Rational Numbers Explained Standard Form And Comparison

Let's kick things off by tackling the first question, which dives into the concept of expressing rational numbers in their standard form. So, what exactly is standard form, you might ask? Well, in the world of rational numbers, the standard form is like the number's most simplified and presentable outfit. It's the form where the denominator is a positive integer, and the numerator and denominator have no common factors other than 1. In simpler terms, we're talking about reducing the fraction to its lowest terms while ensuring the bottom number is positive.

Now, let's get our hands dirty with the given fraction: $\frac{-108}{120}$. Our mission, should we choose to accept it, is to transform this fraction into its standard form. The first thing we need to do, guys, is to find the greatest common divisor (GCD) of the numerator and the denominator. This is the largest number that divides both 108 and 120 without leaving a remainder. There are a couple of ways we can find the GCD. One way is to list out the factors of each number and identify the largest one they have in common. Another way, which is often quicker for larger numbers, is to use the Euclidean algorithm. But for now, let's stick to the factor listing method for clarity.

The factors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108. The factors of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120. Peering at these lists, we can spot that the greatest common factor is 12. So, 12 is our magic number! Now, to reduce the fraction, we divide both the numerator and the denominator by 12. This gives us $\frac{-108 \div 12}{120 \div 12} = \frac{-9}{10}$. Ta-da! We've successfully reduced the fraction. But hold on, we're not quite done yet. We need to make sure the denominator is positive. In this case, it already is, so we're good to go. Therefore, the standard form of $\frac{-108}{120}$ is $\frac{-9}{10}$. See? It's like giving the fraction a makeover, transforming it into its most sleek and simplified self. This skill of expressing rational numbers in standard form is crucial, guys, because it allows us to easily compare and manipulate fractions, making our mathematical lives much easier.

Now, let's switch gears and dive into the second part of our challenge: filling in the blanks with the correct symbols to compare rational numbers. We're talking about the classic greater than (>), less than (<), and equal to (=) symbols. This is like a showdown between fractions, where we need to determine which one is the mightier, or if they're evenly matched. Comparing rational numbers is a fundamental skill in mathematics. It allows us to understand the relative sizes of different fractions and is essential for ordering and performing operations with them. To effectively compare rational numbers, especially when they have different denominators, we often need to find a common denominator. This allows us to directly compare the numerators and determine which fraction represents a larger or smaller value.

(i) \frac{-2}{3} \text{ __ } \frac{8}{-12}

Let's start with the first comparison: $\frac{-2}{3}$ versus $\frac{8}{-12}$. At first glance, these fractions might seem quite different. But don't let them fool you! The key here is to get them onto a level playing field by finding a common denominator. The denominators we're dealing with are 3 and -12. To make things easier, let's first address that negative sign in the second fraction's denominator. We can rewrite $\frac{8}{-12}$ as $\frac{-8}{12}$ by simply moving the negative sign to the numerator. This doesn't change the value of the fraction, but it makes our calculations a bit smoother.

Now, we need to find the least common multiple (LCM) of 3 and 12. The LCM is the smallest number that both 3 and 12 divide into evenly. In this case, the LCM is 12. So, we want to rewrite both fractions with a denominator of 12. The fraction $\frac{-8}{12}$ already has the denominator we want, so we can leave it as is. For the first fraction, $\frac{-2}{3}$, we need to multiply both the numerator and the denominator by the same number to get a denominator of 12. Since 3 times 4 equals 12, we multiply both -2 and 3 by 4. This gives us $\frac{-2 \times 4}{3 \times 4} = \frac{-8}{12}$.

Now, we have $\frac{-8}{12}$ and $\frac{-8}{12}$. It's crystal clear that these two fractions are equal! They're the same fraction dressed in slightly different clothes. So, the correct symbol to fill in the blank is the equals sign (=). This first comparison highlights the importance of finding a common denominator when comparing rational numbers. By transforming the fractions to have the same denominator, we can directly compare their numerators and easily determine their relationship.

