Comparing Rational Numbers Math with Mr J

Welcome to Math with Mr. In this video, I'm going to cover how to compare rational numbers. Now, we're going to start with four fairly basic and straightforward examples and then move to four more examples that will be little closer in value and will take some additional thought and work. You'll see what mean as we go through these examples. Now, when we compare rational numbers, it's very helpful to write both numbers in the same form. So, either write both numbers in decimal form or both numbers in fractional form. Remember, decimals can be written as fractions and fractions can be written as decimals. For numbers 1 through 4, this won't be necessary. We will be able to compare without doing this. Again, these comparisons are little more straightforward. For numbers 5 through 8, we will need to write both numbers in the same form. Let's jump into our examples starting with number one where we have 3 and 99 hundredths is greater than, less than, or equal to 6 and 1/8. Now, here we have decimal and fraction, but we're able to compare without changing anything. We can just look at the whole numbers. 6 is always greater than 3 regardless of any decimal or fraction that follows. 6 and 1/8 is greater. So, reading this comparison from left to right, 3 and 99 hundredths is less than 6 and 1/8. Next, for number two, we have -51 is greater than, less than, or equal to -44. So, we are working with two negatives here. Now, we need to be careful with negatives. Although 51 is greater than 44 when they are positive, that's not the case for negatives. -44 is greater than -51. Think about number line. -44 is further right, closer to zero. -44 is greater than -51. So, reading the comparison from left to right, -51 is less than -44. Moving on to number three, we have two is greater than, less than, or equal to -7 and 3/4. So, here we have positive and negative. positive is always going to be greater than negative. So, two is greater than -7 and 3/4. Again, positive is always going to be greater than negative. So, there isn't much to this one. Next, let's move on to number four where we have 9 and half is greater than, less than, or equal to 9 and 5/10. For this one, we have the same whole number. So, we need to take look at the fractional part and the decimal part. Now, 1/2 and 0.5, 5/10 are equal. They both equal 1/2. So, these are actually equal. 9 and half is equal to 9 and 5/10. So, by recognizing that 1/2 is equal to 0.5, 5/10, we can see that these are equal. But, let's say that we did not recognize that 1/2 and 5/10 are equal. How would we compare these? Well, we either need to write both of these as decimals or both as fractions. Let's write the decimal as fraction. So, 9 and 5/10 looks like this as fraction. So, 9 and 5 over 10, 9 and 5/10. Now, 5/10, that fractional part, can be simplified. The greatest common factor between 5 and 10 is 5. So, let's divide both of these by 5, the numerator and denominator, and we get 9 and well, 5 / 5 is 1 and then 10 / 5 is 2. So, we get 9 and half. So, we have 9 and half and 9 and half. So, these are equal. So, again, for number four, we can either recognize that 1/2 and 0.5 are equal and therefore make this comparison, 9 and half is equal to 9 and 5/10, or we need to write both numbers in the same form and that will show these are equal as well. Let's move on to numbers 5 through 8. So, here are numbers 5 through 8. Let's start with 5 where we have 8 and 5/8 is greater than, less than, or equal to 8 and 7/12. Now, for this comparison, we have the same whole number. 8 So, we need to compare the fractional part of these mixed numbers. Now, in order to compare these fractions, we need common denominator. So, common denominator for 8 and 12. The least common denominator for 8 and 12 is 24. So, let's rename with that common denominator of 24. And we'll start with 8 and 5/8. So, how do we get the denominator 8 to equal 24? Well, 8 * 3 = 24. Whatever we do to the denominator, we must do to the numerator in order to keep this equivalent. 5 * 3 gives us numerator of 15. So, 8 and 15/24 is equivalent to 8 and 5/8. We just have that denominator of 24 now. Now, let's rename 8 and 7/12. How do we get the denominator of 12 to equal 24? Well, 12 * 2 is 24. Whatever we do to the denominator, we must do to the numerator in order to keep this equivalent. 7 * 2 gives us numerator of 14. So, 8 and 14/24 is equivalent to 8 and 7/12. We just have that denominator of 24 now. Now that we have common denominator, these fractions are much easier to compare. 15/24 is greater than 14/24. So, that means our original comparison here, 8 and 5/8, is greater than 8 and 7/12. So, the main takeaway from this example, when comparing fractions, make sure you have common denominator. That's going to make it much easier to compare. Now, another option for this one is to change both to decimals in order to compare. So, we're going to go through both here. And as far as which way will work best or be the simplest, it really depends on your situation. It all depends on the numbers being compared, if you're working with calculator or not, and personal preference comes into play as well. I'll cover both fractions and decimals for each example. That way you can choose what works best for you. Now, as far as writing these as decimals, we divide the numerator by the denominators. So, for this one, we do 5 / 8. For this one, we do 7 / 12. We keep the whole number 8 the same. So, it's going to be 8. And then whatever we get from 5 / 8 and 7 / 12. So, we're converting the fractional parts to decimals. You can do these division problems by hand or with calculator. So, again, it really depends on your situation. Now, as far as 8 and 5/8, we have 8 and then 