Welcome to Math with Mr. J. 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 a little closer in value and will take some additional thought and work. You'll see what I 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 a 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 1, where we have 3 and 99 hundredths is greater than, less than, or equal to 6 and 1 eighth.
Now here we have a decimal and a 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 eighth is greater.
So reading this comparison from left to right, 3 and 99 hundredths is less than 6 and 1 eighth. Next, for number 2, we have negative 51 is greater than, less than, or equal to negative 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. Negative 44 is greater than negative 51. Think about a number line.
Negative 44 is further right, closer to zero. Negative 44 is greater than negative 51. So reading the comparison from left to right, negative 51 is less than negative 44. Moving on to number three, we have two is greater than, less than, or equal to negative seven and three-fourths. So here we have a positive and a negative.
A positive is always going to be greater than a negative. So two is greater than negative seven and three-fourths. Again, a positive is always going to be greater than a negative, so there isn't much to this one. Next, let's move on to number four, where we have nine and a half is greater than, less than, or equal to nine and five-tenths. For this one, we have the same whole number, nine.
So we need to take a look at the fractional part and the decimal part. Now, one half and 0.5, five-tenths. are equal. They both equal one half.
So these are actually equal. Nine and a half is equal to nine and five tenths. So by recognizing that one half is equal to 0.5, five tenths, we can see that these are equal. But let's say that we did not recognize that one half and five tenths 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 a fraction. So 9 and 5 tenths looks like this as a fraction. So 9 and 5 over 10. 9 and 5 tenths.
Now 5 tenths, 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. denominator and we get nine and well five divided by five is one and then ten divided by five is two so we get nine and a half so we have nine and a half and nine and a half so these are equal so again for number four we can either recognize that one half and 0.5 are equal and therefore make this comparison nine and a half is equal to nine and five tenths 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 five through eight.
So here are numbers 5 through 8. Let's start with 5 where we have 8 and 5 eighths is greater than, less than, or equal to 8 and 7 twelfths. 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 a common denominator. So a common denominator 4, 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 eighths.
So how do we get the denominator 8 to equal 24? Well, 8 times 3 equals 24. Whatever we do to the denominator, we must do to the numerator. in order to keep this equivalent. 5 times 3 gives us a numerator of 15. So 8 and 15 24ths is equivalent to 8 and 5 eighths.
We just have that denominator of 24 now. Now let's rename 8 and 7 twelfths. How do we get the denominator of 12 to equal 24? Well 12 times 2 is 24. Whatever we do to the denominator we must due to the numerator in order to keep this equivalent. Seven times two gives us a numerator of 14. So eight and 14 24ths is equivalent to eight and 7 12ths.
We just have that denominator of 24 now. Now that we have a common denominator, these fractions are much easier to compare. 15 24ths is greater than 14 24ths.
So that means Our original comparison here, 8 and 5 eighths, is greater than 8 and 7 twelfths. So the main takeaway from this example, when comparing fractions, make sure you have a 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 a 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 five. divided by 8. For this one we do 7 divided by 12. We keep the whole number 8 the same. So it's going to be 8 point and then whatever we get from 5 divided by 8 and 7 divided by 12. So we're converting the fractional parts to decimals. You can do these division problems by hand or with a calculator.
So again it really depends on your situation. Now as far as 8 and 5 eighths we have 8 and then five divided by eight gives us 0.625. So this is 8.625. So eight and 625 thousandths is greater than, less than, or equal to. And then eight and 7 12ths, well, we get 8.7 divided by 12 gives us 583, and that three repeats.
We get 8.58 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 negative one and ninety nine hundredths is greater than less than or equal to negative one and nine tenths. So here we have a fraction and a decimal.
Let's write these as decimals first in order to compare and then take a look at fractions. Negative one and ninety nine hundredths looks like this as a decimal. So negative one. point nine nine. So comparing these two decimals, negative one and nine tenths is going to be greater.
Think of these on a number line. Negative 1 and 9 tenths 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, negative 1.99, so negative 1 and 99 hundredths, goes to the hundredths place. So let's use a placeholder 0 to have negative 1.9, negative 1 and 9 tenths, go to the hundredths place as well.
So negative 1 decimal 9 0. So negative 1 and 90 hundredths. That placeholder 0 does not change the value of our number. So we can do this.
