Strategies for adding and subtracting fractions with unlike denominators
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- Let's see if we can figure out with 3/4 minus 5/8 is. And we have 3/4 depicted right over here. You could view this entire bar as a whole, and we see that it is divided into four equal sections, and that three of them are shaded in. So those three that are shaded in, those represent 3/4 of the whole. So you see that right over there, and then this bar down here, you could view this as another whole. This is another whole right over here and you could see this divided into eight equal pieces, and five of them are shaded in. So that represents, that represents the 5/8. So we want to have 3/4, this green shaded area, and we want to take away the 5/8. So how could we do it? And even when you look at it visually, it might jump out at you. Whenever we add or subtract fractions, we like to think in terms of having the same denominator. Are we going to deal in fourths or eighths or 16s, or whatever else? So let's think about having a common denominator. And a good common denominator is going to be a common multiple of the two denominators right over here, and ideally their least common multiple. And one way that I like to tackle that, there's many ways to do it, is look at the larger of the two denominators, look at eight, and then keep looking at increasing multiples of eight until you find one that's also divisible by four, perfectly divisible by four. But with eight, you immediately say, "Well, eight is divisible by four," and that's clearly divisible by itself as well, so eight is actually the least common multiple of four and eight. So you can rewrite both of these, both of these fractions as something over eight. So the 3/4, you can write it as something over eight, and then subtracting from that, the 5/8, if you want to write that as something over eight, well, that's just going to be 5/8. And then you can figure out your actual answer. So how can we rewrite 3/4 to something over eight? Well, there's a couple of ways to think about doing it. One way, look. I had four in the denominator, now I'm going to have twice as many equal sections. I'm multiplied by two, so I'm going to have twice as many of the sections actually shaded in. So times two, 3/4 is the same thing as 6/8. And we can also see that visually. If we're going to have twice as many equal sections, here we have everything in fourths, but I'm going to divide, I'm going to turn this into twice as many equal sections so I have eighths. So let's do that. So let me... So you have this right here. Let me divide that. Let me divide that. Let me divide that, and then let me divide that, and now I went from fourths to eights. I have one, two, three, four, five, six, seven, eight equal sections, and we see that six of them are shaded in, that 3/4 is the same thing as 6/8. But regardless, now we can subtract. We have 6/8 and we want to take away five of the eighths. So we have 6/8, and we want to take away one, two, three, four, five of them, and those five of them correspond to these purple five right over here. We're taking away one, two, three, four, five. We're taking these away. So if you're just looking at the green, we started with 6/8, we're taking away one, two, three, four, five of them, and you can see that corresponds to the 5/8 down here and what are you left with? Well, you're just going to be left with, you're just going to be left with this 1/8 right over there. So it's just going to be 1/8. And you could see that numerically up here. If I have six of something, in this case it's 6/8, and I want to subtract five of that something, in this case 5/8, I'm going to be left with one of that something, or 1/8.