5 / 8 gives us 0.625. So, this is 8 .625. So, 8 and 625 thousandths is greater than, less than, or equal to and then 8 and 7/12. Well, we get 8 .7 / 12 gives us 5 8 3 and that 3 repeats. We get 8.5 and that 3 continues on forever, repeats. So, we can put that bar above that 3 to show that. So, 8 and 625 thousandths is greater than 8.58 and then that 3 repeats. We get the same result either way, one way with fractions, one way with decimals. Next, let's move on to number six where we have -1 and 99 hundredths is greater than, less than, or equal to -1 and 9/10. So, here we have fraction and decimal. Let's write these as decimals first in order to compare and then take look at fractions. -1 and 99 hundredths looks like this as decimal. So, -1.99. So, comparing these two decimals, -1 and 9/10 is going to be greater. Think of these on number line. -1 and 9/10 is further to the right and closer to 0. It's greater in value. And something else to keep in mind, when we compare two decimals, something that can be really helpful is to make both decimals go to the same place. For example, -1.99, so -1 and 99/100 goes to the hundredths place. So, let's use placeholder 0 to have -1.9, -1 and 9/10 go to the hundredths place as well. So, -1 decimal 9 0, so -1 and 90/100. That placeholder 0 does not change the value of our number, so we can do this. -1 and 90/100 is greater. So, reading this comparison from left to right, -1 and 99/100 is less than -1 and 9/10. Now, let's take look at the fractional comparison here. So, we already have fraction with -1 and 99/100 is greater than, less than, or equal to -1 and 9 tenths. Now, here we have -1 and -1, so let's compare the fractional part of these mixed numbers. And let's rewrite these with common denominator in order to do so. The least common denominator between 100 and 10 is 100. So, -1 and 99 hundredths already has denominator of 100. And then, let's rename -1 and 9/10 with that denominator of 100. So, how do we get 10 to equal 100? 10 * 10. Whatever we do to the denominator, we must do to the numerator. 9 * 10 is 90. So, here these fractions are much easier to compare. We have -1 and 99/100 is less than -1 and 90/100. So, we can fill this in right here as well. -1 and 99/100 is less than -1 and 9/10. Moving on to number seven, we have -2 and 45/100 is greater than, less than, or equal to -2 and 9/20. So again, decimal and fraction. Let's work with fractions first. So, we need to change -2 and 45/100 to fraction. So, it's going to look like this. -2 and 45 hundredths. Now, in order to compare these fractions, we need common denominator. The least common denominator between 100 and 20 is 100. So, let's rename -2 and 9/20 with denominator of 100. -2 and 45/100 already has that denominator of 100, so we can leave it. So, as far as -2 and 9/20, how do we get 20, the denominator, to equal 100? Well, 20 * 5 = 100. Whatever we do to the denominator, we must do to the numerator in order to keep it equivalent. 9 * 5 is 45. So, we get -2 and 45/100 is equal to -2 and 45/100. So, these are equal. -2 and 45/100 is equal to -2 and 9/20. So, let's write these in decimal form as well. So, -2 and 45/100 is greater than, less than, or equal to, and then as far as -2 and 9/20, well, -2 decimal and then 9 / 20 gives us 0.45. So, that's -2 and 45/100 as well. So, we can see that these decimals are equal. Lastly, moving on to number eight, we have 16/3 is greater than, less than, or equal to 5 and 3/10. So, we have fraction, an improper fraction, and decimal. Now, the first thing that we're going to do here is convert the improper fraction to mixed number. So, we do that by dividing the numerator 16 by the denominator three. So, 16 / 3. How many whole groups of three in 16? Well, five. That gets us to 15, so we have remainder of one. That is the numerator of the fractional part, and then we keep the denominator of three the same. So, that equals 5 and 1/3 as mixed number. So, now let's compare, and we will start with decimal form. So, we need to change 5 and 1/3 to decimal. 5 and 1/3 written as decimal is 5, and then 1 / 3 gives us three repeating, so we can put three with bar above it to show that. So, it looks like 5.333, and that continues on forever. Again, we can put bar above the three to show that that digit repeats. So, this comparison is close. We have 5.3 repeating, and then 5 and 3/10. But, 5.3 repeating is going to be greater. Let's take look over here to the side. So, if we have 5. three repeating, this continues on, and then 5 and 3/10. We can see that this decimal is greater. We have those threes that go on forever. So, this is going to be greater. 5.3 repeating is greater than 5 and 3/10. So, looking at the original comparison, 16/3 is greater than 5 and 3/10. Let's wrap up by writing these in fractional form as well. So, let's squeeze these in. We have 5 and 1/3 is greater than, less than, or equal to and then 5 and 3/10. We need common denominator in order to compare. The least common denominator between 3 and 10 is 30. So, let's rename these with that denominator of 30. Let's start with 5 and 1/3. So, how do we get three to equal 30? Well, 3 * 10 = 30. 1 * 10 gives us numerator of 10. So, 5 and 10/30. As far as 5 and 3/10, how do we get 10 to equal 30? Well, 10 * 3. 3 * 3 gives us numerator of nine. So, now since we have the same whole number of five, we need to compare the fractional part, and since we have common denominator here of 30, these are much easier to compare. 10/30 is greater than 9/30. So, 5 and 10/30 is greater than 5 and 9/30. So, let's fill this in right here. 5 and 1/3 is greater than 5 and 3/10. So, there you have it. There's how to compare rational numbers. hope that helped. Thanks so much for watching. Until next time, peace.