Arithmetic (all content)
Understanding division of fractions
Dividing fractions can be understood using number lines and jumps. To divide a fraction like 8/3 by another fraction like 1/3, count the jumps of 1/3 needed to reach 8/3. Alternatively, multiply 8/3 by the reciprocal of the divisor (3/1) to get the same result. This concept applies to other fractions, such as dividing 8/3 by 2/3. 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?(18 votes)
- You have a very good question. Think about it this way: a fraction itself is a division problem, the numerator divided by the denominator. When you multiply by a value greater than 1, the original amount becomes greater; when you multiply by a value less than 1, the original value becomes smaller.
You want to find a way to "move the dividend into the denominator of the divisor". The easiest way to do this is to change the dividend into its inverse. For whole numbers, the value moves to the denominator, and the "invisible 1" (since anything divided by 1 is itself) goes to the numerator. To find the numerator you multiply the numerators together; to find the denominator you multiply the denominators together.
The same works for values which are not whole numbers.
Here are two examples:
10÷(2/5). The 2 is a whole number, which is already being divided by 5. Division by a number greater than 1 means the value is being "reduced" (and 10÷2 means the 10 is getting reduced to one-half). But at the same time, the "reduction power of the 2" is also being reduced, reduced to one-fifth. So in order to compensate for that, you will need to multiply the 10 by a 5 as well. In the end, there will be a 5 in the numerator and a 2 in denominator, and the quotient of the problem is 25.
When you divide by a fraction greater than 1, the original value is still reduced. Let's say you have 60÷(5/4). Without the "1/4", the 60 would be divided by 5; but the "1/4" reduces the "reduction power", and the quotient will be 48 in the end.
Hope this helps in better understanding and clearing things up.(23 votes)
- I noticed in the case of 8/3 / 1/3 (best I can do to type a fraction division math problem on here, but basically eight thirds (8/3) divided by one third (1/3)), and likewise 8/3 / 2/3, that the denominators were both 3 in both cases, and the answer was in the same as the first numerator (8 in both cases) divided by the second numerator (1 in the first case, 2 in the second case), but a whole number and not a fraction. Is this a common pattern when dividing fractions with common denominators or are there exceptions, and would finding common denominators and dividing the numerators be one method of dividing fractions, or would it just be extra work compared to inverting and multiplying? Might be something to play around with.(17 votes)
- Yes the pattern you noticed will always work, but you are wise to suspect atet he most efficient way to divide fractions is usually to simply multiply by the reciprocal. I always remind my students to simplify the product before they multiply numerators and denominators as they are already partially factored which makes finding common factors to eliminate easier.(4 votes)
- This doesn't make sense!(5 votes)
- I noticed in the case of 8/3 / 1/3 (best I can do to type a fraction division math problem on here, but basically eight thirds (8/3) divided by one third (1/3)), and likewise 8/3 / 2/3, that the denominators were both 3 in both cases, and the answer was in the same as the first numerator (8 in both cases) divided by the second numerator (1 in the first case, 2 in the second case), but a whole number and not a fraction. Is this a common pattern when dividing fractions with common denominators or are there exceptions, and would finding common denominators and dividing the numerators be one method of dividing fractions, or would it just be extra work compared to inverting and multiplying? Might be something to play around with.(3 votes)
- To correctly type it, you should have put in parentheses like you did in the description (8/3)/(1/3). It is a pattern if you could reduce the two numerators and/or two denominators before flipping. Your examples do work out to become whole numbers because the denominators cancel to be 1. You could extend your correct theory to harder problems such as (8/3)/16/9) Note 8/16=1/2 and 3/9=1/3, so the answer will end up as 3/2 (since 1/3 will reciprocate). 8/3 * 9/16 does this exact thing. If you have (8/9)/(16/3) you have 8/16=1/2 and 9/3=3 (reciprocate to get 1/3), so answer would be 1/2 * 1/3 =1/6.
This is quite an astute observation, good math.(4 votes)
- Its kinda confusing but im getting the hang of it thanks for the video!(4 votes)
- ok so you saying that 8/3 = something(2 votes)
- Yeah, you would need to divide the top from the bottom.(4 votes)
- How can I divide something like, 22 5/9 by 1/2?(1 vote)
- To divide a mixed number by a fraction, you will have to convert it into an improper fraction. This means 22 5/9 would turn into 203/9.
Dividing 203/9 by 1/2, is the same as multiplying 203/9 by 2/1.
203/9 x 2/1 is 406/9. This can be converted back into a mixed number: 45 1/9
Hope this helped.(6 votes)
- For example, If equation is 3/4 ÷ 2/5, How can it be represented as a number line?(3 votes)
- Why is it so complicated is there an easier way? Because in a test I cant do all that so is there an easier way?(3 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)
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.