Comparing Rational Numbers Explained (with & without a Number Line) | Math with Mr. J

[Music] welcome to math with Mr J in this video I'm going to cover how to compare rational numbers we will take a look at comparing rational numbers using a number line and then also comparing rational numbers without one let's start by using a number line so we will take a look at some basic comparisons and we will use number lines to help us determine greater than less than or equal to now remember when it comes to number lines if we look at zero positive numbers are to the right of zero and negative numbers are to the left of zero and keep in mind as far as moving along a number line numbers increase in value as we move to the right and decrease in value as we move to the left so when we compare rational numbers using a number line the number further to the right on the number number line will be greater in value let's jump into number one where we have a very simple comparison is 2 greater than less than or equal to 4 well obviously 2 is less than four but I wanted to start very simple because we're going to build off of this and this leads into number two where we have negatives so as far as the number line two is right here and four is right here four is greater than two it's further right on the number line so 2 is less than 4 let's move on to number two where we have -2 is greater than less than or equal to -4 well -2 is right here on the number line and -4 is right here so which number is further to the right and therefore greater in value well -2 -2 is greater than -4 at first glance -4 may seem greater because 4 is greater than two but we have to think we are working with negatives here -2 is further right on the number line it's greater in value so -2 is greater than -4 moving on to number three we have 6 and 34s is greater than less than or equal to 6 and 510 so we have a number in fractional form and a number in decimal form remember fractions can be written as decimals and decimals can be written as fractions so when we compare rational numbers it's always a good idea to have the numbers in the same form so either both in fractional form or both in decimal form this will make it easier to compare for number three let's put 6 and 510 in fractional form 6 and 510 is 6 and a half anytime you have 0.5 510s that's a half and that always works out to be a half because 0.5 510 in fractional form is 5/ 10 which we can simplify the greatest common factor between five and 10 is five so let's divide our numerator and denominator by five in order to simplify 5 / 5 is 1 and 10 / 5 is 2 so 51.5 equal 12 now as far as the number line both of these numbers are between 6 and 7 let's start with 6 and 34s this is 6 and 1/4 6 and 24s which is 6 and a half and then 6 and 34s so 6 and 34s is right here and then we have 6 and 510 which is 6 and a half that's right here and I'll write the original decimal form so we can see that 6 and 34s is further to the right on the number line therefore greater in value so 6 and 34s is greater than 6 and 510 next let's move on to number four we have 3 is greater than less than or equal to 17 fths three is right here on the number line and then 17 fths 17 over 5 that's an improper fraction so we need to convert conver it to a mixed number that will make this much simpler to put on the number line and compare so to convert this to a mixed number let's do it underneath here we need to divide the numerator 17 by the denominator five so how many whole groups of five in 17 well three that's going to be the whole number of our mixed number that gets us to 15 so we have a remainder of of two that's the numerator of the fractional part and then we keep the denominator of five the same so 175s equals 3 and 25s as a mixed number 3 and 2 fths is greater than 3 but less than 3 and a half 3 and 25s is about right here and I'll write the original improper fraction so 3 and 25ths 17 fths is greater than three it's further right on the number line so looking at the comparison 3 is less than 17 fths moving on to number five we have NE 2500s is greater than less than or equal to 9/10 now here we are comparing a negative to a positive so the positive is automatically going to be greater but let's put these on the number line to show that and remember it's helpful to have both numbers in the same form so let's do decimal form for this one 9/10 in fractional form so 9 over 10 is written like this in decimal form as far as the number line - 2500s is going to be to the left of 0 it's negative and in between 0 and - 510 -2 now - 2500s is -4 so it is about right here right in between 0 and - 510 so NE 25 hundredths which again is 1/4 and then 9/10 is to the right of zero and in between 510 and 1 9/10 is pretty close to one it's only 1/10th away so let's put 9/10 right here and I'll write it in the original fractional form so we can see that 9/10 is further right on the number line and therefore greater in value so as far as the comparison -25 hundredths is less than 9/10 lastly let's move on to number six where we have have -8 and 710 is greater than less than or equal to -8 and 85 H hundredths now both of these numbers are in between -8 and -9 as far as the number line 8 and 710 will be well we have 8 and 1/10th 210 310 410 510 which is8 and a half 610 and 710 so8 and 710 is right here and then -8 and 8500s this will be right in between 8 and 8/10 and 8 and 9/10 8 and 8/10 is right here and -8 and 9/10 is right here and if we write a zero in the hundreds place for both of these so 100's place and remember we can write zeros to the right of a decimal that doesn't change the value so these are equivalent and that will help us Place 8 and 85 hundreds because that goes to the hundredths place and now those decimals do as well and we know that 85 is in between 80 and 90 so -8 and 8500s is about right here 8 and 7/10 is further to the right on the number line it's greater in value so -8 and 710 is greater than 8 and 85 hundredths so there's how to use a number line when comparing rational numbers let's move on to comparing without one here are our first four examples of comparing rational numbers without a number line 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 