Dividing a whole number by a decimal

Let's divide 518 by-- so we're going to divide it by 0.7. So we're dividing this whole number by a decimal. So we could also write this as 518 divided-- let me write a little bit bigger than that, since we have to do some work with it. 518 divided by-- I'll do the division sign in white-- divided by 0.7. So the first thing we do, since we have this decimal here, we're dividing by a decimal. So try to turn this into a whole number somehow. Well, the best way to turn this into a whole number is to multiply this by 10, which is essentially multiplying, shifting the decimal point over to the right. So this would become a 7. But we can't just do that only for what we're dividing by. We also have to do that to the 518, so that a value does not change. So we need to multiply both of these times 10. So if we move the decimal over to the right with the 0.7 to turn into a 7, we also need to move the decimal over to the right for 518. Now, you're probably saying, well, I don't see a decimal in 518. Well, there is one. You just didn't have to write it, because it's 518.00-- and we can add as many zeroes as we want. So if we move the decimal to the right, it becomes 5,180. So really what we're saying is 518 divided by 0.7 is the same thing as 5,180 divided by 7. Notice all we did by moving the decimal one place to the right, is we multiplied both of these numbers by 10, which is not going to change the actual value of the decimal. One other way of thinking about this, if you wanted to write this as a fraction, this is the same thing as 518 over 0.7. You multiply both the numerator and denominator by 10, you will get 5,180 over 7. So let's clean this up a little bit, just so we remember what we did. So we moved the decimal over to the right, one. So now this is just a 7. The decimal is there. In fact, we really don't have to write the decimal anymore. It's just a 7.0-- you could imagine 7.0, so we can just write this as a 7. And then the 518, the decimal is now out here. So this is 5,180. And let's increase the sign right over here. Now, this is just a straight-up long division problem. How many times does 7 go into 5? Well, it goes 0 times. 0 times 7-- actually, let's just cut to the chase. 7 doesn't go into 5. It does go into 51. 7 times 7 is 49. So it goes 7 times. 7 times 7 is 49. Subtract 51 minus 49 is 2. And now we can bring down this 8. 7 goes into 28 four times. 4 times 7 is 28. Subtract, you get a 0. Now we can bring down another 0. We want at least get to the decimal place. So we bring down another 0 right over here. When I say get to the decimal place, we could put the decimal place up here too, just to make sure we're keeping track of the right place values or that we can have the decimal in the right place. So notice, I'm very particular. When I'm doing 7 goes into 51, I put the 7 right above the 1 in the 51's place. When I'm saying 7 goes into 28, I'm putting the 4 right above the 8 in this one's place when we're doing the division. So now we say how many times does 7 go into 0? Well, it goes 0 times. 0 times 7 is 0. Subtract, you have no remainder. So we could keep going, and we'll just keep getting zeroes like this. But we see that this is equal to 740.