6th grade (Eureka Math/EngageNY)
- Understanding division of fractions
- Dividing a fraction by a whole number
- Divide fractions by whole numbers
- Meaning of the reciprocal
- Dividing a whole number by a fraction
- Dividing a whole number by a fraction with reciprocal
- Divide whole numbers by fractions
- Dividing fractions: 2/5 ÷ 7/3
- Dividing fractions: 3/5 ÷ 1/2
- Dividing fractions
- Dividing mixed numbers
- Divide mixed numbers
- Writing fraction division story problems
- Interpret fraction division
- Dividing whole numbers & fractions: t-shirts
- Area with fraction division example
- Dividing fractions word problems
- Dividing fractions review
Using a number line, we'll explain why multiplying by the inverse is the same as dividing. Created by Sal Khan.
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- OK, here we go with my question. I easily understand how to multiply and divide fractions. I have watched these videos over and over and still do not understand conceptually WHY I flip the reciprocal and multiply across to get the answer. Is there another source to read or watch to explain why these steps work and what is actually happening?(14 votes)
- This is a great question and I'm glad you are thinking at a deeper level in mathematics. The properties of numbers are useful for explaining why we flip the second fraction and multiply, when we divide fractions.
For b, c, and d nonzero, the problem a/b divided by c/d is the answer to the question "what number times c/d is a/b?" This is because of division's inverse relationship with multiplication (basic definition of division), but now applied to fractions.
Because any number times 1 is itself, d/c * c/d = dc/cd = cd/cd = 1, and factors can be grouped in any order (associative property of multiplication), we have
a/b = a/b * 1 = a/b * (d/c * c/d) = (a/b * d/c) * c/d.
So a/b * d/c is the answer to the question "what number times c/d is a/b?"
Therefore, a/b divided by c/d equals a/b * d/c.(2 votes)
- how do you divide fraction without a number line(2 votes)
- With KCF (Keep, change, flip). For example, if you had 3/5 divided by 10/6, you would keep the first number(3/5) the same; change the sign(÷) to the opposite, in this case it is ×; then flip the second fraction(10/6), it will be 6/10. Now your equation should be 3/5 × 6/10. then you multiply across. You would do 3 × 6, which is 18. Then 5 × 10, which is 50. So your answer is 18/50, or 9/25 (simplified). Hope this helps!(3 votes)
- why we have to multiply it with the reciprocal when dividing is the case ?(3 votes)
- The goal is to make the division expression look like just one number, perhaps a fraction or mixed number, but, still just one number. Multiplying by the reciprocal and multiplying by 1 result in "the product of the first fraction and the reciprocal of the second(1 vote)
- ok so you saying that 8/3 = something(2 votes)
- I noticed that in the video the fraction 1/3 became 3/1 after taking the reciprocal. Why did it not happen in the case of 8/3?(3 votes)
- wouldn't 8/3 divided by 1/3 = 8/1 because 3 goes into 3, 1 time I'm just confused?(2 votes)
- Yes, you’re on the right track! The only remaining step is to simplify 8/1 to get 8.
Have a blessed, wonderful day!(0 votes)
Let's think about what it means to take 8/3 and divide it by 1/3. So let me draw a number line here. So there is my number line. This is 0. This is 1. And this is 2. Maybe this is 3 right over here. And let me plot 8/3. So to do that, I just need to break up each whole into thirds. So let's see. That's 1/3, 2/3, 3/3, 4/3, 5/3, 6/3, 7/3, 8/3. So right over here. And then of course, 9/3 would get us to 3. So this right over here is 8/3. Now, one way to think about 8/3 divided by 3 is what if we take this length. And we say, how many jumps would it take to get there, if we're doing it in jumps of 1/3? Or essentially, we're breaking this up. If we were to break up 8/3 into sections of 1/3, how many sections would I have, or how many jumps would I have? Well, let's think about that. If we're trying to take jumps of 1/3, we're going to have to go 1, 2, 3, 4, 5, 6, 7, 8 jumps. So we could view this as-- let me do this in a different color. I'll do it in this orange. So we took these 8 jumps right over here. So we could view 8/3 divided by 1/3 as being equal to 8. Now, why does this actually make sense? Well, when you're dividing things into thirds, for every whole, you're now going to have 3 jumps. So whatever value you're trying to get to, you're going to have that number times 3 jumps. So another way of thinking about it is that 8/3 divided by 1/3 is the same thing as 8/3 times 3. And we could either write it like this. We could write times 3 like that. Or, if we want to write 3 as a fraction, we know that 3 is the same thing as 3/1. And we already know how to multiply fractions. Multiply the numerators. 8 times 3. So you have 8-- let me do that that same color. You have 8 times 3 in the numerator now, 8 times 3. And then you have 3 times 1 in the denominator. Which would give you 24/3, which is the same thing as 24 divided by 3, which once again is equal to 8. Now let's see if this still makes sense. Instead of dividing by 1/3, if we were to divide by 2/3. So let's think about what 8/3 divided by 2/3 is. Well, once again, this is like asking the question, if we wanted to break up this section from 0 to 8/3 into sections of 2/3, or jumps of 2/3, how many sections, or how many jumps, would I have to make? Well, think about it. 1 jump-- we'll do this in a different color. We could make 1 jump. No, that's the same color as my 8/3. We could do 1 jump. My computer is doing something strange. We could do 1 jump, 2 jumps, 3 jumps, and 4 jumps. So we see 8/3 divided by 2/3 is equal to 4. Now, does this make sense in this world right over here? Well, if we take 8/3 and we do the same thing, saying hey, look, dividing by a fraction is the same thing as multiplying by a reciprocal. Well, let's multiply by 3/2. Let's multiply by the reciprocal of 2/3. So we swap the numerator and the denominator. So we multiply it times 3/2. And then what do we get? In the numerator, once again, we get 8 times 3, which is 24. And in the denominator, we get 3 times 2, which is 6. So now we get 24 divided by 6 is equal to 4. Now, does it make sense that we got half the answer? If you think about the difference between what we did here and what we did here, these are almost the same, except here we really just didn't divide. Or you could say you divided by 1, while here you divided by 2. Well, does that make sense? Well, sure. Because here you jumped twice as far. So you had to take half the number of steps. And so in the first example, you saw why it makes sense to multiply by 3. When you divide by a fraction, for every whole, you're making 3 jumps. So that's why when you divide by this fraction, or whatever is in the denominator, you multiply by it. And now when the numerator is greater than 1, every jump you're going twice as far as you did in this first one right over here. And so you would have to do half as many jumps. Hopefully that makes sense. It's easy to think about just mechanically how to divide fractions. Taking 8/3 divided by 1/3 is the same thing as 8/3 times 3/1. Or 8/3 divided by 2/3 is the same thing as 8/3 times 3/2. But hopefully this video gives you a little bit more of an intuition of why this is the case.