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### Course: Arithmetic (all content) > Unit 5

Lesson 15: Adding and subtracting fractions with unlike denominators- Adding fractions with unlike denominators
- Add fractions with unlike denominators
- Subtracting fractions with unlike denominators
- Subtracting fractions with unlike denominators
- Adding and subtracting 3 fractions
- Solving for the missing fraction
- Add and subtract fractions

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# Adding fractions with unlike denominators

Learn how to add two fractions with different denominators. It can be challenging to combine fractions when the denominators don't match. It is important to find a common denominator. Finally, the resource shows how to find a common multiple of the two denominators in order to convert the fractions so they can be added together.

## Want to join the conversation?

- How do I find the common denominator for 3/4+3/8??(90 votes)
- First, you'll want to figure out whether or not the larger denominator (3/
**8**) is divisible by the smaller one (3/**4**). In this case, the 8 is divisible by 4, (8/4 = 2), so you're going to multiply the smaller one (4) by 2, bringing it to 8. Now, you have to remember, whenever you scale fractions, you have to multiply both, so now it's 6/8+3/8 = 9/8 = 1 1/8. In other cases where the larger denominator isn't divisible by the smaller one, find the LCM (least/smallest common multiple), and scale both fractions so the denominators are equal. Hope this helped!(200 votes)

- Can the Rule of Four be used for what he teaches at3:39? Or is this something different?(36 votes)
- let's say that we have the fraction nine-tenths and I want to add to that the fraction 1/6 1/6 what is this what is this going to equal so when you first look at this you say oh I have different denominators here it's not obvious how I add these and you'd be right and and the way to actually move forward is to find a common denominator to convert both of these fractions into fractions that have a common denominator so how do you think about a common denominator well common denominator is going to be a common multiple of these two denominators of 10 and 6 so what's a common multiple of 10 and 6 and it's usually simplest to find the least common multiple and a good way of doing that is start with the larger denominator here 10 and say okay is 10 divisible by 6 no okay no is 20 divisible by 6 no is 30 divisible by 6 yes 30 is divisible by 6 so I'm just going to the multiples of 10 and saying well what is the smallest multiple of 10 that is divisible by 6 and that's going to be 30 so I can rewrite both of these fractions as something over 30 so 9 over 10 how would I write that as something over 30 well i multiplied the denominator I'm multiplying the denominator by 3 so I've just multiplied the denominator by 3 so if I don't want to change the value of the fraction I have to do the same thing to the numerator I have to multiply that by 3 as well because now I'm just multiplying the numerator by 3 and the denominator by 3 and that doesn't change the value of the fraction so 9 times 3 is 27 so once again 9 tenths and 27 thirtieths represent the same number I've just written it now with the denominator of 30 and that's useful because I can also write 1/6 with the denominator of 30 let's do that so 1/6 is what over 30 I encourage you to pause the video and try to think about it so what did we do to go from 6 to 30 we had to multiply by 5 so if we multiply the denominator by 5 we have to multiply the numerator by 5 as well so 1 times 5 1 times 5 is 5 so 9 tenths is the same thing as 27 thirtieths and one six is the same thing as five xxx and now we can add now we can add and it's fairly straightforward we have a certain number of thirtieths added to another number of thirtieths so27:30it's plus five thirty it's well that's going to be twenty seven that's going to be twenty seven plus five plus five plus five thirtieths plus five xxx which of course is going to be equal to what is this thirty-two xxx thirty-two over thirty and if we want we could try to reduce this fraction we have a common factor of 32 and 30 they're both the divisible by let's see they're both divisible by two so if we divide the numerator and the denominator by two numerator divided by two is sixteen denominator divided by two is 15 so this is the same thing as 16 15 and if I want to write this as a mixed number 15 goes into 16 one time with the remainder one so this is the same thing as one and one 15 let's do another example let's say that we wanted to add we wanted to add one half to two eleven twelfths to 11 over 12 and I encourage you to pause the video and see if you can work this out well like we saw before we want to find a common denominator if these had the same denominator we could just add them immediately but we want to find a common denominator because right now they're not the same well what we want to find is a multiple a common multiple of two and twelve and ideally we'll find the lowest common multiple of two and twelve and just like we did before let's start with the larger of the two numbers 12 and we could just say well this 12 times 1 is 12 so that you could view that as the lowest multiple of 12 and is that divisible by 2 yeah sure 12 is divisible by 2 so 12 is actually the least common multiple of 2 and 12 so we could write both of these fractions as something over 12 so 1/2 is what over 12 well to go from 2 to 12 you multiply by 6 so we'll also multiply the numerator by 6 and we see one half and six twelves these are the same thing one is one is half of two six is half of twelve and how would we write eleven twelve as something over 12 well it's already written as something over twelve eleven twelve already has 12 in the denominator so we don't have to change that eleven twelve and now we're ready to add so this is going to be equal to six this is going to be equal to six plus 11 six plus 11 over 12 over 12 we have six twelve plus eleven twelve is going to be six plus 11 over 12 which is equal to six plus 11 is 17 12 if we want to write it as a mixed number that is what 12 goes into 17 one time with a remainder 5 so 1 + 5 over 12 let's do one more of these this is this is strangely fun all right let's say that we wanted to add we're going to add 3/4 - we're going to add 3/4 - 1 v 2 1 over 5 what is this going to be and once again pause the video and see if you can work it out well we have different denominators here and we want to find this we want to rewrite these so they have the same denominator so we have to find a common multiple ideally the least common multiple so what's the least common multiple of 4 and 5 we'll start with the larger number and let's look at its multiples and keep increasing them until we get one that's divisible by 4 so 5 is not divisible by 4 10 is not divisible by 4 or perfectly divisible by 4 is what we care about 15 is not perfectly divisible by 4 20 is divisible by 4 in fact that is 5 times 4 that is 20 so what we could do is we could write both of these fractions as having 20 in the denominator or 20 as a denominator so we could write 3/4 is something over 20 so to go from 4 to 20 in the denominator we multiply it by 5 so we also do that to the numerator we multiply 3 times 5 to get 15 all I did to go from 4 to 20 multiplied by 5 so to do the same thing to the numerator 3 times 5 is 15 3/4 is the same thing as 15 xx and over here 1/5 what is that over 20 well to go from five to 20 you have to multiply by 4 so to do the same thing in the numerator I have to multiply this numerator times 4 to get 420 it's so now I've rewritten this instead of 3/4 plus 1/5 it's now written as 1520 it's + 4 xx and what is that going to be well that's going to be 15 plus 4 is 1920 it's 1920 it's and we're done(2 votes)

