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I can calculate the probability of compound events.

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LT 1C: Define Experimental and Theoretical Probability

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LT 1C: Define Experimental and Theoretical Probability

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I can define experimental and theoretical probability.

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I can express experimental and theoretical probabilities as fractions.

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Develop a probability model and use it to find probabilities of events.

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Compound Probability of Independent Events

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Compound Probability of Independent Events

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Let's think about the situation where we have a completely fair coin here. So let me draw it. I'll assume it's a quarter or something. Let's see. So this is a quarter. Let me draw my best attempt at a profile of George Washington. Well, that'll do. So it's a fair coin. And we're going to flip it a bunch of times and figure out the different probabilities. So let's start with a straightforward one. Let's just flip it once. So with one flip of the coin, what's the probability of getting heads? Well, there's two equally likely possibilities. And the one with heads is one of those two equally likely possibilities, so there's a 1/2 chance. Same thing if we were to ask what is the probability of getting tails? There are two equally likely possibilities, and one of those gives us tails, so 1/2. And this is one thing to realize. If you take the probabilities of heads plus the probabilities of tails, you get 1/2 plus 1/2, which is 1. And this is generally true. The sum of the probabilities of all of the possible events should be equal to 1. And that makes sense, because you're adding up all of these fractions, and the numerator will then add up to all of the possible events. The denominator is always all the possible events. So you have all the possible events over all the possible events when you add all of these things up. Now let's take it up a notch. Let's figure out the probability of-- I'm going to take this coin, and I'm going to flip it twice-- the probability of getting heads and then getting another heads. There's two ways to think about it. One way is to just think about all of the different possibilities. I could get a head on the first flip and a head on the second flip, head on the first flip, tail on the second flip. I could get tails on the first flip, heads on the second flip. Or I could get tails on both flips. So there's four distinct, equally likely possibilities. And one way to think about is on the first flip, I have two possibilities. On the second flip, I have another two possibilities. I could have heads or tails, heads or tails. And so I have four possibilities. For each of these possibilities, for each of these two, I have two possibilities here. So either way, I have four equally likely possibilities. And how many of those meet our constraints? Well, we have it right over here, this one right over here-- having two heads meets our constraints. And there's only one of those possibilities. I've only circled one of the four scenarios. So there's a 1/4 chance of that happening. Another way you could think about this-- and this is because these are independent events. And this is a very important idea to understand in probability, and we'll also study scenarios that are not independent. But these are independent events. What happens in the first flip in no way affects what happens in the second flip. And this is actually one thing that many people don't realize. There's something called the gambler's fallacy, where someone thinks if I got a bunch of heads in a row, then all of a sudden, it becomes more likely on the next flip to get a tails. That is not the case. Every flip is an independent event. What happened in the past in these flips does not affect the probabilities going forward. The fact you got a heads on the first flip in no way affects that you got a heads on the second flip. So if you can make that assumption, you could say that the probability of getting heads and heads, or heads and then heads, is going to be the same thing as the probability of getting heads on the first flip times the probability of getting heads on the second flip. And we know the probability of getting heads on the first flip is 1/2 and the probability of getting heads on the second flip is 1/2. And so we have 1/2 times 1/2, which is equal to 1/4, which is exactly what we got when we tried out all of the different scenarios, all of the equally likely possibilities. Let's take it up another notch. Let's figure out the probability-- and we've kind of been ignoring tails, so let's pay some attention to tails. The probability of getting tails and then heads and then tails-- so this exact series of events. So I'm not saying in any order two tails and a head. I'm saying this exact order-- the first flip is a tails, second flip is a heads, and then third flip is a tail. So once again, these are all independent events. The fact that I got tails on the first flip in no way affects the probability of getting a heads on the second flip. And that in no way affects the probability of getting a tails on the third flip. So because these are independent events, we could say that's the same thing as the probability of getting tails on the first flip times the probability of getting heads on the second flip times the probability of getting tails on the third flip. And we know these are all independent events, so this right over here is 1/2 times 1/2 times 1/2. 1/2 times 1/2 is 1/4. 1/4 times 1/2 is equal to 1/8, so this is equal to 1/8. And we can verify it. Let's try out all of the different scenarios again. So you could get heads, heads, heads. You could get heads, heads, tails. You could get heads, tails, heads. You could get heads, tails, tails. You can get tails, heads, heads. This is a little tricky sometimes. You want to make sure you're being exhaustive in all of the different possibilities here. You could get tails, heads, tails. You could get tails, tails, heads. Or you could get tails, tails, tails. And what we see here is that we got exactly eight equally likely possibilities. We have eight equally likely possibilities. And the tail, heads, tails is exactly one of them. It is this possibility right over here. So it is 1 of 8 equally likely possibilities.

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Compound probability of independent events (video) | Khan Academy

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Let's think about the situation where we have a completely fair coin here. So let me draw it. I'll assume it's a quarter or something. Let's see. So this is a quarter. Let me draw my best attempt at a profile of George Washington. Well, that'll do. So it's a fair coin.

