Division strategies for decimal quotients

In this video, we're going to 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 3 divided by 2. 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.

All right, now let's try to work this together. And like all things in mathematics, there are multiple strategies that you can use to try to figure out what 3 divided by 2 is. One strategy is, well, let's just rewrite this as a fraction. So 3 divided by 2, you could write that as 3 over 2. Now, or you could say this is the same thing as 3 halves. But how can we express this as a decimal? Well, you might recognize that 3 halves is the same thing as two halves plus one half. Let me write that down.

So this is the same thing as 2 over 2, so that's 2 over 2 plus 1 over 2. Plus one half, I could break this up into two over two plus one half. Now, two over two is just one, and so this is going to be equal to one and one half. Now this might, you might immediately say, hey, a half, I could write that as 5 tenths, and that would be exactly right! You could just, if you want to spell out every step, we could say this is equal to 1, and since when we write it in decimal form we express things as tenths or hundredths or thousandths, so one half is the same thing as five tenths. And if we wanted to express that as a decimal, this would be equal to one and five tenths.

One and five tenths. Now, I did every step here; as you get more practice, you say, okay, this is the same thing as three halves. Three halves, two goes into three one time, and there's a half left over. So writing this as a mixed number, it's one and a half, and a half written as a decimal is five tenths, so this is one and five tenths.

Now, another way that we could have 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 thirty tenths. Thirty tenths. And we'd be dividing by two; we're going to be dividing. We're going to be dividing it by 2. So 30 tenths divided by 2, well that's going to be equal to 15 tenths. This is equal to 15 tenths, which is equal to 1 and 5 tenths or one and five tenths.

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.

All right, let's figure out what 34 divided by 4 is. Divided by 4. 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. All right, so as we just said, we could re-express this as a fraction. This is the same thing as 34 divided by 4, 34 divided by 4, or 34 over 4. Now, what is this going to be equal to? Well, 4 goes into 34 8 times, so it's going to go eight times, and you're gonna have two left over.

So this is eight and two fourths. Eight, actually let me write it well. All right, eight and two fourths. Let's do that two in that blue just for fun; eight and eight and two over four. Eight and two fourths. 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 going to have two fourths left over.

Another way, if you want to see all the steps, you could say, hey, I could rewrite this as 32 over 4 plus 2 over 4. The 32 over 4 is our 8, so eight and two-fourths. Well, two-fourths, that's the same thing as one-half. That's the same thing as one-half, and if we want to express that in terms of tenths, this is equal to eight, and one-half is the same thing as five-tenths. Eight and five-tenths, which if we want to express it as a decimal is of course eight and five-tenths, or eight point five, and we are done.

Let's do another one of these, and actually let's do one of them where we are dividing into a decimal, where a decimal itself is being divided. So let's say we wanted to calculate 8.4 divided by 7. Pause this video and see if you can figure it out. So when you look at this, you might immediately say, well, okay, I know 84 is divisible by 7. We know, if you know your 7 times tables, we know that 7 times 12 is equal to 84, or that 84 divided by 7 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 7, well, 84 of anything divided by 7 is going to be 12 of that thing. So it's going to be 12 tenths, and 12 tenths we can rewrite as 1 and 2 tenths or 1.2. 1.2, and we are done. So this is equal to 1.2.

Another way that we could have thought about this is we could have said, you know what? 84 tenths is the same thing as 84 over 10. In fact, you would read this as 84 tenths. And now you want to divide this by 7, so you want to divide this by 7. This is the same thing when you divide by something; it's the same as multiplying by the reciprocal. So it's 84 over 10 times 1 over 7, which is equal to... Let me write it this way; this is equal to 84 over 10 times 1 over 7.

Now we could simplify this. If we divide the numerator and denominator by 7, 84 divided by 7 is 12, and 7 divided by 7 is 1. 12 divided by 10; this is going to be equal to 12 tenths, which is 1.2. We could write this as 1 and 2 tenths or 1.2.

Let's do one more example that's kind of related. Let's say we want to figure out what 7 divided by 70 is. Pause this video and try to figure it out. Well, we can rewrite this as we've been doing as 7 over 70. Instead of writing it as 70, let me write that as 7 times 10. And what's valuable about this is we can divide the numerator and the denominator by 7.

We get, if we divide the numerator by 7, we get a 1. If we divide the denominator by 7, we get 10. 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 over 10, which if you express it as a decimal, you go to the tenths place. You say, hey, 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 7 divided by 7 divided by the blue 7, and then you divide by 10. If you're dividing by 7 times 10. We're dividing. Actually, let me write that down. If you're dividing by 7 times 10, and this essentially, another way of writing what we have over here. This is going to be equal to 7 divided by 7 divided by 10 divided by 10.

Well, 7 divided by 7 is 1, so you get 1 divided by 10, which is 1 over 10. 1 over 10.