Multiplying Fractions

[One speaker]

Narrator: Welcome to the video on multiplying fractions. We’re going to start out immediately with an example. 2/3 times, I’m showing it with a dot this time, 3/5.

[Writes 2/3 · 3/5.]

The rule for multiplying fractions is simply to multiply across the numerator and multiply across the denominator [Draws two arrows. One from 2 pointing to 3 and the other from 3 pointing to 5.] and in the case of multiplication, we don’t need to worry about having the denominators be the same.

So two times 3 equals 6, and 3 times 5 equals 15. [Writes 6/15.] If we reduce 6/15 it is equal to 2/5, but right now let’s just focus on multiplication.

Let’s do another example: 5/8 times 3/2 or 3 over 2. [Writes 5/8 · 3/2=.] So we multiply straight across so this is the same as-- I’m just going to rewrite it this time, but we’re doing the exact same thing-- 5 times 3, divided by 8 times 2. This equals, well 5 times 3 is 15, and 8 times 2 is 16.

[Writes = 15/16.]

So now that we see how it works, let’s discuss a little bit about what is going on. In this case i’m going to do it with the example with 1/2 times 1/2. [Writes 1/2 · 1/2.] Well, according to our rules of multiplying fractions this is equal to 1, because 1 times 1 is 1, over 4, because 2 times 2 is 4. So 1/4. [Writes 1/4.]

Remember a fraction actually indicate division. So 1/2 times 1/2 is really the same as 1/2 times 1 divided by 2. [Writes 1/2 · 1  2.] And since there is only multiplication and division going on we can kind of move these around. 1/2 times 1 is still 1/2, because multiplying anything by one it stays the same. And then we’re dividing by 2. [Writes 1/2  2.]

Well 1/2, let’s draw a picture here. I have a circle and I have half of it right here, [Draws a circle divided in two with one half shaded..] and then I divide that by 2. So I divide the half that I have by 2, and that leaves me with must this portion. [Divided the shaded half into halves.] Well, notice this is the same as if we had one out of four original pieces. So 1/2 divided by 2 is the same as 1/2 times 1/2. Which equals 1/4. This represents 1/4, because a half of a half is the same as a fourth.

Let’s do another one. This time 1/4 times 1/2. Well, according to the algorithm, this should equal...1 times 1 is 1 and 1 times 4 times 2 is 8. [Writes 1/4 · 1/2 = 1/8.] So we started out with a fourth and let’s just...we’re going to draw it in a square this time. That’s supposed to be a square and we’re cutting it into fourths. So we have 1/4. This is the piece that we have. [Draws 2 x 2 square with one section shaded in.] That’s the chunk we have, and now we’re going to multiply it by one and divide by two.

So we’re actually, when multiplying by 1/2 is the same as dividing by 2. So I’m going to divide this by two, and we’ll be left with this piece in the blue. [Divides shaded section into halves and shades in one half.] Well that is the same as if we had the whole thing divided into eight pieces. [Divides square into a 4x2 rectangle.] 1, 2, 3, 4, 5, 6, 7, 8, eight pieces. That’s equal to one of the eight pieces. So 1/4 times 1/2 equals 1/8.

So you may have already caught on to this, but another important aspect of multiplying fractions is when you’re multiplying a fraction and an integer. Let’s say 3 times 3/4. [Writes 3 · ] 3 times 3/4 is the same as 3 divided by one, [Changes 3 to .] because 3 divided by 1 is still three. So we can treat this 3 as though it’s a fraction, and just multiply the numerators. 3 times 3 is 9 and 1 times 4 is 4.

[Writes .]

The same is true going the other direction. 2/3 times 5 is the same as 2/3 times 5 divided 1. So all we have to do is take the 2 times 5 equals 10 and three times 1 on the bottom, in the denominator, is three. So 10/3.

[Writes 2/3 · 5 = 2/3 · 5/1 = 10/3.]

[End of video.]