Reasoning through multiplying decimal word problems

- [Instructor] We're told that Juan runs 1.7 kilometers every morning. Juan runs the same amount every day for six days. How many kilometers did Juan run in six days? Pause this video and see if you can figure this out before we do this together. All right, so Juan is going 1.7 kilometers per day and he's doing it for six days. So it feels like we should just be able to multiply by 6. Now in the future we're gonna learn many different ways of multiplying decimals, but let's just see if we can reason through this. So instead of just thinking about 1.7 times 6, we know how to multiply say something like 17 times 6. So let's do that first and then let's see if we can reason our way to what 1.7 times 6 is based on whatever 17 times 6 is. So 17 times 6, we've done this before, or things like this. 6 times 7 is 42. That is 2 ones and 4 tens. And then 1 times 6, that's really 10 times 6 right over there. That's 6 plus 4, so then that gives you 10. So 17 times 6 is 102. So based on that, what do you think 1.7 times 6 is? Well, there's a couple of ways to think about it. One way to think about it is you could estimate. You could say, all right, a whole 6 would be 6, and then you have seven tenths of a 6. So I don't know, that could be like 4 or 5, someplace in between there. So you would feel that if you have that, this should be a little bit over 10. And if you put the decimal right over here, that would be a little bit more than 10. So maybe you could say, hey, I'm gonna put the decimal right over here. So instead of 102, I have 10.2. This seems to make sense, if we would've just estimated. 2 times 6 is 12, 1 times 6 is 6, and this is going to be closer to 2 times 6 'cause it's 0.7. So this one feels right. Another way that you could have thought about this is when you go from 1.7 to 17, you're multiplying by 10. So this number right over here is going to be 10 times larger than this number, and that actually works out as well. So you could divide by 10 right over here, so it would move the decimal place over. Let's do another example of this. The answer is actually 10.2 here. Let's see. The cost to jump at the trampoline park is 8.25 per hour. Riley jumped for four hours. How much money did Riley spend to jump? Pause this video again and see if you can figure this out. Well, similarly, we're gonna multiply $8 and 25 cents times 4. And as I said, in the future, we're gonna learn ways to do this very almost automatically. But let's reason through it right now. So let's do this first without the decimals, 825 times 4. And then we're gonna figure out where we might wanna put the decimal. But before I even do that, let's just think about this is, if it was 8 times 4, that would be 32. If it was 9 times 4 it would be 36. So whatever this is, it should be someplace in between 32 and 36 and it should probably be closer to 32 'cause we're only a quarter of the way between 8 and 9. So let's just multiply this and then think about where we could put the decimal to be between 32 and 36. 4 times 5 is 20, regroup the 2. 2 times 4 is 8, plus 2 is 10. Regroup the 1. 8 times 4 is 32, plus 1 is 33. So we get 3,300 here. So what do we think this is going to be? It's going to have the same digits, but where do you think that decimal's going to be? Well, we already said that this should be someplace between 32 and 36. And if we wanna do that, well, you can't put the decimal there. That'd be 3,300. You can't put it there. That'd be 330. You can't put it here. That'd be 3.3. If you put it right over here, it seems to make a lot of sense. And just like we reasoned through the last example, it does make a lot of sense, because to go from 8.25 to 825, you're moving the decimal over two spots. That's the same thing as multiplying by a hundred. So this answer is gonna be a hundred times larger than whatever this answer is. So another way to think about it, yet you have to undo that. You're going to divide by a hundred and that would get you to the same place that we got just by reasoning through it.