Hi and welcome to Math Antics. This video is all about dividing fractions,
but in order to understand how dividing fractions works, we first need to learn about something called reciprocals.
A reciprocal is just a fancy math term for what you get when you switch the top and bottom numbers of a fractions.
For example, if you have the fraction 1 over 2 and then switch the top and bottom numbers, you’ll end up with 2 over 1.
2 over 1 is the reciprocal of 1 over 2.
And, 1 over 2 is the reciprocal of 2 over 1.
And an interesting thing about reciprocals is; multiplying a fraction by its own reciprocal will always give you ‘1’.
That’s because you’ll have the same multiplication problem on the top and bottom,
so you’ll end up with a whole fraction which is always ‘1’.
Okay, that’s nice, but what do reciprocals have to do with dividing fractions?
Well, reciprocals let us do a really cool trick that makes dividing fractions easy!
Whenever you have to divide something by a fraction, you can just multiply it by the reciprocal of that fraction instead,
and you’ll get the correct answer.
And that’s great news because multiplying fractions is so simple.
This trick of multiplying by the reciprocal works because fractions are really just mini division problems,
so when you multiply something by 1 over 2, it’s the same as dividing by 2, since 2 is below the fraction’s division line.
AND… dividing by 2 is the same as dividing by 2 over 1,
because you can turn any number into a fraction by just writing a ‘1’ as the bottom number, right?
But look… reciprocals! That’s why multiplying by 1 over 2 is the same as dividing by 2 over 1.
And it’s true the other way around too.
So really, it’s kind of like you never have to divide fractions.
You can just re-write your division problems so that you’re multiplying by the reciprocal instead.
Then when you multiply, you’ll get the answer for the original division problem.
As always, let’s see a couple examples of how this works so you’ll really understand.
Let’s try this problem: 3 over 4, divided by 2 over 7.
Okay, so the first thing we want to do is re-write our problem.
Instead of dividing by 2 over 7, we can multiply by the reciprocal instead.
The reciprocal of 2 over 7, is 7 over 2, so our problem becomes 3 over 4, times 7 over 2.
Oh, I should mention a mistake that a lot of students make when they first learn to divide fractions.
Sometimes students take the reciprocal of the first fraction (the one that’s being divided),
or even the reciprocal of both fractions… but you only want to take the reciprocal of the second fraction (the one you are dividing BY).
Okay, now that our problem has been changed to multiplication, it’s easy!
Just multiply the tops (3 times 7 equals 21)
and multiply the bottoms (4 times 2 equals 8)
and we have the answer to our fraction division problem.
So, 3 over 4, divided by 2 over 7, is 21 over 8.
So that’s pretty easy, but let’s try one more example.
Let’s try 15 over 16, divided by 9 over 22.
Again, the first thing we want to do is re-write our problem.
We’ll change the ‘divided by 9 over 22’ into ‘times 22 over 9’.
Now all we have to do is multiply, but since these numbers are kinda big,
I’m going to use my calculator to help. Let’s see here… so we have… alright!
On the top we have, 15 times 22 equals 330,
and on the bottom, we have16 times 9 equals 144.
So the answer to our division problem is 330 over 144.
Of course that could be simplified for your final answer on a test, but we cover simplifying fractions in another video.
Alright, that’s how you divide fractions; you just multiply by the reciprocal and you have your answer.
But, there’s one more thing I want to show you.
You already know that the line between the top and bottom number of a fraction is just another form of the division symbol.
Well, that means you'll sometimes see fraction division problems written like this…
This shows the top fraction (2 over 3) being divided by the bottom fraction (4 over 5).
It’s really just that we have a fraction made up from other fractions.
The top number is a fraction and the bottom number is a fraction.
It just looks a little confusing because we have all these fractions lines here.
But, we can make it look a lot better.
Let’s re-write this as a multiplication problem by taking the reciprocal of the bottom number (the fraction that we are dividing BY)
and multiplying it by the fraction on top.
There, that looks easier to do, and it’s really the same problem!
We just need to multiply to get the answer.
So 2 times 5 equals 10, and 3 times 4 equals 12.
Okay, so there you have it. What sounded really hard turns out to be as easy as flipping fractions upside-down.
If you can multiply fractions, then you can divide fractions too!
Don’t forget to practice what you’ve learned by doing the exercises for this section.
Thanks for watching and see ya next time.
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