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# Understanding division of fractions

Using a number line, we'll explain why multiplying by the inverse is the same as dividing. Created by Sal Khan.

## Want to join the conversation?

• 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?
• 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. [R]
• 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.
• 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.
• Its kinda confusing but im getting the hang of it thanks for the video!
• 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.
• 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.
• This doesn't make sense!
• If you need help then read this.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.
• For anyone having a hard time with this;

Whenever you divide 2 fractions, you leave the first fraction the way it is, but you flip the second fraction.

For example:
2/3 divided by 4/5, becomes 2/3 multiplied by 5/4.
(2/3)*(5/4) is (10/12), which is (5/6) once simplified.
Thus, our answer would be 5/6.
Remember, once you flip the second fraction, you need to multiply the fractions, not divide them.
However, if you divide more than two fractions, you do need to flip every fraction except for the first fraction, and then you would multiply everything together (and simplify if needed).

Hope this helps :)
• ok so you saying that 8/3 = something
(1 vote)
• Yeah, you would need to divide the top from the bottom.