Today we’re gonna take a look at how to add and subtract unlike fractions using the least common multiple. Before we jump in to adding and subtracting unlike fractions, let’s quickly do a vocabulary review. There are three things that I wanna review with you; the first one is the least common multiple. The least common multiple sometimes called the LCM of two numbers is the smallest multiple that both of the numbers have in common. Let me show you an example. Here are the multiples of three and the multiples of seven. The ones boxed in red are common multiples of three and seven. But the least common multiple is twenty-one because it’s the smallest of the common multiples. Keep this in mind as it’s an important tool in adding unlike fractions.
Next step, order of operations. Order of operations is the order in which we solve problems. We start with parentheses. Then move to anything with exponents. Next comes multiplication and division, but remember we do this from left to right, so multiplication and division from left to right. And finally, we have addition and subtraction also as we’re moving from left to right. And last but not least, let’s talk quickly about simplifying fractions. When I say simplify fractions, I mean reduce them to their lowest terms. We simplify fractions when there’s a larger number on the top and a smaller number on the bottom, so when the numerator is larger than the denominator. Here’s an example of when we have a numerator that’s bigger than our denominator. And in this video, any time we have an improper fraction, a fraction that has a numerator that’s larger than the denominator, we’re gonna simplify it and turn it into a mixed number. So here to simplify this fraction seven-fifths, I turned it into the mixed number one and two-fifths.
Okay! Enough review. Let’s start adding and subtracting unlike fractions. I’m sure you’ve seen something like this before where you add four-tenths and five-tenths for a final answer of nine-tenths, but today we wanna add something like this: one-fourth plus one-third. To do that, I’m gonna start by finding the least common multiple of four and three. The least common multiple for four and three is twelve. We’re going to use the least common multiple to create a common denominator for one-fourth and one-third; the common denominator is their least common multiple of twelve. So what we’re gonna do is ask the question how many twelfths is one-fourth. and how many twelfths is one-third. We answer this question by thinking what can I multiply four by that equals twelve and then again what can I multiply three by that equals twelve. four times three is twelve, and then three times four equals twelve. But in order to keep the fractions equal, if we multiply the bottom by three or by four, we have to multiply the top by that same value. We would need to multiply the top of the one-fourth fraction by three and the numerator of the one-third fraction by four, so one times three is three and one times four is four. We’ve now found equivalent fractions that have a common denominator and can be added easily three-twelfths plus four-twelfths equals seven-twelfths. I wanna stop quickly and show you why we can multiply that those fractions by three and by four and come out with equivalent fractions. On the other screen, we multiplied one-fourth; we multiplied the numerator by three and the denominator by three. We also multiplied one-third by four over four; we multiplied the numerator by four and the denominator by four. If you look at these fractions in red, three over three is the same thing as multiplying something by one; four over four is the same as multiplying something by one. Here we can see that we’re not actually changing the value of these fractions. When you multiply something by one, you get itself. So we’re making it look a little different, but the value is actually the same. These are equivalent fractions: three-twelfths is equivalent to one-fourth, and four-twelfths is equivalent to one-third.
The next example is a subtraction problem, but we’re gonna follow this same process. Start by searching for the least common multiple. Here we see that six and fifteen have a least common multiple of thirty. So I know that I need a new denominator for both of these pieces of thirty. To change the denominator of six to thirty, I need to multiply by five; and to change the denominator of fifteen to thirty, I need to multiply by two. But remember that we need to do that for the numerator and the denominator. five times five is twenty-five; eleven times two? twenty-two; twenty-five minus twenty-two is three, so we get three over thirty. But three over thirty is not in its most simple form, so we also wanna simplify this fraction. Both the numerator and denominator here are divisible by three, so the most simple form of this fraction is one over ten, one tenth as our final answer.
Our next example is a little bit harder because we have three fractions and we have both addition and subtraction in the problem. So with order of operations, we do addition and subtraction in order from left to right. But before we can do any adding or subtracting, let’s start with the least common multiple of five, nine, and three. The least common multiple of five, nine, and three is forty-five. You might notice that I didn’t write out all the multiples of three, and that’s because three has a multiple of nine. So you can see that on the three list nine is one of its multiples, so we don’t have to keep writing them out because we know that all the multiples of nine are already multiples of three. So we know that multiplying nine times five gets us forty-five, which means we need to multiply two fifths by nine over nine. We need to multiply four ninths by five over five. And finally three times fifteen equals forty-five, so we need to multiply two-thirds by fifteen over fifteen. Don’t forget to carefully copy over where we’re adding and where we’re subtracting. Now we’ll follow the order of operations, working from left to right to add eighteen forty-fifths plus twenty forty-fifths and then we’ll take away thirty forty-fifths. Adding the first two gives us thirty-eight forty-fifths. We now just need to take away thirty from that thirty-eight. Here our final answer is eight forty-fifths, and it is in its most simplified form so we leave it just like that.
Our last example is gonna be another problem where we have three different denominators, but the steps are the same. I wrote out all the least common multiples and saw that six and fifteen have the least common multiple of thirty. Again you’ll notice that I didn’t include the three because six is a multiple of three and, therefore, any multiple of six will also be a multiple of three. So I’m gonna use thirty as my least common multiple. five-sixths needs to be multiplied by five over five, two-thirds needs to be multiplied by ten over ten, and
eleven-twelfths [eleven-fifteenths] needs to be multiplied by two over two. After we multiply, we get twenty-five over thirty plus twenty over thirty plus twenty-two over thirty. Adding them all up, we get sixty-seven over thirty. But remember, at the beginning of the video and all throughout, we’ve been wanting to simplify these fractions so we need to simplify sixty-seven over thirty. We know that sixty-seven over thirty equals sixty over thirty plus seven over thirty, so our mixed number will look like this: two and seven-thirtieths because sixty over thirty simplifies to two, and we leave our seven-thirtieths as the remainder. So here are the steps we always follow when we’re adding and subtracting unlike fractions: we find their least common multiple, multiply by equivalent fractions, add or subtract always using the order of operations, and then simplify.