we'll 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 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- 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 one where we have 3 and 99 hundredths is greater than less than or equal to 6 and 1/8 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 six is always greater than three regardless of any decimal or fraction that follows 6 and 1 18 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 a 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 2 is greater than less than or equal to -7 and 34s 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 -7 and 34s 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 9 and 510 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 1 12 and 05 510 are equal they both equal 1/2 so these are actually equal 9 and a half is equal to 9 and 510 so by recognizing that 12 is equal to 0.5 510 we can see that these are equal but let's say that we did not recognize that 1/ 12 and 510 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 510 looks like this as a fraction so 9 and 5 over 10 9 and 510 now 510 that fractional part can be simplified the greatest common factor between five and 10 is five so let's divide both of these by five the numerator and denominator and we get 9 and well 5 / 5 is 1 and then 10 / 5 is 2 so we get N9 and a half so we have 9 and 1/2 and 9 and A2 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 a half is equal to 9 and 510 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 number num 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 12ths 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 5/8 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 a numerator of 15 so 8 and 15 24s is equivalent to 8 and 58 we just have that denominator of 24 now now let's rename 8 and 7 12th 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 a numerator of 14 so 8 and 14 24ths is equivalent to 8 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 24s is greater than 14 24s so that means our original comparison here 8 and 5/8 is greater than 8 and 7 12ths 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 / 8 for this one we do 7 / 12 we keep the whole number 8 the same so it's going to be 8 point 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 a 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 .625 so this is 8.625 so 8 and 625,000 is greater than less than or equal to and then 8 and 7 12ths well we get 8 7 / 12 gives us 5 83 and that three repeats we get 8.58 and that three continues on forever it repeats so we can put that bar above that three to show that so 8 and 625,000 is greater than 8.58 and then that three 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 910 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 -1 and 99 hundredths looks like this as a decimal so - 1.99 so comparing these two decimals -1 and 9/10 is going to be greater think of these on a number line -1 1 and 9/10 is further to the right and closer to zero 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 hundredths goes to the hundred's place so let's use a placeholder zero to have 1.9 -1 and 9/10 go to the hundred's place as well so1 decimal 9 0 so -1 and 90 H hundredths that placeholder zero does not change the value of our number so we can do this -1 and 90 hundredths is greater So reading this comparison from left to right -1 and 9900s is less than -1 and 910 now let's take a look at the fractional comparison here so we already have a fraction with -1 and 99 hundredths is greater than less than or equal to -1 and 910 now here we have -1 and -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 -1 and 99 hundredths already has a 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 H hundredths is less than -1 and 90 H hundredths so we can fill this in right here as well --1 and 99 hundredths is less than -1 and 910 moving on to number seven we have -2 and 4500s is greater than less than or equal to -2 and 9 20ths so again a decimal and a fraction let's work with fractions first so we need to change -2 and 4500s to a fraction so it's going to look like this -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 -2 and 9/ 20ths with a denominator of 100 -2 and 4500s already has that denominator of 100 so we can leave it so as far as -2 and 92th how do we get 20 the denominator to equal 100 well 20 * 5 equal 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 4500 is equal to -2 and 4500s so so these are equal -2 and 4500s is equal to -2 and 9 20ths so let's write these in decimal form as well so -2 and 4500s is greater than less than or equal to and then as far as -2 and 92s well -2 decimal and then 9 / 20 gives us 45 so that's -2 and 4500s 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 310 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 three so 16 divided by three how many whole groups of three in 16 well five that gets us to 15 so we have a 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 a mixed number so now let's compare and we will start with decimal form so we need to change 5 and 1/3 to a decimal 5 and 1/3 written as a decimal is five and then 1 / 3 gives us three repeating so we can put a three with a bar above it to show that so it looks like 5333 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 3/10 but 5.3 repeating is going to be greater let's take a look over here to the side so if we have 53 repeating this continues on and then five 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 310 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 five and 3/10 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 1/3 so how do we get 3 to equal 30 well 3 * 10 = 30 1 * 10 gives us a numerator of 10 so 5 and and 103th as far as 5 and 310 how do we get 10 to equal 30 well 10 * 3 3 * 3 gives us a numerator of 9 so now since we have the same whole number of five we need to compare the fractional part and since we have a common denominator here of 30 these are much easier to compare 1030s is greater than 930s so 5 and 1030s is greater than 5 and 930s so let's fill this in right here 5 and 1/3 is greater than 5 and 310 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