Relating fractions and division

0:00- [Instructor] So just as a little bit of review, we've seen that if we have three holes, that's the same thing as three. And we also know that if we take three divided by one, that is also the same thing as three. So you might have already suspected or maybe you've already learned this, that if you have a fraction like this where you have one number over another number, that's the same thing as dividing the number.

0:28So three over one is the same thing as three divided by one. We could see that with other examples that we've come across. If I were to say eight-fourths, you might recognize that that is the same thing as two. But if I were to say eight divided by four, that is also equal to two.

0:51So once again, eight over four, same thing as eight divided by four. Now, this isn't just true with numbers, it could be with expressions. For example, if I have one plus two over three plus four, how do you think I could write this as division? Well, you might be tempted to say one plus two divided by three plus four, but would this be exactly right?

1:24Because if someone were to write it like this, order of operations would mean that you're going to do the division first, and that's definitely not what this fraction represents. So what you're going to need to make sure is that you're saying we're taking the numerator and we are dividing it by the denominator.

1:42Now, we're going to have our order of operations correct. So let me give you another example. If I were to say nine over four plus five, If I were to say nine over four plus five, why don't you pause this video and think about how you would write that as division?

2:00All right, now once again, we could write nine divided by four plus five, we could write nine divided by four plus five, but this wouldn't be right because order of operations would say you do the division before you do the addition. So nine over four plus five isn't the same thing as this, which is what the order of operations would suggest doing it.

2:20So we have to be careful. We have to use parenthesis. We have to say it's nine divided by all of this by four plus five, by four plus five, just like that. Let's do another one, and let's actually go the other way. Let's say that I have nine minus one divided by four.

2:42How would you write that as a fraction? Well, you're taking this thing and dividing it by the other thing. Nine minus one is being divided by four. So nine minus one would be the numerator, and then the denominator would be equal to four. Now, just as a little bit of a different variation, what would the following be?

3:06Nine minus one divided by four. Write that as a fraction or maybe an expression that involves a fraction. All right, once again, you might be tempted, hey, isn't that what we just wrote down? But no, look, order of operations would tell you to do division before you do the subtraction.

3:25So this would be equivalent to this right over here. So that would be nine minus. If we wanted to write one divided by four as a fraction, we would write one over four. And you can see very clearly that this and this are very different expressions.