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

Lesson 17: Adding and subtracting fractions with unlike denominators word problems

# Adding fractions word problem: paint

Learn how to add and subtract fractions with unlike denominators through a real-world problem. Watch as the problem is broken down step-by-step, practice finding common denominators, and apply this knowledge to determine if the sum or difference of the fractions meets a specific requirement. Created by Sal Khan.

## Want to join the conversation?

• question when looking for a common denominator
must it be the lowest you can find or can it be just any?
(2 votes)
• A common denominator cannot be "any" number, because it has to fit both fractions. However, it can be "any" fitting number. The reason you want it to be a smaller number is that it makes adding and simplifying easier. (see simplifying fractions) I hope this helped!
(22 votes)
• you are confusing me about splitting the bar graph
(5 votes)
• Splitting the bar graph really isn't at all confusing.It means that you divide something into fractions so you can understand better.The best object to use is a bar graph.Sal divided the first two into different fractions.Then, when you converted the two into fractions with same denominators,you add them together.When you add it,you get the answer,which Sal demonstrated in the third bar graph. So it is not confusing at all.Thanks for reading this reply!
(12 votes)
• thank u for helping me i think ima get a 100
(8 votes)
• Hope you do
(0 votes)
• "Since 5 and 2 are both prime numbers, the smallest number (least common multiple) is just going to be their product."

Is there an explanation for this?
(3 votes)
• Yes there is. Your getting into number theory for why that is the case. As well as formal math logic and proofs. There is a key theorem involved which would require far more math to put into this small space. But
Fundamental Theorem of Arithmetic:
Every integer greater than 1 (because 1 isn't prime), either is prime itself or is the product of a unique combination of primes.

Now I'm not going to shove a formal proof down you. Just some thoughts. LCM is the smallest value that can be cleanly divided by both numbers. So if x = 2 * 3 * 7 = 42 and y = 2 * 2 * 5 = 20
I'll use | to seperate, but when say 2 occurs on one side and 2 * 2 = 2^2 on the other, eliminate the one with the smaller exponent.
3 * 7 | 2 * 2 * 5 - the two on the left can be eliminated.
3 * 7 | 2 * 2 * 5 well the LCM is now 2*2*3*5*7=420

I'll also expand it so that you know it's not just prime * prime that is the smallest x * y for LCM.

Two numbers are relatively prime if they share no prime numbers in common. Thus if x and y are relatively prime or prime then LCM = x * y

Example: x = 6 = 2 * 3, y = 65 = 5 * 13
2 * 3 | 5 * 13 no way to eliminate any primes so the LCM is 6 * 65 = 390
(2 votes)
• At 4.40 from this video I see how to do this but what do you do if theres are three numbers to add or subtract. How do you do that?
(2 votes)
• How come when you have prime numbers, the smallest common multiple is them times each other? (Example: 5 and 2.) At - , Sal said that when both numbers were prime, you just needed to multiply them times each other. Why does that work?
(1 vote)
• That works because the 2 numbers he's using have only 2 factors each(1&5) and (1&2) and since these 2 numbers have only these factors or they are considered prime,you will find that the smallest common multiple is just multiplying those 2 numbers together.
(1 vote)
• thanks for the help. But when i am doing a harder problem is there an easy and quick way to check if you got it right?
(1 vote)
• You can estimate. Sometimes this doesn't work, but it usually does. Sal, at the end of the video, estimated to check his answer (2/5 is less than half, so when added 1/2, it won't equal a whole).
(1 vote)
• its harder to understand with bars.
(1 vote)
• it really isnt because it helps you visulive the problem or equasion better
(1 vote)
• How do you solve 11/15 - (-3/5) =
(1 vote)
• Well first recognize that 11/15- (-3/5) is also equal to 11/15+3/5 (because when two negatives are by each other, they become a positive). So from here on it should be easy to solve, change 3/5 to 9/15. So 11/15 + 9/15 = 20/15. Simplifying it would make it 4/3, change it to a mixed fraction would make it 1 1/3.
Hope this helped!
(1 vote)
• can anyone explain amit stamp collection problem?
I need to know why we have to subtract to get spain's collection?
(1 vote)

