to $\frac{1}{21}$, thus validating the inequality. Through this exercise, we've reinforced our understanding of fraction addition, multiplication, and comparison – fundamental skills in mathematics. Great job working through this problem with me, guys! You've not only solved a specific mathematical problem but also strengthened your overall mathematical toolkit. Keep practicing and exploring, and you'll continue to build your mathematical confidence and abilities.
We also delved into a detailed discussion of why this inequality holds, emphasizing the importance of understanding the principles behind the operations. We explored the significance of finding common denominators when adding fractions and the relative simplicity of multiplying fractions. Additionally, we highlighted the crucial skill of comparing quantities to draw conclusions and validate mathematical statements. This deeper understanding will help you apply these concepts in various contexts and tackle more complex problems in the future. Remember, it's not just about getting the right answer; it's about understanding why the answer is correct. This conceptual grasp is what truly empowers you mathematically.
Moreover, we explored several real-world applications of fractions and inequalities, from cooking and financial planning to science and engineering. These examples demonstrate the practical relevance of the mathematical concepts we've discussed and show how they're used in everyday life and various professional fields. Whether you're measuring ingredients for a recipe, managing your budget, or designing a bridge, fractions and inequalities are essential tools. This connection between abstract math and real-world applications makes the learning process more meaningful and engaging. It also highlights the value of mathematical literacy in navigating and understanding the world around us. So, keep those fractions and inequalities in mind – they're more useful than you might think! With that, we wrap up our exploration of this intriguing mathematical problem. I hope you've enjoyed this journey and gained a deeper appreciation for the beauty and utility of mathematics. Keep up the fantastic work, and I look forward to our next mathematical adventure together!
