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Lesson 8: Dividing fractions

# Meaning of the reciprocal

Reciprocals are essential in understanding fractions. To find the reciprocal of a fraction, simply swap the numerator and the denominator. The product of a number and its reciprocal always equals 1. Reciprocals help us determine how many times a specific fraction fits into the whole number 1, making them valuable in various mathematical operations. Created by Sal Khan.

## Want to join the conversation?

• I don't really think I understand
• Well, the reciprocal is just swap the fraction, like the reciprocal of 1/2 is 2/1. Hope this helped!
• Hey guys Honestly there is a way easier way it is called KCF which is aka keep, change, flip (KFC IS KENTUCKY FRIED CHICKEN).

1/2 divided by 5
keep 1/2 change dividing into multiplying flip 5 into 1/5 now solve which = 1/10

I hope this helps
-QUEEN⚔
• That is what I'm learning in class.
• I don't get it, what does reciprocals mean please?
• The reciprocal is simply just the fraction but flipped upside down, so 2/3 will become 3/2, hope it answered your question!
• Sharing this visualisation I had to doodle to pin down reciprocals :
``0            2/3 ×   1|------l------l------| (=)0             1  ÷  3/2``
• ya idont get it
• Why is it called the reciprocals
• It's from the latin "reciprocus" which means returning, hope it helps!
• Remember this cheat sheet. KCF: Keep Change Flip. Sounds like off-brand KFC...
• How does simply flipping a fraction immediately give you the amount of times that fraction goes into 1?
• Because the fraction itself is already a division. The numerator divided by the denominator. So, if you do the opposite division action, then that means everything was restored (or no division action was made at all). That is why flipping the fraction immediately gives the inverse, the amount of times that fraction goes into one.
• If the reciprocal of 5/4 is 4/5, would 8/10 be considered 5/4 reciprocal?
• Kind of, but the reciprocal is simplified, so not really.
• I need everyone to see that the reciprocal is not just swapping the numerator and denominator around. It's that when the 2 fractions are multiplied they have a total of 1. Even Sal said it himself. " At , when you multiply it by the reciprocal, you get one."

## Video transcript

- [Instructor] Let's talk a little bit about reciprocals. Now, when you first learn reciprocals some folks will immediately tell you, Hey, just swap the numerator and the denominator. So for example, if I have the fraction 2/3, the reciprocal of 2/3, if I swap the numerator and the denominator, is 3/2. If I had the fraction 5/6, the reciprocal of that is going to be six over five. And that's all well and good, but what does this actually mean? Well, one interpretation of a reciprocal is it's the number that when you multiply it by the original number, you get one. So 2/3 times 3/2 we'll see is equal to one or 5/6 times 6/5 is equal to one. Another way to think about reciprocals are, how many of that number or how many of that fraction fit into one? So if I were to take one and I divide it by 2/3, one interpretation of this is saying how many 2/3 fit into one. If I take one divided by 5/6, an interpretation of this is how many 5/6 fit into one. And we'll see that 3/2 of a 2/3 fit into one. And we'll see that in a second. Or that 6/5 of a 5/6 fit into one. So let's start with a very straightforward example. Let's say that I have the fraction 1/2. So if I have 1/2. If that whole rectangle is a whole, this is 1/2 here. So if I were to ask how many one halves fit into one, so one divided by 1/2. How many one halves fit into one? Well, I have one 1/2 right it over here. And then I would have another 1/2 right over there. So we have two 1/2. So this is equal to two. Now you might be saying, wait, two doesn't look like I just swapped the numerator and the denominator but you have to realize that two is the same thing as two wholes. So the reciprocal of 1/2 is indeed two over one. Or if you take two over one, or if you have two 1/2, that is indeed going to be equal to one. But now let's work on 2/3, things that are a little bit more nuanced. So 2/3 here, I can shade that in. That's 1/3 and then 2/3. So this right over here is 2/3. Now how many of these fit into one? If we were to say what's one divided by 2/3? Well, we can clearly get a whole 2/3 into one and then we can get another third, which is half of a 2/3. So we can have a whole 2/3 and then half of a 2/3, or one and a half 2/3. So we could say one divided by 2/3 is equal to one and a half. Well, one and a half is the exact same thing as three halves. So once again, you can see that 3/2 times 2/3 is equal to one or that 3/2 of a 2/3 fit into one. Let's do another example. If we were to think about 3/2. So 3/2 would be, let's see that's a half, that's two halves, and then this is three halves right over here. So let me mark all of that. So this whole thing right over here is 3/2. Now how many 3/2 fit into a whole? Well, you can see that you can't even fit a whole 3/2 into a whole. You can only fit two of the three halves. So one half, two halves of the three halves. So what you can see here is that this is 2/3 of the 3/2. So if you say one divided by 3/2, how many 3/2 can fit into one? Well, you can only fit 2/3 of a 3/2 into one. And this is interesting because the reciprocal of 2/3 is 3/2, and the reciprocal of 3/2 is 2/3.