## Video transcript

Cindy and Michael need 1 gallon of orange paint for the giant cardboard pumpkin they are making for Halloween. Cindy has 2/5 of a gallon of red paint. Michael has got 1/2 a gallon of yellow paint. If they mix their paints together, will they have the 1 gallon they need? So let's think about that. We're going to add the 2/5 of a gallon of red paint, and we're going to add that to 1/2 a gallon of yellow paint. And we want to see if this gets to being 1 whole gallon. So whenever we add fractions, right over here we're not adding the same thing. Here we're adding 2/5. Here we're adding 1/2. So in order to be able to add these two things, we need to get to a common denominator. And the common denominator, or the best common denominator to use, is the number that is the smallest multiple of both 5 and 2. And since 5 and 2 are both prime numbers, the smallest number's just going to be their product. 10 is the smallest number that we can think of that is divisible by both 5 and 2. So let's rewrite each of these fractions with 10 as the denominator. So 2/5 is going to be something over 10, and 1/2 is going to be something over 10. And to help us visualize this, let me draw a grid. Let me draw a grid with tenths in it. So, that's that, and that's that right over here. So each of these are in tenths. These are 10 equal segments this bar is divided into. So let's try to visualize what 2/5 looks like on this bar. Well, right now it's divided into tenths. If we were to divide this bar into fifths, then we're going to have-- actually, let me do it in that same color. So it's going to be, this is 1 division, 2, 3, 4. So notice if you go between the red marks, these are each a fifth of the bar. And we have two of them, so we're going to go 1 and 2. This right over here, this part of the bar, represents 2/5 of it. Now let's do the same thing for 1/2. So let's divide this bar exactly in half. So, let me do that. I'm going to divide it exactly in half. And 1/2 literally represents 1 of the 2 equal sections. So this is one 1/2. Now, to go from fifths to tenths, you're essentially taking each of the equal sections and you're multiplying by 2. You had 5 equals sections. You split each of those into 2, so you have twice as many. You now have 10 equal sections. So those 2 sections that were shaded in, well, you are going to multiply by 2 the same way. Those 2 are going to turn into 4/10. And you see it right over here when we shaded it initially. If you Look at the tenths, you have 1/10, 2/10, 3/10, and 4/10. Let's do the same logic over here. If you have 2 halves and you want to make them into 10 tenths, you have to take each of the halves and split them into 5 sections. You're going to have 5 times as many sections. So to go from 2 to 10, we multiply by 5. So, similarly, that one shaded-in section in yellow, that 1/2 is going to turn into 5/10. So we're going to multiply by 5. Another way to think about it. Whatever we did to the denominator, we had to do the numerator. Otherwise, somehow we're changing the value of the fraction. So, 1 times 5 is going to get you to 5. And you see that over here when we shaded it in, that 1/2, if you look at the tenths, is equal to 1, 2, 3, 4, 5 tenths. And now we are ready to add. Now we are ready to add these two things. 4/10 plus 5/10, well, this is going to be equal to a certain number of tenths. It's going to be equal to a certain number of tenths. It's going to be equal to 4 plus 5 tenths. And we can once again visualize that. Let me draw our grid again. So 4 plus 5/10, I'll do it actually on top of the paint can right over here. So let me color in 4/10. So 1, 2, 3, 4. And then let me color in the 5/10. And notice that was exactly the 4/10 here, which is exactly the 2/5. Let me color in the 5/10-- 1, 2, 3, 4, and 5. And so how many total tenths do we have? We have a total of 1, 2, 3, 4, 5, 6, 7, 8, 9. 9 of the tenths are now shaded in. We had 9/10 of a gallon of paint. So now to answer their question, will they have the gallon they need? No, they have less than a whole. A gallon would be 10 tenths. They only have 9 tenths. So no, they do not have enough of a gallon. Now, another way you could have thought about this, you could have said, hey, look, 2/5 is less than 1/2, and you could even visualize that right over here. So if I have something less than 1/2 plus 1/2, I'm not going to get a whole. So either way you could think about it, but this way at least we can think it through with actually adding the fractions.