Adding and Subtracting Fractions With Unlike Denominators

Adding and subtracting fractions with unlike denominators. Adding or subtracting fractions with like denominators is pretty easy. You just add or subtract the numerators and keep the denominator the same. For example, if you have one fourth plus two fourth, add one and two and keep four the same. The answer is three quarters. Subtraction works the same. If you have three fifth minus one fifth, you subtract the numerators on the top and keep the denominator at five. Three minus one is two. The answer is two fifths and you can make sedition and subtraction. For example, one third plus two thirds minus one third. On the bottom, the denominator remains three and on the top, for the numerators, you add one and two, which is three, and then subtract one, which leaves two. The final answer is two thirds. That's pretty straightforward. But what if the denominators are different? How do you add or subtract fractions with unlike denominators? What if you had something like three eighths plus one fourth? Or three fourths minus one third. How do we go about solving these problems? If only there was some way to make these fractions with different denominators have the same denominators. Well, there is. But before we get to that, let's consider what a denominator is. Let's make the problem a bit more real. To start figuring this out, let's get some wooden planks and cut them into equal segments. The first plank we won't cut. We'll leave it so we can see how long a whole uncut plank is. The next plank we'll cut into two sections. Each section is one half of a whole plank. We'll keep doing this until we've added planks that are in thirds, quarters, fifth, sixth, seventh, and eighths. Now, we'll put them aside and look at a fraction addition problem. Three eighths plus one fourth, we'll drag back our uncut board to remind us how long an uncut plank is and put it at the top. Let's grab 31 eighth plank segments and one fourth segment, and we'll line them all up. There's three eighths plus a quarter. In order to add these fractions, we need to make all our denominators the same without changing the sizes of any of our plank segments. Let's look at one fourth. Let's try and make it a fraction with eight in the denominator. We can see that one fourth segment is the same as two of the one eighth segments. So let's pretend to cut the one quarter plank segment in half. We'll show our virtual cut by drawing a dotted line. This makes our addition problem much easier. Instead of three eighths plus one fourth, it's now three eighths plus two eighths. We can just add the two numerators and keep the eight in the denominator, which gives us five eighths. But what if it's not obvious how to make the denominator the same? For instance, if we're trying to add two thirds and one fifth, one way to do this is a method that involves finding a common multiple of the two denominators. But what is a multiple? A multiple is a number that you get by multiplying a number by another number. Three times one is three, so three is a multiple of itself. Three times two is six, so six is a multiple of three. We can keep going. Nine, 12, and 15 are also multiples of three. 18 and 21 are also multiples of three. We could keep doing this forever, but let's stop here for now. Now we know what a multiple is. How do we use this to solve our problem? How does this help us virtually slice up our boards so that the two fractions have the same denominator? The process we are going to learn involves finding the smallest shared multiple of our two denominators, or as you may have heard in the phrase, the least common denominator of the two fractions in our problem. In this case, our denominators are three and five. Well, we've already found a bunch of multiples of three, three, six, nine, 12, 15, 18, 21, and so on. Now let's look at some multiples of five. Five times one is five, no match with the multiples of three there. Five times two is ten, still no match. Five times three is 15. Ding ding ding. Yeah, we've got a match. The smallest multiple that is common to both denominators is 15. The least common denominator of three and five is 15. Now we'll look at our planks again. Here's one plank uncut virtually or otherwise, here's a plank we virtually cut up into 15 segments. Here are our boards that are cut into thirds and fifths. For our fraction addition problem, we'll need two of the thirds and one of the fifths. Notice how both of our one third board segments and the one fifth board segment all line up evenly to the one 15th board segment. This is because 15 is the lowest common multiple of both three and five. Now, we can simply count up the one 15th segments that add up to the same length as the two thirds plus one fifth segments and have our answer. One, two, three, four, five. Wait a minute. All this board cutting counting is a lot of work. Now that we understand what's going on, surely there's an easier way of figuring out this problem and there is. Let's go back to our multiples. We want to change the denominators three and five into 15, and we want to do that without changing the size of our two fractions. Let's start by looking at the two thirds fraction. We can see that each third is the same size as 5/15. 21 third segments is going to be 10/15. But to get there without the boards and segment counting, we need to think about what's going on here mathematically. The number we are aiming for in the new denominator is 15, we need to make that three into a 15. So what's a fraction we can multiply two thirds by to get 1015? Let's start with the denominator. The easiest way to find a number to multiply three by to get 15 is to divide 15 by three. This gives us five. For the numerator, what can we multiply the two by to get ten? 10/2 is five. Hey, that's the same number we're multiplying in the denominator, and that will always be the case. If we think about it, this makes sense. Anytime you have a fraction where the numerator and the denominator are the same value, then the value of the whole fraction is one. We know that by multiplying two thirds by five fifths, we aren't changing the value of the fractions. We are just making an equivalent fraction with proportionally different values in the numerator and denominator. Now we'll try this with the other fraction in the problem. We can see that one fifth will end up being the same as three 15th. But let's hide the boards and the answer. We know the least common multiple is 15, that's going to be the number in the denominator of the answer. To get the number we need to multiply by five to get 15, we divide 15 by five. Five goes into 153 times. The number and the denominator of the fraction that we will be multiplying the original fraction by is three. We put the same number three into the numerator of the fraction, we're multiplying the original fraction by. Remember that the new fraction is the same value as the one since its numerator and denominator are the same. Now, multiply the original fraction by the new one we've made. And there is our new equivalent fraction 3/15. When we bring back the boards and rearrange things, we see that we've got the right answer. Wow, I'm exhausted. I feel like I've been sawing wooden planks, both real and virtual all day. Let's see if we can solve a problem without sawing. Here's a new problem, five sixth plus three fourths. To start, we need to find the least common denominator of six and four. The first few multiples of six are six, 12, 18 and 24. The first few multiples of four are four, eight, and Bingo, 12. The least common denominator of four and six is 12. For the first fraction, in order to change the six into 12, we multiply it by two. 12/6 is two. To keep the size of the fraction the same, we multiply the numerator by two also. Five sixth can be written as 10/12. For the second fraction, we need to multiply the denominator of four by three to get 12. Again, we multiply both the denominator and the numerator by three and get 9/12. Now that the two fractions have the common denominator, we can just add the numerators. Ten plus nine is 19. The answer is 19 12th. There's our answer and not a board or handsaw in sight. Since the numerator is bigger than the denominator, we simplify this fraction to the mixed number of 1 7/12. Oh, yeah. And subtraction works the same way. Only you subtract the two numerators instead of adding them.