Multiplication of Fractions

Keep in mind that the division bar in a fraction indicates division. For example,(1)

If the division doesn’t come out evenly, we generally just leave the division problem as a fraction.

Keep in mind also that division and multiplicaton go together well, and as a result, multiplying and dividing
fractions is relatively easy. Two underlying properties help us out. We have, for example, that

Since we can reverse a multiplication without affecting the answer, we say that multiplication is commutative.
The other thing that helps us is that we can think of division as multiplication. For example,

and since multiplication is commutative, we have

Suppose we want to multiply two fractions. For example, consider

We have two numbers on top, the 2 and the 5, and these are to be multiplied. We have two numbers on the
bottom, the 3 and the 4, and these are to be divided. Other than this, the order doesn’t matter. We have

We can just multiply the 2 and 5, then divide by the 3 and 4. We generally end up multiplying the tops and
multiplying the bottoms.
Here’s another example.

We’ll talk about reducing fractions in a bit. Don’t worry about that now.

1. Quiz 03, Part I
 

2. Reducing fractions

Reducing fractions works by doing a multiplication backwards. For example, in the following, we can factor
(unmultiply) the numbers in the fraction.

In the second step, we can see that we’re multiplying by a 2 and dividing by a 2.that’s the
same as multiplying by 1, and that doesn’t do anything.
Here’s a couple of other examples.

Since we’re multiplying by 5 and dividing by 5, that’s the same as multiplying by 1.
It’s usually best to factor the numbers as much as you can. You’ll be able to see things more reliably.

We’re multiplying by a 2 and 3, and we’re dividing by a 2 and 3.

3. Dividing by fractions

Consider the example

Since dividing by 5 is the same as multiplying by 1/5 , we can rewrite this as a multiplication.

On the other hand, if we’re dividing by a fraction, this involves dividing by a division.

In general, we just invert the fraction that we’re dividing by.

It works the same way when we have a fraction with fractions inside. The middle fraction bar is just another
division.

It’s customary to reduce any fractions you get as an answer.

4. Quiz 03, Part II

For the rest of these problems, multiply or divide, and then reduce as much as you can.

 

5. Homework 03

For the rest of these problems, multiply or divide, and then reduce as much as you can.