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# Adding fractions with unlike denominators

CCSS.Math:

## Video transcript

let's say that we have the fraction nine-tenths and I want to add to that the fraction 1/6 1/6 what is this what is this going to equal so when you first look at this you say oh I have different denominators here it's not obvious how I add these and you'd be right and and the way to actually move forward is to find a common denominator to convert both of these fractions into fractions that have a common denominator so how do you think about a common denominator well common denominator is going to be a common multiple of these two denominators of 10 and 6 so what's a common multiple of 10 and 6 and it's usually simplest to find the least common multiple and a good way of doing that is start with the larger denominator here 10 and say okay is 10 divisible by 6 no okay no is 20 divisible by 6 no is 30 divisible by 6 yes 30 is divisible by 6 so I'm just going to the multiples of 10 and saying well what is the smallest multiple of 10 that is divisible by 6 and that's going to be 30 so I can rewrite both of these fractions as something over 30 so 9 over 10 how would I write that as something over 30 well i multiplied the denominator I'm multiplying the denominator by 3 so I've just multiplied the denominator by 3 so if I don't want to change the value of the fraction I have to do the same thing to the numerator I have to multiply that by 3 as well because now I'm just multiplying the numerator by 3 and the denominator by 3 and that doesn't change the value of the fraction so 9 times 3 is 27 so once again 9 tenths and 27 thirtieths represent the same number I've just written it now with the denominator of 30 and that's useful because I can also write 1/6 with the denominator of 30 let's do that so 1/6 is what over 30 I encourage you to pause the video and try to think about it so what did we do to go from 6 to 30 we had to multiply by 5 so if we multiply the denominator by 5 we have to multiply the numerator by 5 as well so 1 times 5 1 times 5 is 5 so 9 tenths is the same thing as 27 thirtieths and one six is the same thing as five xxx and now we can add now we can add and it's fairly straightforward we have a certain number of thirtieths added to another number of thirtieths so 27:30 it's plus five thirty it's well that's going to be twenty seven that's going to be twenty seven plus five plus five plus five thirtieths plus five xxx which of course is going to be equal to what is this thirty-two xxx thirty-two over thirty and if we want we could try to reduce this fraction we have a common factor of 32 and 30 they're both the divisible by let's see they're both divisible by two so if we divide the numerator and the denominator by two numerator divided by two is sixteen denominator divided by two is 15 so this is the same thing as 16 15 and if I want to write this as a mixed number 15 goes into 16 one time with the remainder one so this is the same thing as one and one 15 let's do another example let's say that we wanted to add we wanted to add one half to two eleven twelfths to 11 over 12 and I encourage you to pause the video and see if you can work this out well like we saw before we want to find a common denominator if these had the same denominator we could just add them immediately but we want to find a common denominator because right now they're not the same well what we want to find is a multiple a common multiple of two and twelve and ideally we'll find the lowest common multiple of two and twelve and just like we did before let's start with the larger of the two numbers 12 and we could just say well this 12 times 1 is 12 so that you could view that as the lowest multiple of 12 and is that divisible by 2 yeah sure 12 is divisible by 2 so 12 is actually the least common multiple of 2 and 12 so we could write both of these fractions as something over 12 so 1/2 is what over 12 well to go from 2 to 12 you multiply by 6 so we'll also multiply the numerator by 6 and we see one half and six twelves these are the same thing one is one is half of two six is half of twelve and how would we write eleven twelve as something over 12 well it's already written as something over twelve eleven twelve already has 12 in the denominator so we don't have to change that eleven twelve and now we're ready to add so this is going to be equal to six this is going to be equal to six plus 11 six plus 11 over 12 over 12 we have six twelve plus eleven twelve is going to be six plus 11 over 12 which is equal to six plus 11 is 17 12 if we want to write it as a mixed number that is what 12 goes into 17 one time with a remainder 5 so 1 + 5 over 12 let's do one more of these this is this is strangely fun all right let's say that we wanted to add we're going to add 3/4 - we're going to add 3/4 - 1 v 2 1 over 5 what is this going to be and once again pause the video and see if you can work it out well we have different denominators here and we want to find this we want to rewrite these so they have the same denominator so we have to find a common multiple ideally the least common multiple so what's the least common multiple of 4 and 5 we'll start with the larger number and let's look at its multiples and keep increasing them until we get one that's divisible by 4 so 5 is not divisible by 4 10 is not divisible by 4 or perfectly divisible by 4 is what we care about 15 is not perfectly divisible by 4 20 is divisible by 4 in fact that is 5 times 4 that is 20 so what we could do is we could write both of these fractions as having 20 in the denominator or 20 as a denominator so we could write 3/4 is something over 20 so to go from 4 to 20 in the denominator we multiply it by 5 so we also do that to the numerator we multiply 3 times 5 to get 15 all I did to go from 4 to 20 multiplied by 5 so to do the same thing to the numerator 3 times 5 is 15 3/4 is the same thing as 15 xx and over here 1/5 what is that over 20 well to go from five to 20 you have to multiply by 4 so to do the same thing in the numerator I have to multiply this numerator times 4 to get 420 it's so now I've rewritten this instead of 3/4 plus 1/5 it's now written as 1520 it's + 4 xx and what is that going to be well that's going to be 15 plus 4 is 1920 it's 1920 it's and we're done