Evaluating Fraction Expressions A Step By Step Guide

Hey guys! Today, we're diving into a fun little math problem that involves fractions. Don't worry, it's not as scary as it sounds! We're going to break it down step-by-step, so even if fractions aren't your favorite thing, you'll be able to follow along and understand exactly what's going on. The expression we're tackling is: [$\\frac{1}{5}+\\frac{1}{7}$]+[$\\frac{1}{3} \\times \\frac{1}{7}$]. Sounds intimidating? Nah, we got this!

Let's Start with Addition: $\frac{1}{5} + \frac{1}{7}$

Our first order of business is to tackle the addition part: $\frac{1}{5} + \frac{1}{7}$. Remember, we can't just add fractions straight away unless they have the same denominator (the bottom number). So, what do we do? We need to find a common denominator. The easiest way to do this is to find the least common multiple (LCM) of 5 and 7. In this case, the LCM of 5 and 7 is 35 (since 5 x 7 = 35). Now, we need to rewrite each fraction with 35 as the denominator. For $\frac{1}{5}$, we multiply both the numerator (top number) and the denominator by 7: $\frac{1}{5} \\times \\frac{7}{7} = \\frac{7}{35}$. For $\frac{1}{7}$, we multiply both the numerator and the denominator by 5: $\frac{1}{7} \\times \\frac{5}{5} = \\frac{5}{35}$. Great! Now we have $\frac{7}{35} + \frac{5}{35}$. Since the denominators are the same, we can simply add the numerators: 7 + 5 = 12. So, $\frac{7}{35} + \frac{5}{35} = \frac{12}{35}$. We've successfully added our first set of fractions! This might seem like a lot of steps, but with practice, it becomes second nature. Remember the key is finding that common denominator! Think of it like trying to add apples and oranges – you need to convert them to a common unit (like 'fruit') before you can add them together. Fractions are the same way; you need a common denominator before you can add (or subtract) them.

Now, let's consider why this works. When we multiply $\frac{1}{5}$ by $\frac{7}{7}$, we're essentially multiplying by 1 (because any number divided by itself is 1). So, we're not changing the value of the fraction; we're just changing the way it looks. The same goes for multiplying $\frac{1}{7}$ by $\frac{5}{5}$. This is a crucial concept in fraction manipulation. Understanding this principle will help you tackle more complex problems later on. Furthermore, recognizing patterns and shortcuts can significantly speed up your calculations. For instance, in this case, you might notice that to find the common denominator, you can simply multiply the original denominators together (5 x 7 = 35). Then, to adjust the numerators, you can cross-multiply: 1 x 7 = 7 and 1 x 5 = 5. This shortcut works well for adding or subtracting two fractions, but it's essential to understand the underlying principle of finding the least common multiple, especially when dealing with more than two fractions. Remember, math is not just about memorizing formulas; it's about understanding the 'why' behind them. This understanding will empower you to solve a wider range of problems and adapt your strategies as needed. So, let’s recap what we’ve done so far: We identified the addition operation as the first step, found the least common multiple of the denominators, converted the fractions to equivalent fractions with the common denominator, and finally, added the numerators. This systematic approach is applicable to many fraction problems, so keep it in mind as we move forward.

Next Up: Multiplication! $\frac{1}{3} \times \frac{1}{7}$

The next part of our expression involves multiplication: $\frac{1}{3} \times \frac{1}{7}$. Luckily, multiplying fractions is much simpler than adding them! To multiply fractions, we simply multiply the numerators together and the denominators together. So, 1 x 1 = 1, and 3 x 7 = 21. Therefore, $\frac{1}{3} \times \frac{1}{7} = \frac{1}{21}$. Easy peasy, right? There's no need to find a common denominator here; just multiply straight across. This is one of the beautiful things about fraction multiplication – it's a straightforward process. However, it's important to remember that this simplicity doesn't extend to addition and subtraction. You always need that common denominator for those operations. Now, let’s delve a bit deeper into the concept of multiplying fractions. When you multiply fractions, you're essentially finding a fraction of a fraction. For example, $\frac{1}{3} \\times \\frac{1}{7}$ means we're finding one-third of one-seventh. Visualizing this can be helpful. Imagine a pie cut into 7 slices, and you have one of those slices ($\frac{1}{7}$ of the pie). Now, you want to take one-third of that slice. You would divide that slice into three equal parts and take one of those parts. This resulting piece is $\frac{1}{21}$ of the whole pie. This visual representation can make the concept of multiplying fractions more intuitive. Furthermore, understanding the relationship between multiplication and division is crucial. Multiplying by a fraction is the same as dividing by its reciprocal. The reciprocal of a fraction is simply flipping the numerator and denominator. For example, the reciprocal of $\frac{1}{7}$ is $\frac{7}{1}$ (which is just 7). So, multiplying by $\frac{1}{7}$ is the same as dividing by 7. This concept can be useful in simplifying more complex expressions. In our current problem, the multiplication step was quite simple, but in other cases, you might encounter larger numbers or mixed numbers. Remember to convert mixed numbers to improper fractions before multiplying. An improper fraction is a fraction where the numerator is greater than or equal to the denominator (e.g., $\frac{5}{2}$). Converting to improper fractions makes the multiplication process easier. Also, look for opportunities to simplify before multiplying. If there are any common factors between the numerators and denominators, you can divide them out to make the numbers smaller and the multiplication simpler. This is called canceling. So, we've successfully multiplied our fractions! We found that $\frac{1}{3} \\times \\frac{1}{7} = \\frac{1}{21}$. This step was relatively straightforward, but it's important to have a solid understanding of the underlying concepts. Remember, multiplying fractions is about finding a fraction of a fraction, and it involves simply multiplying the numerators and denominators. With these principles in mind, you'll be well-equipped to tackle more challenging multiplication problems.