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Compound Probability of Dependent Events

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Compound Probability of Dependent Events

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Let's get you started with a great explanation of dependent probability using a scenario involving a casino game.

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- Let's say that you've got a bag, and in that bag you put a bunch of marbles. So, let's say you put 50 of these magenta marbles. So one, two, three, four, five, six, seven, I'm not gonna draw all of them but you get the general idea. There are going to be 50 magenta marbles, and there's also going to be 50 blue marbles. And what you do is, you have these 100 marbles in there, half of them magenta, half of them blue. And before picking a marble out, and you're gonna be blindfolded when you pick a marble out, you shake the bag really good to, so you think, mix them up a little bit. And so, if you were to say theoretically, what is the probability, if you stuck your hand in and you're not looking, what is the probability of picking a magenta? I feel the need to write magenta in magenta. What is the probability of picking a magenta marble? Well, theoretically there's 100 equally likely possibilities, there's 100 marbles in the bag. And 50 of them involve picking a magenta. So, 50 out of 100, when this is the same thing as a 1/2 probability. So you could say, well, "Theoretically, "there is a 1/2 probability, I just did the math." If you say these are 100 equally likely possibilities, 50 of them are picking magenta. Now let's say you actually start doing the experiment. So you literally take a bag with 50 magenta marbles, 50 blue marbles, and then you start picking the marbles, and then you see what marble color you picked, and you put it back in, and then you do it again. And so, let's say that after every time you put your hand in the bag and you take something out of the bag, and you observe what it is, we're gonna call that an experiment. So, after 10 experiments, let's say that you you have picked out seven magenta and three blue. So, is this strange that out of the first 10 experiments, you haven't picked out exactly half of them being magenta, you've picked out seven magenta, and then the other three were blue. Well no, this is definitely a reasonable thing. If the true probability of picking out a magenta is 1/2, it's definitely possible that you could still pick out seven magenta, that just happened to be what your fingers touched. And this isn't a lot of experiments, it's completely reasonable that out of 10, yeah, you could have, later on in statistics we'll define these things in more detail, but there's enough variation in where you might pick that you're not going to always get, especially with only 10 experiments, you're not definitely going to get exactly 1/2. Instead of having five magenta, it's completely reasonable to have seven magenta. So, this really wouldn't cause me a lot of pause. I still wouldn't question what I did here when I calculated this theoretical probability. But let's say you have a lot of time on your hands. And let's say after 10,000 trials here, after 10,000 experiments, and remember the experiment; you're sticking your hand in the bag without looking, your fingers kind of feeling around, picks out a marble, and you observe the marble and you record what you found. And so, let's say after 10,000 experiments, you get 7,000 magenta. Actually I'm gonna do slightly different numbers, so let me make it even more extreme. Let's say you get 8,000 magenta and you have 2,000 blue. Now this is interesting, because here what you're seeing experimentally seems to be very different. And now you have a large number of trials right over here, not just 10. 10 is completely reasonable that, hey you know, I got seven magenta and three blue instead of five and five, but now you've done 10,000. You would've expected if this was the true probability, you would've expected that half of these would've been magenta, only 5,000 magenta and 5,000 blue, but you got 8,000 magenta. Now, this is within the realm of possibility if the true probability of picking a magenta is 1/2, but it's very unlikely that you would've gotten this result with this many experiments, this many trials if the true probability was 1/2. Here your experimental probability is showing, look, out of 10,000 trials, experimental probability here is you had 10,000 trials, or 10,000 experiments I guess you could say. And and in 8,000 of them, you got a magenta marble. And so, this is going to be 80%, or 8/10. So, there seems to be a difference here. The reason why I would take this more seriously is that you had a lot of trials here, you did this 10,000 times. If the true probability was one half, it's very low likelihood that you would've gotten this many magenta. So, when you think about it you're like, "What's going on here, what are "possible explanations for this?" This, I wouldn't have fretted about, after 10 experiments, not a big deal. But after 10,000, this would have caused me pause. Well, why would this happen, I mixed up the bag every time. And there're some different possibilities; maybe the blue marbles are slightly heavier, and so when you shake the bag up enough, the blue marbles settled to the bottom, and you're more likely to pick a magenta marble. Maybe the blue marbles have a slightly different texture to them, in which case, maybe they slip out of your hands, or they're less likely to be gripped on, and so you're more likely to pick a magenta. So, I don't know the explanation, I don't know what's going on in that bag, but if I thought theoretically that the probability should be 1/2, because half of the marbles are magenta, but I'm seeing through my experiments that 80% of what I'm picking out, especially if I did 10,000 of them, if I did this 10,000 times, well, this is going to cause me some pause. I would really start to think about whether it's truly equally likely for me to pick out a red, a magenta versus a blue. Something else must be going on.

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Theoretical and experimental probabilities (video) | Khan Academy

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Let's say that you've got a bag, and in that bag you put a bunch of marbles. So, let's say you put 50 of these magenta marbles. So one, two, three, four, five, six, seven, I'm not gonna draw all of them but you get the general idea.

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Probability models example: frozen yogurt

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Model the probability of a frozen yogurt line having 0, 1, or 2 people in it.

