Division with fractions

Now, we are going to divide fractions. It's not as difficult as it looks. But if you haven't learned yet how to multiply fractions yet, do that first because you're going to need it. Let's begin with a simple example. You have three liters of lemonade and you want to pour it into glasses with one quarter of a liter each.

How many glasses can you fill? In other words, how many quarters are there in three whole. If you count them on the screen, you find that there are 12 quarters in three whole ones. Three liters will fill 12 glasses. Now, how many of the same glasses can half a liter of lemonade fill?

In other words, how many quarters are there in one half? As you see, there are two quarters in one half. One half liter of lemonade can fill two glasses. It is not always so easy to find the quotient of two fractions. Therefore, we'll look at a way to divide the fractions that always works.

To compare, we'll use the same example. How many quarters are there in one half or how much is one half divided by one quarter? It can be hard to divide by fraction so we want to remove the fraction from the denominator. How do we do that? We expand the fraction in the denominator so that the denominator becomes one.

To turn a quarter into one, we expand it by four or four oneths. This gives one times four equals four and four times one equals four which gives four fourths which is one. Then we do the same thing with the fraction in the numerator. Expand one half by four or four over one which is one times four equals four and two times one equals two which in turn gives us four halves equals two. The quotient is two over one equals two.

Dividing a number by a quarter or one over four is the same as multiplying this number by four or four over one. Four over one is the inverse or reciprocal of one over four. Therefore, dividing a number by a fraction is the same thing as multiplying this number by the reciprocal of that fraction. Let's apply this rule to another example. 5/6 divided by 4/5.

What do we do? Yes. Instead of dividing 4/5, we can multiply by its reciprocal. How do we find the reciprocal? If you flip 4/5 upside down, you see that the reciprocal of 4/5 is 5/4.

Now, we can multiply 5/6 by 5/4. For the numerator, we get 5 times 5 equals 25. For the denominator, we get 6 times 4 equals 24. Hence, the quotient is 25/24 or 1 1/24 as a mixed number. Memorize the following rule.

Dividing by a fraction is the same as multiplying by a reciprocal. You get the reciprocal of that fraction by flipping it. Just switch the numerator and denominator. For example, 4/1 is the reciprocal of 1/4. When you multiply a number by its reciprocal, the product is always one. For example, 1/4 times 4/1 equals 4/4 equals 1.