Dividing Fractions

[One speaker] 

[The title Dividing Fractions shows on screen, transitioning to a blank screen used as a virtual whiteboard.]

Narrator: Welcome to the video on dividing fractions. Let's look at an example first.

3/4 divided by 3/8. So 3/4 divided by 3/8 [the instructor writes the equation on the screen]. Now you may have been taught to turn over this other fraction and then multiply or flip the other fraction and multiply.

What I want you to do--and what I want you to think of--is that we're actually multiplying by the inverse--the multiplicative inverse, or in other words, the reciprocal [this phrase is flashed on screen].

So it's not just a matter of flipping or turning over. It's actually based on the idea of the inverse. The multiplicative inverse of 3/8 is 8/3 [writes the fraction 8/3 below the previously written equation and the answer 3/4 to the equation 3/4 divided by 3/8]. So let's rewrite this as a multiplication of the inverse. [Writes the equation she says on the screen as she says it.] So this [ ¾ / ⅜ ] equals 3/4 times now the inverse, which is 8/3 [ ¾ * 8/3]. Now we can do the multiplication like we normally would for a fraction, where we multiply the numerator and we multiply the Denominator.

So this is equal to 8 times 3 is 24 and the bottom--4 times 3 is 12. [Writes 24/12.] Now we can actually reduce this, so 24 divided by 12. Well 24 is the same as 2 times 12, and 12 is the same as 1 times 12--1 times 12. [Writes .] So even though we didn't break this down into the prime factorization, we did break it down into the factors and both of these numbers--24 and 12--have a factor of 12 in them, or in other words are divisible by 12.

So we can rewrite this as 2 divided by 1 times 12 divided by 12, or in other words, this 12 divided by 12 turns into a 1. [Writes .] So let's just take that and say 2 divided by 1 times 1. [Changes into 1.] Well, 2 divided by 1 is the same as 2 because anything divided by 1 is just itself. So 2 divided by 1 is 2. So this equals 2 times 1. Well anything times 1 is just itself. So 2 times 1 equals 2. [Writes = 2 · 1 = 2.] So our answer--way down here--is actually 2. [Draws box around 2.] So 3/4 divided by 3/8 equals 2.

Now let's go back through exactly what we did. So we took the 3/4 and instead of dividing by 3/8--instead we're multiplying by the reciprocal or the multiplicative inverse, and that is what gives us our answer.

Let's look at why this works. So 3/4 divided by 3/8. [Writes down equation.] We can rewrite this in fraction form. Remember, 2 divided by 2 is the same as 2 divided by 2 in fraction form. [Writes 2  2 = and then erases it.] So let's write this equation in fraction form. It gets a little bit big. I'll have to try to write smaller. Three-fourths divided by--we're going to make this line a bit , just so that we see the difference--divided by 3/8.

[Writes .] 

The reason I'm showing this to you is to show you why multiplying by the inverse is the same as dividing. So remember, anything times one is going to equal itself. [Writes · 1 next to big fraction.] Well what if we make this one look different? What if we make it two divided by two? [Changes 1 to .] Well it's still going to be equal to this, it will just look a little bit different. But because this is ultimately equal to one, we're still--we still haven't actually changed the value of the fraction.

So what if we want to get rid of the denominator? All we need to do would be to multiply by the reciprocal of this because this number 3/8, times its reciprocal, equals 1. So if we multiply by 8/3, then this denominator will equal 1, because 3/8 times 8/3 equals 1.

[Writes .]

But anything that we do down here, we've got to put in the numerator as well, because then we're still multiplying by one. So we have to multiply by 8/3 in the numerator.

[Writes .]

So this--this whole fraction on top of a fraction is still equal to one, because it's 8/3 divided by 8/3. It's just a much more complicated version of one.

[Draws box around  then writes =1 next to it.]

Well now we can use the concept of the reciprocal, and we know that 3/8 times its reciprocal, equals 1. So all we're left with is one in the denominator, and what we're left with in the numerator is 3/4 times the reciprocal of 3/8, which is 8/3.

[Writes .]

And anything divided by one is just itself, so this equals 3/4 times 8/3.

[Writes .]

And that is why 3/4 divided by 3/8 is the same as 3/4 times the reciprocal of 3/8, which is 8/3. And in this form, all we have to do is multiply across the numerators and the denominators to get our answer.

[end of video]