Division strategies for decimal quotients

- [Instructor] In this video, we're gonna come up with some strategies for division when the quotient isn't a whole number, when it's going to be a decimal, so let's try to compute three divided by two. Pause the video and see if you can figure out what that is going to be, and I'll give you a hint, it's not going to be a whole number. Alright, now let's try to work this together. And like all things in mathematics, there's multiple strategies that you can use to try to figure out what three divided by two is. One strategy is, well, let's just rewrite this as a fraction, so three divided by two, you can write that as, you could write that as three over two. Now, or you could say, this is the same thing as 3/2. But how can we express this as a decimal? Well, you might recognize that 3/2 is the same thing as 2/2 plus 1/2, let me write that down. So this is the same thing as two plus one over two, and I'm really doing every step here to hopefully make things clear, which is the same thing as two over two, so that's two over two, plus one over two, plus 1/2. I could break this up into two over two plus 1/2. Now, two over two is just one, and so this is going to be equal to 1 1/2. Now, you might immediately say, "Hey, 1/2, I could write that as 5/10," and that would be exactly right. You could just, we don't wanna spell out every step, we could say this is equal to one, and when we write it in decimal form, we express things as tenths or hundredths or thousandths, so 1/2 is the same thing as 5/10, and if we wanna express that as a decimal, this would be equal to one and five tenths. Now I did every step here, as you get more practice, you say okay this is the same thing as 3/2. 3/2, two goes into three one time, and there's a 1/2 left over, so writing this as a mixed number, it's 1 1/2, and 1/2 written as a decimal is 0.50, so this is 1.50. Now another way that we could've thought about this is "Okay, I'm not getting a whole number, "when I divide three divided by two. "Maybe I'll get something in terms of tenths, "so let me express each of these in terms of tenths." So three is how many tenths? Well, three is 30 tenths, and we'd be dividing by two, we're gonna be dividing it by two, so 30 tenths divided by two, well that's going to be equal to 15 tenths. This is equal to 15 tenths, which is equal to 10 tenths and five tenths, or 1.50. So both of these are equally legitimate strategies for figuring out what three divided by two is. I like the first one a little bit, it leverages what we know about fractions, but let's do another example. Let's do a few more examples, this is fun. Let's figure out what 34 divided by four is, and like before, pause this video and try to figure it out and try to see if you can use some of the strategies that we used in the last video. Alright, so as we just said, we can re-express this as a fraction, this is the same thing as 34 divided by four, 34 divided by four, or 34/4. Now what is this going to be equal to? Well, four goes into 34 eight times, it's gonna go eight times, and you're gonna have two left over, so this is eight and 2/4. Eight, let me write it both out. Alright, eight and 2/4, let me do that two in that blue just for fun. Eight and two over four, 8 2/4, so how do I do this again? I said four goes into 34 eight times, and then I have two left over, so I'm gonna have 2/4 left over. Another way, if you wanna see all the steps, you can say, "Hey I can rewrite this as 32 over four "plus two over four." The 32 over four is our eight, so 8 2/4. Well 2/4, that's the same thing as 1/2. That's the same thing as 1/2, and if we wanna express that in terms of tenths, this is equal to eight and, 1/2 is the same thing as 5/10, 8 5/10, which if we wanna express as a decimal is of course eighth and five tenths, or 8.50, and we are done. Let's do another one of these, and actually let's do one of them where we're dividing into a decimal, where a decimal itself is being divided. Let's say we wanted to calculate 8.4 divided by seven. Pause this video and see if you can figure it out. When you look at this, you might immediately say well I know 84 is divisible by seven. We know if you know your seven times tables, we know that seven times 12 is equal to 84, or that 84 divided seven is equal to 12, but this isn't 84, this is 8.4, so how do we think about it? Well one way, we can think about it in terms of tenths, 8.4 is the same thing as 84 tenths, and so 84 tenths divided by seven, well 84 of anything divided by seven is going to be 12 of that thing, so it's gonna be 12 tenths, and 12 tenths we can rewrite as one and 2/10, 1.20 and we are done, so this is equal to 1.2. Another way that we could have thought about this is we could've said, you know what, 84 tenths is the same thing as 84 over 10. If fact, you would read this as 84/10, and now you wanna divided this by seven. So you wanna divide this by seven, this is the same thing, when you divide by something, it's the same as multiplying by the reciprocal, so it's 84 over ten times 1/7, which is equal to, we can change, let me write it this way. This is equal to 84 over 10 times seven, over ten times seven, and now we could simplify this. If we divide the numerator and the denominator by seven, 84 divided by seven is 12, seven divided by seven is one. 12 divided by ten, this is gonna be equal to 12/10, which is 1.2, we could write this as 1 2/10, or 1.2. Let's do one more example that's kind of related. Let's say we wanna figure out what seven divided by 70 is. Pause this video and try to figure it out. Well, we can rewrite this, as we've been doing, as seven over 70. Instead of writing it as 70, let me write that as seven times ten, and what's valuable about this is we can divide the numerator and the denominator by seven. If we divide the numerator by seven, we get a one, we divide the denominator by seven, we get a one. Remember, we can do the same thing to the numerator and the denominator. If we multiply or divide by the same value, we're not changing the value of the actual fraction. And so you're left with 1/10, which if you express it as a decimal, you go to the tenths place, you say I have one of those tenths, that's one tenth, so this is one tenth. Another way to think about it is this is the same thing as, and really this is what we wrote over here, but you could write this as seven divided by seven, divided by the blue seven, and then you divide by 10. If you're dividing by seven times ten, actually let me write that down. If you're dividing by seven times ten, and this essentially another way of writing what we have over here, this is going to be equal to seven divided by seven divided by 10. Well, seven divided by seven is one, so you get one divided by 10, which is one tenth.