Divide whole numbers with decimal quotients: 78÷12

Let's say that we wanted to compute 78 divided by 12. What would this be? Pause this video and try to figure that out. Well, one way to think about it is, this is the same thing as 78 divided by 12, heh. I know when I said it in English, it sounds the same, but here, on the right, I've written it as 78/12, or 78 divided by 12. Now, how can we re-express this? Well, let's think about it. Is there a multiple of 12 that's near 78 without going above it? Well, we know that 12 times five is 60, 12 times six is 72, 12 times seven is 84, so 12 times seven is too high, but we can write this numerator as, instead of 78, I can write it as a multiple of 12 plus whatever's left over. So, 72 plus six, that's the same thing as 78, so it's that divided by 12, and, so, I can write this as, I can write the 72/12, so this is equal to 72 over 12 plus six over 12, plus six over 12 or plus 6/12. Now, what's 72 over 12? We know that 72 is the same thing as six times 12. So, this is going to be six times 12 divided by 12, well, that's just going to simplify to six, and then what is six 12ths? Well, you might recognize six as half of 12 or you could divide the numerator and the denominator both by six. Either way, you are going to get 1/2. So, you can view this as six plus 1/2 or you could view this as six and a half, and a lot of times, or as you get more used to this, you won't go do all of these steps, but I want to make sure you really understand what's going on. So, this is the same thing as six and a half, and if I wanted to express that as a decimal, that's the same thing as six ones and then how many tenths? Well, 1/2 is the same thing as 5/10, so six and 5/10. So, there you go! That's one way of trying to compute what 78 divided by 12 is. Let's do another example. Let's say we wanted to compute what 20 divided by 80 is. Pause this video and see if you can figure that out. Well, we could use a similar technique. We could say, hey, this is the same thing as 20 80ths, or we could write 20 the numerator and 80 the denominator, so it's the same thing as 20 divided by 80, and then we could think about, well, how can we simplify this fraction, or re-express it in some way? Well, let's see. We can write the numerator as... We could write this numerator as equal to one times 20, and then we could write the denominator as four times 20, as four times 20. And, so, you could just view this as being equal to 1/4, this is equal to 1/4, times 20 over 20. Times 20 over 20, well, what's 20 over 20? Well, that's just one. That's just one, so this all just becomes 1/4. Now, how would we express that as a decimal? Well, let's see, it's hard to express 1/4 in terms of tenths 'cause four doesn't divide easily into 10, but you can express it in terms of hundredths. So, 1/4 is the same thing as 25 over a hundred. That's the same thing as 25/100. Lemme write it down here so that I get more space. So, 1/4 is the same thing as 25 over 100, and you could get that by multiplying the numerator and the denominator here by 25, and the reason why I care about hundredths is we know how to represent hundredths as decimals. This is going to be the same thing as zero ones and 25/100. And, so, 20 divided by 80 is 0.25, or 25/100. So, once again, these are all different strategies for thinking about how we can divide numbers that result in decimals.