- At1:35to1:38. Since Sal says that 9/10 and 27/10 are the same number, could you have written only 9/10 or 27/30.(20 votes)
- Good Question! You can do it either ways! But sal is not switching the numbers in this video! You could do both but only can do 27/30 when switching numbers!(11 votes)

- I don't understand what he did in3:13, someone please explain(12 votes)
- the fraction was a improper fraction so he made it a proper fraction by saying 15 goes into 16 1 time and there's a re

remainder of 1(14 votes)

- I do not understand what if it is a whole number times a fraction(12 votes)
- If there is a whole number, you would make the whole number look like this: (this is just an example) 3/1. If the whole number is 3, then you just make the whole number over one. 3/1.

I hope this helped you!!

P.S. Sorry for the three-year-late response!!(13 votes)

- what is the best way to multiply fractions(0 votes)
- Say you are multiplying 7/8 times 4/9 it would also be written as 7 times 4 which equals 28 and 8 times 9 equals 72 giving you an answer of 28/72. If you didn't understand you multiply the numerator by the numerator and the denominator by the denominator then combine the two to get your answer.(31 votes)

- How do you make fractions into decimals.(12 votes)
- To convert fractions into decimals, divide the denominator into 100, then multiply the answer by the numerator, and finally add the decimal point, i guess. For example, 3/5 is equal to 0.6 because 5 going into 100 is 20 which would make the decimal 0.2. The numerator is 3 and 0.2 x 3 = 0.6. I hope this helped!(8 votes)

- 6 13/21

+ 5 3/7 fraction addition(9 votes)- Yes, it is fraction +, but it might take a bit to solve.(0 votes)

- For improper fractions, for example 32/30, does the denominator always have to be equal? If yes, why?(4 votes)
- When you're adding/subtracting, the denominators have to be equal regardless of whether the fraction is proper or improper. We can easily think of an improper fraction as a mixed number, right? We have a whole number plus a fractional number. For example, if we add two improper fractions that are converted to mixed numbers, it could look like this:

5/3 + 28/5 = ?

1 2/3 + 5 3/5 = ?

