How to Multiply and Divide Fractions
Multiplying and Dividing Fractions
Many students have a real fear of fractions. However, if you can remember what a fraction represents and a few mathematical rules on how to work with them algebraically, you will be able to face fractions with confidence. In this video, we will review how to multiply and divide fractions. Let’s get started.
We should start by defining exactly what a fraction is. A fraction represents a ratio of a “part” to a “whole”, or part over whole. The value above the division line is referred to as the numerator, and the value below the division line is the denominator.
To multiply fractions, simply multiply “straight across”, meaning the “numerator times the numerator” divided by the “denominator times the denominator”. Let’s look at a couple of quick examples:
Here, we want to multiply two thirds by two fifths. As we said earlier, we’re going to multiply straight across. So we’re going to have 2 times 2, over, 3 times 5. Which is equal to four over fifteen. So our answer is four-fifteenths.
Now let’s try another one. We’re going to try four-sevenths times three-elevenths.
Again it’s the same concept. We’re going to multiply 4 times 3, divided by 7 times 11. Which gives us 12 over 77, twelve-seventy-sevenths.
Pretty simple, right? Now let’s take a look at dividing fractions.
Dividing fractions involves a slightly different process. Before we jump into the mechanics of the process, let’s start by looking at an intuitive example of dividing a fraction by two. The effect of dividing by 2 is simply cutting the fraction in half, or simply multiplying the fraction by 1 over 2.
So, four-fifths divided by 2 is really the same as saying four-fifths times one-half. Then it’s going to be multiplied across just like we did before. So we have 4 times 1 is four, over 5 times two is ten. Which then simplifies to 2 over 5.
So in other words, two-fifths is half the size of four-fifths.
Similarly, dividing a fraction by 3 would result in a fraction that is one-third the size of the original:
Two-fifths divided by is the same as saying two-fifths times one-third, which gives you two-fifteenths.
So, two-fifteenths is one third the size of two-fifths.
Before we generalize this process, let’s review some important terminology. Consider the relationship between 2 and one-half. These numbers are called reciprocals of one another, which means that the numerator of one number is the denominator of the other, and vice versa. Remember that “2” can be written as a fraction by writing it over “1”, like this: 2 over 1. Therefore 2 over 1 and one-half are reciprocals. The same is true of 3 and one-third, because 3 can be written as 3 over 1.Therefore 3 and one-third are reciprocals.
With this in mind, what pattern do you see in the process for dividing fractions?
The process of dividing fractions is the same as multiplying the first fraction by the reciprocal of the second. A “shorthand” version of this wordy explanation that may help you remember the division process is “Keep, Change, Flip”:
You “Keep” the first fraction as is;
Then you “Change” the operation from division to multiplication;
And you “Flip” (or take the reciprocal of) the second fraction.
Once this adjustment is made, simply follow the rules for multiplying fractions by multiplying the numerators and dividing by the product of the denominators.
Here is an example using the “keep, change, flip” process:
Say we want to divide three-fifths by seven-fifths. We’ll keep the first fraction as is, change the operation from division to multiplication, and flip the second number. Now we just multiply our numerators, 3 times 5 is fifteen, over, 5 times 7 is thirty-five. And then from there we simplify to three-sevenths.
I hope this video was helpful! Thanks for watching, and happy studying!