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Lesson 8: Dividing fractions

# Dividing fractions: 3/5 ÷ 1/2

Dividing two fractions is the same as multiplying the first fraction by the reciprocal of the second fraction. The first step to dividing fractions is to find the reciprocal (reverse the numerator and denominator) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. Finally, simplify the fractions if needed. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• when you are dividing with mixed numbers do you turn them into improper fractions like when you are multiplying with mixed numbers?
• It is very similar. The main difference between multiplying and dividing is that you change it to the reciprocal. The reciprocal is what you multiply the number by to make it 1. For example, 5/3 divided by 2/3 would be the same as 5/3 * 3/2.
• How do you divide fractions that include whole numbers with fractions?
• If you mean perfect fractions it would be best to turn them into an improper fraction first. So if you were to solve for 2/3 ÷ by 2 1/2 then turn it into 2/3 ÷ 5/2 then take the recipricol 2/3 X 2/5 = 4/15
• Will you only and always just use the reciprocal of the second fraction?? Please help...Have a test on this soon!! ( If you need me to clarify pls let me know!!) Thanks!!☺️
• Yes, only the reciprocal of the second fraction is used. For instance, 3/5 ÷ 1/2 = 1 1/5, which is not the same as 5/3 ÷ 1/2 = 3 1/3 or 5/3 ÷ 2/1 = 5/6.
• how if its 1/3 divided by 4
• Well, since 4 = 4/1, you are asking what is 1/3 divided by 4/1. To divide with fractions, you swap the numerator and denominator on the second number, so that is 1/3 times 1/4. 1x1 is 1 and 3x4 is 12, so 1/3 divided by 4= 1/12
• I don't understand -. why do you need the multiplication sign when you can put that there in the first place and what is a reciprocal in mixed numbers, example you are solving 1 2/3 divided by 1 3/4 (IF! because I don't understand this) what is the reciprocal of 1 3/4?? please explain this to me!!
• I'm not sure what you mean by "when you can put that there in the first place" but I think i can explain mixed numbers.

There's no easy way to get a reciprocal with mixed numbers as far as I'm aware. So you want to turn any mixed numbers into impoper fractions. In fact, as you get into higher math you use mixed numbers less and less and improper fractions more and more, to the point I just automatically turn all mixed numbers into improper fractions most of the time.

Anyway, as a quick demonstration I'll turn the two numbers you gave into improper fractions and give their reciprocal.

1 2/3 = 3/3 + 2/3 = 5/3 so the reciprocal is 3/5

1 3/4 = 4/4 + 3/4 = 7/4 so the reciprocal is 4/7

If you wanna work through your other question I'd be happy to, otherwise hope that helps.
• when dividing fractions how do you do it with a whole number
• when there is a whole number you change like if it is 4's then change it to 4/4
• This is a bit unrelated, but it applies to a certain facet of a problem. Does a fraction need to be simplified before you add/subtract/multiply/divide it with another fraction?
For example, 3/8 x 2 - 1/4. Using the PEMDAS method, we multiply 3/8 x 2 first.This equals 6/8. Now, does this 6/8 need to be simplified before we subtract it by 1/4?
• The fractions don't have to be simplified before doing addition or subtraction; we are just converting the fraction(s) into forms which they have common denominators. When fractions have common denominators, you add/subtract the numerators to get the "new numerator's value", and keep the value of the denominator.
Also, sometimes it is better to not simplify fractions so common denominators can be kept.
For the 6/8 minus 1/4, simplifying the 6/8 to 3/4 will give it the same denominator as 1/4, and the only calculation that needs to be done is 3 minus 1 (we "copy and paste" the 4 into the denominator). You can also multiply the 1/4 by 2/2, which gives 2/8, and you can subtract 6/8 by 2/8. However, if the problem requires you to write answer in the simplest form, simplifying fractions in advance may make things easier (e.g. you won't have to divide both the numerator and denominator by a very large value.)
• But why do we have to switch the numerator and denominator and why do we multiply?
• Dividing is the opposite of multiplying, so you flip the fraction upside down to make it the opposite (or reciprocal) of the original fraction. Then, since you have the opposite already of that fraction, you do the opposite of dividing, which is multiplying.
• what is reciprocal
• A reciprocal of a number is one divided by that number. For fractions, this means that the reciprocal is flipping the numerator and the denominator. FOr example, the reciprocal of 1/2 would be 2/1, or 2. The reciprocal of -7 would be -1/7, as you can rewrite -7 as -7/1 and then flip the numerator and denominator. Multiplying by the reciprocal of a number is the same as dividing by that number, and vice versa for division.
(1 vote)
• such old comments makes me feel super young :C
• Indeed. Holy heck, there are comments from 11 years! This is a very old website.

## Video transcript

Divide and write the answer as a mixed number. And we have 3/5 divided by 1/2. Now, whenever you're dividing any fractions, you just have to remember that dividing by a fraction is the same thing as multiplying by its reciprocal. So this thing right here is the same thing as 3/5 times-- so this is our 3/5 right here, and instead of a division sign, you want a multiplication sign, and instead of a 1/2, you want to take the reciprocal of 1/2, which would be 2/1-- so times 2/1. So dividing by 1/2 is the exact same thing as multiplying by 2/1. And we just do this as a straightforward multiplication problem now. 3 times 2 is 6, so our new numerator is 6. 5 times 1 is 5. So 3/5 divided by 1/2 as an improper fraction is 6/5. Now, they want us to write it as at mixed number. So we divide the 5 into the 6, figure out how many times it goes. That'll be the whole number part of the mixed number. And then whatever's left over will be the remaining numerator over 5. So what we'll do is take 5 into 6. 5 goes into 6 one time. 1 times 5 is 5. Subtract. You have a remainder of 1. So 6/5 is equal to one whole, or 5/5, and 1/5. This 1 comes from whatever is left over. And now we're done! 3/5 divided by 1/2 is 1 and 1/5. Now, the one thing that's not obvious is why did this work? Why is dividing by 1/2 the same thing as multiplying essentially by 2. 2/1 is the same thing as 2. And to do that, I'll do a little side-- fairly simple-- example, but hopefully, it gets the point across. Let me take four objects. So we have four objects: one, two, three, four. So I have four objects, and if I were to divide into groups of two, so I want to divide it into groups of two. So that is one group of two and then that is another group of two, how many groups do I have? Well, 4 divided by 2, I have two groups of two, so that is equal to 2. Now, what if I took those same four objects: one, two, three, four. So I'm taking those same four objects. Instead of dividing them into groups of two, I want to divide them into groups of 1/2, which means each group will have half of an object in it. So let's say that would be one group right there. That is a second group. That is a third group. I think you see each group has half of a circle in it. That is the fourth. That's the fifth. That's the sixth. That's the seventh, and then that's the eighth. You have eight groups of 1/2, so this is equal to 8. And notice, now each of the objects became two groups. So you could say how many groups do you have? Well, you have four objects and each of them became two groups. I'm looking for a different color. Each of them became two groups, and so you also have eight. So dividing by 1/2 is the same thing as multiplying by 2. And you could think about it with other numbers, but hopefully, that gives you a little bit of an intuition.