Putting It All Together: $\frac{12}{35} + \frac{1}{21}$

Now comes the final step: adding the results we got from the addition and multiplication parts. We have to add $\frac{12}{35} + \frac{1}{21}$. Guess what? We're back to needing a common denominator! This time, we need to find the LCM of 35 and 21. One way to find the LCM is to list the multiples of each number until we find a common one. Multiples of 35: 35, 70, 105, 140... Multiples of 21: 21, 42, 63, 84, 105... Aha! 105 is the LCM of 35 and 21. Another way to find the LCM is to use prime factorization. This method is particularly helpful when dealing with larger numbers. First, find the prime factorization of each number: 35 = 5 x 7 and 21 = 3 x 7. Then, take the highest power of each prime factor that appears in either factorization: 3, 5, and 7. Multiply these together: 3 x 5 x 7 = 105. This confirms that 105 is indeed the LCM. Now, we need to rewrite our fractions with 105 as the denominator. For $\frac{12}{35}$, we need to multiply both the numerator and the denominator by 3 (since 35 x 3 = 105): $\frac{12}{35} \\times \\frac{3}{3} = \\frac{36}{105}$. For $\frac{1}{21}$, we need to multiply both the numerator and the denominator by 5 (since 21 x 5 = 105): $\frac{1}{21} \\times \\frac{5}{5} = \\frac{5}{105}$. Now we have $\frac{36}{105} + \frac{5}{105}$. Since the denominators are the same, we can add the numerators: 36 + 5 = 41. So, $\frac{36}{105} + \frac{5}{105} = \frac{41}{105}$. And there you have it! We've successfully evaluated the entire expression. The final answer is $\frac{41}{105}$. This fraction cannot be simplified further because 41 is a prime number, and it doesn't share any common factors with 105. This final step highlights the importance of understanding the order of operations (PEMDAS/BODMAS). We first tackled the addition within the first set of brackets, then the multiplication within the second set, and finally, we added the results together. This order is crucial for arriving at the correct answer. Furthermore, throughout this problem, we've emphasized the importance of showing your work. Writing out each step not only helps you keep track of your calculations but also makes it easier to identify and correct any mistakes. It's also a valuable skill for communicating your mathematical reasoning to others. In addition to the arithmetic aspects, it's worth noting the importance of estimation in mathematics. Before diving into the calculations, you could estimate the final answer. For example, $\frac{1}{5}$ is about 0.2, and $\frac{1}{7}$ is about 0.14, so their sum is around 0.34. $\frac{1}{3} \\times \\frac{1}{7}$ is $\frac{1}{21}$, which is about 0.05. So, the final answer should be somewhere around 0.39. Our calculated answer of $\frac{41}{105}$ is approximately 0.39, which confirms that our calculations are likely correct. Estimation is a powerful tool for checking the reasonableness of your answers.

Final Answer and Key Takeaways

So, guys, the final answer to the expression [rac{1}{5}+rac{1}{7}]+[rac{1}{3} imes rac{1}{7}] is $\frac{41}{105}$. We made it! We started with a seemingly complex expression and broke it down into manageable steps. Remember, the key to solving math problems is to break them down into smaller, more manageable parts. We first tackled the addition, then the multiplication, and finally, we added the results together. We also learned (or re-learned!) some important concepts about fractions, such as finding common denominators and multiplying fractions. Don't be afraid to take on math problems; just remember to take it one step at a time, and you'll get there! And remember, practice makes perfect. The more you work with fractions and other mathematical concepts, the more comfortable and confident you'll become. So, keep practicing, keep asking questions, and keep exploring the wonderful world of mathematics! We've covered a lot of ground in this problem, and hopefully, you've gained a deeper understanding of how to work with fractions. From finding common denominators to multiplying and adding fractions, we've explored the fundamental principles and techniques. Remember, mathematics is not just about getting the right answer; it's about understanding the process and developing problem-solving skills. So, embrace the challenges, celebrate your successes, and never stop learning! And most importantly, have fun with it! Math can be like a puzzle – challenging, but incredibly rewarding when you finally solve it. Keep practicing, and you'll become a math whiz in no time! Keep an eye out for more mathematical explorations, and until next time, happy calculating!