(1 + 2/3) + (5 + 3/5) = ?

Then, using the associative property, we can switch around the parenthesis like this:

(1 + 5) + (2/3 + 3/5) = ?

Now you see we have a completely normal unlike denominator problem, and we just add 6 to the answer. As with any unlike denominator adding/subtracting problem, you have to get the fractions to the same denominator. In this case, it would be a denominator of 15.

= 6 + 10/15 + 9/15

= 6 + 19/15

= 6 + 1 + 4/15

= 7 4/15, or 109/15(11 votes)

- Does anyone know why when adding 2 fractions you would multiply each fraction by 1?

I am reading a book and it does not seem clear to me how or why it works or why you would use it at all rather than the easy cross multiplication method. I just don't want to ignore it just in case there are situations where it needs to be used which I find is always the case that there are rules that work for some combination of things and not others.

The book I am reading has stumped me in the addition of fractions. Here is a snippet of it:

1/7 + 3/4

"Here, we can neither convert sevenths to quarters nor quarters to sevenths, so we’re going to learn a method that you can use to add or subtract any fractions. To get identical bottom numbers, we begin by multiplying each number by 1, written as quarters in one case, and as sevenths in the other."

1x 1/7 + 1x3/4 = 4/4 x 1/7 + 7/7 x 3/4

I have no idea how they got the 4/4 and 7/7 try as I might and how the strange-looking order of the terms after the equal sign even if I overlooked how they got those numbers to begin with.

I thought I worked out how they did it but my brain refused to accept what it thought was either a convoluted approach or that I was creating my own incorrect method lol

Can anyone provide a possible clarification of why they would suggest using this method rather than the easy cross multiplication method?(6 votes)- Some basic info you need to know:

1) The denominator of a fraction tells you the size of each portion. Visualize a pizze that was cut into 8 slice. Each slice is 1/8 of the pizza. If a pizza is cut int 4 slices, then each slice is 1/4 of the pizza and a slice is twice as big as a slice of 1/8

2) To add & subtract fractions we need to work with fractions of the same size. So, we have to force the denominators to have a common value. This is called a "common denominator".

3) Equivalent fractions are created by multiplying both numerator & denominator by the same value. Why? This is based upon the identity property of multiplication: Any number times 1 = the original number. So, by multiplying a fraction by 4/4 = 1, we aren't changing the value of the fraction. We're just converting it to an equivalent value.

Now to your questions - Where did the 4/4 and 7/7 come from?

Since the fractions in the video don't have a common denominator, they need to be converted to have a common denominator. So, we start by find the lowest common multiple(LCM), also called the lowest common denominator (LCD) of the fractions. The LCM for 4 and 7 is 28. There are lessons on finding an LCM at: https://www.khanacademy.org/math/pre-algebra/pre-algebra-factors-multiples/pre-algebra-lcm/v/least-common-multiple-exercise

To convert the fraction 1/7 to have a denominator of 28, you ask yourself the question: what times 7 will create 28? The answer is 4. You do the same thing to determine how to convert 3/4 to have a denominator of 28. What times 4 will create a denominator of 28? The answer is 7. These values 4 and 7 become 4/4 and 7/7 because we have to multiply each fraction by a value = 1 to have an equivalent fraction.

Hope this helps.(7 votes)

