Adding and subtracting fractions with unlike denominators
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- [VOICEOVER] Let's see if we can figure out what 4/3 minus 1/5 is, and if you think you know how to do it, I encourage you to pause the video and give it a go. So, when you first look at this, the thing that might jump out at you is we have different denominators here. So, it's not obvious how to subtract 1/5 from 4/3 when you have these different denominators and the key is, is to rewrite each of these fractions so that they have the same denominator. And how do we figure out what that same denominator is? Well, it's going to be a common multiple of three and five, and, ideally, it's going to be the least common multiple of three and five. So, how can we calculate that? Well, we can start with the larger of the two numbers, say five, and let's go through it's multiples and see when we get to one that's divisible perfectly by three. So, five is not divisible by three, 10 is not divisible by three, 15 is divisible by three. In fact, 15 is three times five. So, I can rewrite both of these fractions as something over 15. So, what's 4/3 if I were to rewrite it as something over 15? Well, to get from three to 15 in the denominator, we have to multiply by five. So, if you multiply the denominator by five, if you don't want to change the value of the fraction, you have to multiply the numerator by five as well. So, you have to multiply the numerator by five as well, four times five is going to be 20. So, 4/3 is the same thing as 20/15. Alright. Now, how would we rewrite 1/5 and something over 15? So, we're going to have 15 in the denominator. Well, to go from five to 15, we had to multiply by three. So, if we multiply the denominator by three, we have to multiply the numerator by three as well. So, times three. One times three is just three. So, 4/3 minus 1/5, we can rewrite that as 20/15 minus 3/15. Now, this becomes a lot more straight forward. What is this going to be? Well, this is going to be a certain number of fifteenths. We have 20/15 and we're taking away three of those fifteenths. So, we are going to have, if you have 20 of something and you take away three of them, you're going to have 17 of those things. In this case, we're talking about fifteenths. So, this is going to be 17/15. And if we wanted to write it as a mixed number, we could say 15 goes into 17 one time with a remainder of two. So, it's one and 2/15. Let's do another example. Let's see if we can figure out what 7/10 minus 5/8 is. Five over eight. And I encourage you to pause this video and see if you can calculate it yourself. Well, just like we saw before, we have different denominators but we need to rewrite them so they have a common denominator, that way, we can subtract. And so, what's a common multiple of 10 and eight and, ideally, the least common multiple. It doesn't have to be but it keeps things a little bit simpler. Well, let's start with the larger of the two numbers and then keep finding in their multiples and find one that is perfectly divisible by the other one, by eight. So, ten isn't perfectly divisible by eight, 20 isn't, 30 isn't, 40 is. 40 is a multiple of 10 and it's a multiple of eight. In fact, it's the least common multiple of 10 and eight. So, we can rewrite both of these fractions as something over 40. So, that's going to be something over 40 minus something over 40. Minus something over 40 is equal to something. So, 7/10 is what over 40? Well, to go from 10 to 40 in the denominator, we multiplied by four. So, we have to do the same thing in the numerator, multiply the numerator by four. Seven times four is 28. So, 7/10 is the same thing as 28/40. Now, let's do the same thing with the other fraction. To go from eight to 40 in the denominator, we had to multiply the denominator by five. Eight times five is 40. So, if we multiply the denominator by five, we have to multiply the numerator by five as well. Five times five is 25. So, 7/10 minus 5/8 is the exact same things as 28/40 minus 25/40 and now this makes a lot of sense. It's going to be a certain number of fortieths. If I have 28/40 and I take away 25 of those fortieths, how many fortieths am I going to have left? Well, I'm going to 3/40 left. 28/40 minus 25/40. So, I'm going to have 3/40. 28 minus 25 is three and we are done. 7/10 minus 5/8 is 3/40.