Solving for the missing fraction

- [Voiceover] So, we have blank plus one-16th is equal to three halves. So, I encourage you to pause the video and just figure out what blank is. What plus one-16th is equal to three-halves? All right. Now, let's work through this together. One thing that might make it a little bit simpler for our brains is if we were to express one-16th and three-halves with a common denominator. When we think about a common denominator, we will look at a common multiple of the denominators here. And lucky for us, 16 is already divisible by 2. It's divisible by 16 as well, so it is the common multiple, or it is the least common multiple of 16 and 2. There are other common multiples, but the smallest one is going to be 16 times 1, which is also divisible by 2. So, let's try both of these, let's try both of these fractions. Let's rewrite this equation where both of these fractions have 16 as their denominators. This one obviously already has it. So, let's write that. So, we're going to have, we're going to have blank plus one-16th, one-16th, is equal to, is equal to, let's see, let's write three-halves as something over 16. Well, to get our denominator from 2 to 16, we have to multiply by 8. So, we have to multiply the numerator by 8 as well. So, 3 times 8 is going to be 24. Now, at this point, you might be able to do it in your head. Blank plus one-16th is equal to 24-16ths. We could say, okay, this could, this could... We could do this as a certain number of 16ths. So, how many 16ths plus one-16th is going to be 24-16ths? Well, 23-16ths. That's 23-16ths, and I add one more 16th, I'm going to have 24-16ths. Another way that you could have thought about it, when we... Actually, you could have even thought about it from the first step, is you could say, look, if blank plus one-16th is equal to three-halves, then you could say that blank, you could say that blank is going to be equal to three-halves, three-halves minus one-16th, minus one-16th. This is another way that you could have tackled it. If blank plus one-16th is three-halves, then blank is going to be equal to three-halves minus one-16th. And we know that this is going to be equal to... Three-halves, we already know, it's the same thing as 24-16ths. So, 24 over 16 minus one-16th, minus one-16th, which we figured out is 23/16ths, which is equal to... I'll just rewrite it again, 23 over 16. Let's do another example. So, here, this is a little bit different. I have blank minus three-fourths is equal to two-thirds. And there's a couple of ways to think about it. If blank minus three-fourths is equal to two-thirds, that, one way to think about it, you can say, blank, that means that blank is going to be the same thing as two-thirds plus three-fourths. Two-thirds plus three-fourths. Plus three-fourths. And what is this going to be? Well, once again, I could rewrite both of these fractions so they have a common denominator. Well, what's the least common multiple of 3 and 4? Well, we could look at the multiples of 4 and keep, and, and keep increasing them until we find one that's perfectly divisible by 3. 4 isn't divisible by 3. 8 isn't divisible by 3. 12 is divisible by 3. In fact, it's 3 times 4 is 12. So, we can rewrite both of these as something over 12. So, this is going to be the same thing as... So, to go from 3 to 12, we multiply it by 4. So, you have to multiply the numerator by 4 as well. So, 2 times 4 is 8. Two-thirds is the same thing as eight-12ths. This is going to be equal to eight-12ths plus, and then three-fourths is the same thing as what over 12? Well, to go from 4 to 12, you have to multiply it by 3, so you have to multiply the numerator by 3 as well. 3 times 3 is 9. And what's that going to be equal to? So, it's equal to this, which is equal to eight-12ths plus nine-12ths is going to be, 8 plus 9 is 17. 17-12ths. So, you get 17-12ths minus three-fourths is equal to two-thirds. Now, another way that you could have done it is you could have just kept it the same and could have said, okay, let me just rewrite, let me just rewrite this, but I'm going to rewrite the fractions so that they have a common denominator. We already said, well, our common denominator we want to use is 12, at least. We could use 24 or something larger than that. But this would have been the least common multiple, so this keeps things a little bit simpler. And three-fourths, we're going to rewrite this as nine-12ths. 9 over 12 is equal to. And two-thirds, we can rewrite as 8 over 12. And you could assume that this is going to be a certain number of 12ths. So, blank-12ths minus nine-12ths is equal to eight-12ths. So, if I have blank of something minus nine of something, and I'm left with eight of that something, that means that I had 17 of that something to begin with. 17-12ths minus nine-12ths is equal to eight-12ths. But either way, you get 17-12ths for our blank.