## Video transcript

- [Voiceover] Let's say that
we have the fraction 9/10, and I want to add to
that the fraction 1/6. What is this, what is this going to equal? So when you first look at this, you say, "Oh, I have different denominators here. It's not obvious how I add these." And you'd be right and the way to actually move forward is to find
a common denominator, to convert both of these fractions into fractions that have a common denominator. So how do you think about
a common denominator? Well, a common denominator's
gonna have to be a common multiple of these two
denominators of 10 and six. So what's a common multiple of 10 and six? And it's usually simplest to
find the least common multiple, and a good way of doing that
is start with the larger denominator here, 10, and say,
okay is 10 divisible by six? No. Okay, now, is 20 divisible by six? No. Is 30 divisible by six?
Yes. 30 is divisible by six. So I'm just going through
the multiples of 10 and saying, "Well what is
the smallest multiple of 10 that is divisible by six?"
And that's going to be 30. So I could rewrite both of these fractions as something over 30. So nine over 10. How would I write that as something over 30? Well I multiply the denominator, I'm multiplying
the denominator by three. So I've just multiplied
the denominator by three. So if I don't want to change
the value of the fraction, I have to do the same
thing to the numerator. I have to multiply that by three as well because now I'm just multiplying
the numerator by three and the denominator by three,
and that doesn't change the value of the fraction. So nine times three is 27. So once again, 9/10 and 27/30 represent the same number. I've just written it now
with a denominator of 30, and that's useful because
I can also write 1/6 with a denominator of 30. Let's do that. So 1/6 is what over 30? I encourage you to pause the video and try to think about it. So what did we do go from six to 30? We had to multiply by five. So if we multiply the denominator by five, we have to multiply the
numerator by five as well, so one times five, one times five is five. So 9/10 is the same thing as 27/30, and 1/6 is the same thing as 5/30. And now we can add, now we can add and it's fairly straightforward. We have a certain number of 30ths, added to another number of 30ths, so 27/30 + 5/30, well that's going to be 27, that's going to be 27 plus five, plus five, plus 5/30, plus 5/30, which of course going to be equal to 32/30. 32 over 30, and if we want, we could try
to reduce this fraction. We have a common factor of 32 and 30, they're both divisible by two. So if we divide the numerator
and the denominator by two, numerator divided by two is 16, denominator divided by two is 15. So, this is the same thing
as 16/15, and if I wanted to write this as a mixed
number, 15 goes into 16 one time with a remainder one. So this is the same thing as 1 1/15. Let's do another example. Let's say that we wanted
to add, we wanted to add 1/2 to to 11/12, to 11 over 12. And I encourage you to pause the video and see if you could work this out. Well like we saw before, we wanna find a common denominator. If these had the same denominator, we could just add them immediately, but we wanna find a common denominator because right now they're not the same. Well what we wanna find is a multiple, a common multiple of
two and 12, and ideally we'll find the lowest common
multiple of two and 12, and just like we did before,
let's start with the larger of the two numbers, 12. Now we could just say
well 12 times one is 12, so that we could view that
as the lowest multiple of 12. And is that divisible by two? Yeah, sure. 12 is divisible by two. So 12 is actually the least
common multiple of two and 12, so we could write both of these fractions as something over 12. So 1/2 is what over 12? Well to go from two to
12, you multiply by six, so we'll also multiply
the numerator by six. Now we see 1/2, and 6/12,
these are the same thing. One is half of two, six is half of 12. And how would we write
11/12 as something over 12? Well it's already written
as something over 12, 11/12 already has 12 in the denominator, so we don't have to change that. 11/12, and now we're ready to add. So this is going to be equal to six, this is going to be equal to six plus 11, six plus 11 over 12. Over 12. We have 6/12 plus 11/12, it's gonna be six plus 11 over 12, which is equal to, six plus 11 is 17/12. If we wanted to write
it as a mixed number, that is what, 12 goes
into 17 one time with a remainder of five, so 1 5/12. Let's do one more of these. This is strangely fun. Alright. Let's say that we wanted to add, We're gonna add 3/4 to, we're gonna add 3/4 to 1/5. To one over five. What is this going to be? And once again, pause the video and see if you could work it out. Well we have different denominators here, and we wanna find, we wanna rewrite these so they have the same denominators, so we have to find a common multiple, ideally the least common multiple. So what's the least common
multiple of four and five? Well let's start with the larger number, and let's look at its
multiples and keep increasing them until we get one
that's divisible by four. So five is not divisible by four. 10 is not divisible by four,
or perfectly divisible by four is what we care about. 15 is not perfectly divisible by four. 20 is divisible by four, in
fact, that is five times four. That is 20. So what we
could do is, we could write both of these fractions as
having 20 in the denominator, or 20 as the denominator. So we could write 3/4
is something over 20. So to go from four to
20 in the denominator, we multiplied by five. So we also do that to the numerator. We multiply by three times five to get 15. All I did to go from four
to 20, multiplied by five. So I have to do the same
thing to the numerator, three times five is 15. 3/4 is the same thing
as 15/20, and over here. 1/5. What is that over 20? Well to go from five to 20,
you have to multiply by four. So we have to do the same
thing to the numerator. I have to multiply this
numerator times four to get 4/20. So now I've rewritten this
instead of 3/4 plus 1/5, it's now written as 15/20 plus 4/20. And what is that going to be? Well that's going to be
15 plus four is 19/20. 19/20, and we're done.