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## Arithmetic

### Course: Arithmetic>Unit 15

Lesson 5: Dividing fractions by fractions

# 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.

## 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 a full 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.
• 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.
• My teacher taught me that dividing fractions can be fun when you flip and multiply the second one. Basically if you have 2/3 dived by 5/7, you flip 5/7 to make it 7/5, then change the whole problem to multiplication, and multiply regularly, making it 2/3 x 7/5= 14/15
• I really don't understand why dividing two fractions is the same thing as multiplying by reciprocal. Please explain further
• It's still hard for me too, but this is how I'm understanding things:

A whole doesn't have to mean the number 1.
When you are dividing by a fraction, that fraction is now your 'whole'.

When you divide by 1/3, you are turning 1 original ‘whole’ into 3 different ‘wholes’.
|— — —|

So you are essentially multiplying the original number by 3.

3/1 is the reciprocal of 1/3, and the math just works out that any fraction can be flipped like this to get the correct answer.

Another way of looking at things would be to just go through the division.

What is 8/3 ÷ 1/3?
It becomes 8 ÷ 1 over 3 ÷ 3
Or 8/1.

For 8/3 ÷ 2/3
It becomes 8 ÷ 2 over 3 ÷ 3, or (8 ÷ 2 = 4) over (3 ÷ 3 = 1)
or 4/1 = 4.

But especially with bigger or more complex numbers multiplying by reciprocals will be easier than figuring out division, so they want us to understand this math concept before moving onward.
• Its kinda confusing but im getting the hang of it thanks for the video!
• "My computer's doing something strange..." R.I.P Sal's Computer
• How can I divide something like, 22 5/9 by 1/2?
• 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.
• ok so you saying that 8/3 = something