Negative 1 and 90 hundredths is greater. So reading this comparison from left to right, negative 1 and 99 hundredths is less than negative 1 and 9 tenths. Now let's take a look at the fractional comparison here. So we already have a fraction with negative 1 and 99 hundredths is greater than, less than, or equal to negative 1 and 9 tenths.
Now here we have negative 1 and negative 1 so let's compare the fractional part of these mixed numbers and let's rewrite these with a common denominator in order to do so. The least common denominator between 100 and 10 is 100 so negative 1 and 99 hundredths already has a denominator of 100 and then let's rename negative 1 and 9 tenths with that denominator of 100. So how do we get 10 to equal 100? 10 times 10. Whatever we do to the denominator, we must do to the numerator.
9 times 10 is 90. So here, these fractions are much easier to compare. We have negative 1 and 99 hundredths is less than negative 1 and 90 hundredths. So we can fill this in right here as well. Negative 1 and 99 hundredths is less than negative 1 and 9 tenths.
Moving on to number 7, we have negative 2 and 45 hundredths is greater than, less than, or equal to negative 2 and 9 twentieths. So again, a decimal and a fraction. Let's work with fractions first. So we need to change negative 2 and 45 hundredths to a fraction. So it's going to look like this.
Negative 2 and 45 hundredths. Now in order to compare these fractions, we need a common denominator. The least common denominator between 100 and 20 is 100. So let's rename negative 2 and 9 twentieths. with a denominator of 100. Negative 2 and 45 hundredths already has that denominator of 100, so we can leave it.
So as far as negative 2 and 9 20ths, how do we get 20, the denominator, to equal 100? Well, 20 times 5 equals 100. Whatever we do to the denominator, we must do to the numerator in order to keep it equivalent. 9 times 5. is 45. So we get negative 2 and 45 hundredths is equal to negative 2 and 45 hundredths.
So these are equal. Negative 2 and 45 hundredths is equal to negative 2 and 9 twentieths. So let's write these in decimal form as well.
So negative 2 and 45 hundredths is greater than, less than, or equal to And then as far as negative 2 and 9 20ths, well, negative 2 decimal and then 9 divided by 20 gives us 0.45. So that's negative 2 and 45 hundredths as well. So we can see that these decimals are equal.
Lastly, moving on to number 8, we have 16 thirds is greater than, less than, or equal to 5 and. three tenths. So we have a fraction, an improper fraction, and a decimal.
Now the first thing that we're going to do here is convert the improper fraction to a mixed number. So we do that by dividing the numerator 16 by the denominator 3. So 16 divided by 3. How many whole groups of 3 in 16? Well, 5. That gets us to 15. So we have a remainder of 1. That is the numerator of the fractional part.
And then we keep the denominator of 3 the same. So that equals 5 and 1 third as a mixed number. So now let's compare and we will start with decimal form.
So we need to change five and one third to a decimal. Five and one third written as a decimal is five. And then one divided by three gives us three repeating.
So we can put a three with a bar above it to show that. So it looks like 5.333 and that continues on forever. Again, we can put a 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 tenths. But 5.3 repeating is going to be greater.
Let's take a look over here to the side. So if we have 5.3 repeating, this continues on, and then 5 and 3 tenths. 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 tenths. So looking at the original comparison, 16 thirds is greater than 5 and 3 tenths. Let's wrap up by writing these in fractional form as well.
So let's squeeze these in. We have 5 and 1 third is greater than, less than, or equal to, and then 5. and 3 tenths. We need a 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 third. So how do we get 3 to equal 30?
Well, 3 times 10 equals 30. 1 times 10 gives us a numerator. of 10, so 5 and 10 30ths. As far as 5 and 3 tenths, how do we get 10 to equal 30?
Well, 10 times 3. 3 times 3 gives us a numerator of 9. So now, since we have the same whole number of 5, we need to compare the fractional part, and since we have a common denominator here of 30, these are much easier to compare. 10 30ths is greater than 9 30ths. So 5 and 10 30ths is greater than 5 and 9 30ths.
So let's fill this in right here. 5 and 1 3rd is greater than 5 and 3 10ths. So there you have it. There's how to compare rational numbers.
I hope that helped. Thanks so much for watching. Until next time, peace.