(ii) \frac{-2}{7} \text{ __ } \frac{-3}{5}

Moving on to the second comparison, we have $\frac{-2}{7}$ versus $\frac{-3}{5}$. Again, we're faced with fractions that have different denominators, so our trusty strategy of finding a common denominator comes into play. This time, we need to find the LCM of 7 and 5. Since 7 and 5 are both prime numbers (meaning they're only divisible by 1 and themselves), their LCM is simply their product, which is 7 times 5, or 35.

So, our goal is to rewrite both fractions with a denominator of 35. For the first fraction, $\frac{-2}{7}$, we need to multiply both the numerator and the denominator by 5 to get a denominator of 35. This gives us $\frac{-2 \times 5}{7 \times 5} = \frac{-10}{35}$. For the second fraction, $\frac{-3}{5}$, we need to multiply both the numerator and the denominator by 7 to get a denominator of 35. This gives us $\frac{-3 \times 7}{5 \times 7} = \frac{-21}{35}$.

Now we're comparing $\frac{-10}{35}$ and $\frac{-21}{35}$. Both fractions have the same denominator, so we can focus on the numerators. Remember, when we're dealing with negative numbers, the number that's closer to zero is actually the larger number. In this case, -10 is greater than -21. Think of it like temperatures: -10 degrees is warmer than -21 degrees. Therefore, $\frac{-10}{35}$ is greater than $\frac{-21}{35}$, which means $\frac{-2}{7}$ is greater than $\frac{-3}{5}$. So, the correct symbol to fill in the blank is the greater than sign (>). This comparison reinforces the concept that understanding the relative sizes of negative numbers is crucial when working with rational numbers.

(iii) \frac{-8}{9} \text{ __ } \frac{-7}{10}

Let's tackle the third and final comparison in this section: $\frac{-8}{9}$ versus $\frac{-7}{10}$. You guessed it – we're back to finding a common denominator! This time, we need the LCM of 9 and 10. The LCM of 9 and 10 is 90 (since they don't share any common factors other than 1, we just multiply them together). So, we're aiming to rewrite both fractions with a denominator of 90.

For the fraction $\frac{-8}{9}$, we need to multiply both the numerator and the denominator by 10 to get a denominator of 90. This gives us $\frac{-8 \times 10}{9 \times 10} = \frac{-80}{90}$. For the fraction $\frac{-7}{10}$, we need to multiply both the numerator and the denominator by 9 to get a denominator of 90. This gives us $\frac{-7 \times 9}{10 \times 9} = \frac{-63}{90}$.

Now we're comparing $\frac{-80}{90}$ and $\frac{-63}{90}$. Again, we have the same denominator, so we can focus on the numerators. Remembering our rule about negative numbers, -63 is greater than -80 (it's closer to zero). Therefore, $\frac{-63}{90}$ is greater than $\frac{-80}{90}$, which means $\frac{-7}{10}$ is greater than $\frac{-8}{9}$. So, the correct symbol to fill in the blank is the greater than sign (>). This final comparison solidifies our understanding of how to compare rational numbers with different denominators. By finding a common denominator, we can transform the fractions into a form where their relative sizes become clear, allowing us to confidently use the greater than, less than, and equal to symbols.

Finally, the discussion category is clearly mathematics. This encompasses all the concepts and skills we've been working with, from expressing rational numbers in standard form to comparing fractions using common denominators. Mathematics is a vast and fascinating field, and rational numbers are just one piece of the puzzle. But they're a crucial piece, forming the foundation for more advanced mathematical concepts and applications. So, by mastering these fundamental skills, we're setting ourselves up for success in our mathematical journey!

In conclusion, we've successfully navigated a series of short answer questions focused on rational numbers. We've learned how to express fractions in their simplest form, and we've honed our skills in comparing fractions using common denominators. These are essential tools in the world of mathematics, and with practice, they'll become second nature. So keep practicing, keep exploring, and keep having fun with math, guys!