Comparing Fractions: Which Sign Makes It True?

Let's dive into the world of fractions and inequalities! This article focuses on determining the correct relationship between two given values: 70 rac{2}{11} and rac{141}{2}. We'll explore how to compare these numbers and identify the sign (>, <, or =) that accurately reflects their relationship. Understanding fraction comparison is a fundamental skill in mathematics, useful in various real-world scenarios, from splitting a pizza to calculating proportions. So, let's break down this problem step by step to ensure a clear understanding.

Understanding the Values: Converting and Simplifying

First, we need to understand the values we're comparing. We have a mixed number, 70 rac{2}{11}, and an improper fraction, rac{141}{2}. To make a direct comparison, it's often helpful to convert them to a similar format. Let's start by converting the mixed number into an improper fraction. To do this, multiply the whole number (70) by the denominator (11) and add the numerator (2). This gives us $(70 * 11) + 2 = 770 + 2 = 772$. So, 70 rac{2}{11} is equivalent to rac{772}{11}.

Now we have two improper fractions: rac{772}{11} and rac{141}{2}. We could convert them both to decimals, but let's stick with fractions for now. To compare fractions, a common method is to find a common denominator. However, in this case, it might be more straightforward to convert rac{141}{2} to a decimal. Dividing 141 by 2 gives us 70.5. Alternatively, we could convert rac{772}{11} to a decimal by dividing 772 by 11. This results in approximately 70.18. Therefore, comparing the decimal approximations reveals which number is bigger. Converting both fractions to decimals makes the comparison more clear. It is really easy to see which one is the largest. We can see that 70 rac{2}{11} equals 70.18 (approximately) while rac{141}{2} equals 70.5. So, we know that $70.18 < 70.5$, which means 70 rac{2}{11} < rac{141}{2}.

Comparing fractions, especially when one is a mixed number, requires a few key steps. First, understanding the different types of numbers (mixed numbers, improper fractions) is essential. Second, converting to a common format (like both fractions or both decimals) is crucial for easy comparison. Third, remember the rules of inequalities: less than (<), greater than (>), and equal to (=). Finally, consider that sometimes, approximation is enough to determine the relationship between two numbers, especially when dealing with decimals.

Step-by-Step Comparison: Methods and Approaches

There are multiple ways to approach the comparison. One method involves finding a common denominator for both fractions. To do this, we can multiply the denominators together, which gives us a common denominator of $11 * 2 = 22$. We then convert both fractions to have this common denominator:

For rac{772}{11}, we multiply both the numerator and the denominator by 2: rac{772 * 2}{11 * 2} = rac{1544}{22}.

For rac{141}{2}, we multiply both the numerator and the denominator by 11: rac{141 * 11}{2 * 11} = rac{1551}{22}.

Now that we have both fractions with a common denominator, we can easily compare their numerators. We can see that $1544 < 1551$. Therefore, rac{1544}{22} < rac{1551}{22}. Thus, 70 rac{2}{11} < rac{141}{2}. This process ensures accurate comparison by providing a clear numerical relationship.

Another approach is to convert both fractions to decimals, as previously mentioned. This often simplifies the comparison. When we convert 70 rac{2}{11} to a decimal, we get approximately 70.18. Converting rac{141}{2} to a decimal results in 70.5. Comparing these decimal values, we immediately see that 70.18 < 70.5. This directly indicates that 70 rac{2}{11} < rac{141}{2}.

Choosing the right method depends on the context and the numbers involved. For simple fractions, a common denominator might be quicker. However, for more complex fractions or when a calculator is available, converting to decimals can be more efficient. The key is to find a method that ensures an accurate and straightforward comparison. Always double-check your calculations to avoid any errors, especially when dealing with larger numbers or multiple steps. Understanding the process of fraction comparison is a useful tool. This skill goes beyond the classroom, such as in cooking or construction, where precise measurements are crucial.

The Correct Sign: Determining the Relationship

After comparing both fractions through either the common denominator or decimal conversion methods, we can now definitively determine the correct sign. As we have seen, 70 rac{2}{11} (approximately 70.18) is less than rac{141}{2} (70.5). Therefore, the correct sign to use is the less-than sign (<).

This means that the statement 70 rac{2}{11} < rac{141}{2} is true. The other options, greater than (>) or equal to (=), would not accurately represent the relationship between these two values. The comparison clearly shows that the mixed number 70 rac{2}{11} is smaller than the improper fraction rac{141}{2}. This conclusion is based on accurate calculations and a thorough understanding of fraction comparison techniques. Always remember to consider which method works best when comparing different fractions. In this case, converting to decimals first would be faster.

In summary, the correct sign that makes the statement true for 70 rac{2}{11} and rac{141}{2} is '<'. This underscores the importance of being comfortable with fraction manipulation and comparison in mathematics. It is a fundamental skill that builds a strong base for more advanced mathematical concepts.

Conclusion: Mastering Fraction Comparisons

In conclusion, comparing 70 rac{2}{11} and rac{141}{2} involved converting a mixed number into an improper fraction or converting each fraction into a decimal. Regardless of the method used, the result is the same: 70 rac{2}{11} is less than rac{141}{2}. The ability to accurately compare fractions is essential for success in mathematics and for solving real-world problems. Whether you're working with recipes, construction, or financial planning, a solid understanding of fractions and their relationships will prove invaluable.

Key takeaways:

• Converting mixed numbers and improper fractions to a common format (fractions with a common denominator or decimals) is crucial for comparison.
• Familiarity with the less than (<), greater than (>), and equal to (=) signs is fundamental.
• Practicing fraction comparison with various examples helps build confidence and proficiency.

By mastering fraction comparisons, you'll be well-equipped to tackle more complex mathematical problems and everyday challenges. Understanding these concepts will save you time. Keep practicing. Remember, the more you practice these techniques, the more natural they will become. Happy calculating!

For further information, you can check out this link:

• Khan Academy (a trusted website for math education) - Khan Academy Fractions