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I can describe how the relationship between experimental and theoretical probability for an experiment changes as the experiment is conducted many times.

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Practice creating probability models and understand what makes a valid probability model.

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I can write fractions as terminating and repeating decimals.

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I can explain why the sum of a number and its opposite is zero.

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I can find the absolute value of a number and/or the sum of numbers.

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I can apply properties of operations as strategies to add, subtract, multiply, and divide rational numbers.

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I can demonstrate that repeated addition can be rewritten using multiplication.

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I can apply the Distributive Property to multiply multi-digit numbers and/or rational numbers.

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Why do you get closer to theoretical probability with the more trials you do in experimental probability?

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Why do you get closer to theoretical probability with the more trials you do in experimental probability?

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When doing experimental trials there is a probability associated with how close a single trial comes to the theoretical probability so by doing more trials and averaging their outcomes you decrease the probability of the average of the experimental trials being far off from the theoretical probability.

For example, suppose you are flipping a coin. You would expect a .5 probability of getting either heads or tails. Let's say you toss the coin two times and get heads both times; this outcome contradicts the theoretical probability, but let's say you flip it two more times and get heads then tails. This is still different from the theoretical probability because .75 or your 4 trials have resulted in heads and .25 in tails. You continue to flip the coin until you have done so 20 times and now you have 13 heads and 7 tails; this outcome is now much closer to the theoretical probability of .5 and as you continue to flip the coin more and more times you will continue to approach the theoretical probability.

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Why is it that the more trials you do, the closer the experimental probability comes to theoretical probability?

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Answer (1 of 2): When doing experimental trials there is a probability associated with how close a single trial comes to the theoretical probability so by doing more trials and averaging their outcomes you decrease the probability of the average of the experimental trials being far off from the t...

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Video transcript

Write 7/8 as a decimal. And so the main realization here is that 7/8 is the same thing as 7 divided by 8, which is the same thing as 7 divided by 8. These are all different ways of writing the same thing. So let's actually divide 8 into 7. And I'll do it down here just so I have some more real estate to work with. I'm going to divide 8 into 7. And I'm going to add a decimal point here, just because we know that this value is going to be less than 1. 7/8 is less than 1. We're going to have some digits to the right of the decimal point. And let me put the decimal point right up here, right above the decimal point in 7. And then we start dividing. And now this really turns into a long division problem. And we just have to make sure we keep track of the decimal sign. So 8 goes into-- it doesn't go into 7 at all, but it does go into 70. So 8 goes into 70 eight times. So it goes into 70 eight times. 8 times 8 is 64. And then you subtract. 70 minus 64 is 6. And then bring down another 0 because we still have a remainder. We want to get to the point that we have no remainders. Assuming that this thing doesn't repeat forever. And there's other ways we can deal with that. 8 goes into 60? Well, let's see. It doesn't go into it eight times because that's 64. 8 goes into 60 seven times. 7 times 8 is 56. And then we subtract again. 60 minus 56 is 4. And now, we can bring down another 0 right over here. And 8 goes into 40? Well, it goes into 40 exactly five times. 5 times 8 is 40. And we have nothing. We have nothing left over. And so we're done. 7 divided by 8 or 7/8 is equal to 7 divided by 8, which is equal to 0.875. But I'll put a leading 0 here just so it makes it clear that this is where the decimal is. 0.875. And we are done.

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Worked example: Converting a fraction (7/8) to a decimal (video) | Khan Academy

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Harder Examples Converting Fractions to Decimals

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Harder Examples Converting Fractions to Decimals

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I'll now show you how to convert a fraction into a decimal. And if we have time, maybe we'll learn how to do a decimal into a fraction. So let's start with, what I would say, is a fairly straightforward example. Let's start with the fraction 1/2. And I want to convert that into a decimal. So the method I'm going to show you will always work. What you do is you take the denominator and you divide it into the numerator. Let's see how that works. So we take the denominator-- is 2-- and we're going to divide that into the numerator, 1. And you're probably saying, well, how do I divide 2 into 1? Well, if you remember from the dividing decimals module, we can just add a decimal point here and add some trailing 0's. We haven't actually changed the value of the number, but we're just getting some precision here. We put the decimal point here. Does 2 go into 1? No. 2 goes into 10, so we go 2 goes into 10 five times. 5 times 2 is 10. Remainder of 0. We're done. So 1/2 is equal to 0.5. Let's do a slightly harder one. Let's figure out 1/3. Well, once again, we take the denominator, 3, and we divide it into the numerator. And I'm just going to add a bunch of trailing 0's here. 3 goes into-- well, 3 doesn't go into 1. 3 goes into 10 three times. 3 times 3 is 9. Let's subtract, get a 1, bring down the 0. 3 goes into 10 three times. Actually, this decimal point is right here. 3 times 3 is 9. Do you see a pattern here? We keep getting the same thing. As you see it's actually 0.3333. It goes on forever. And a way to actually represent this, obviously you can't write an infinite number of 3's. Is you could just write 0.-- well, you could write 0.33 repeating, which means that the 0.33 will go on forever. Or you can actually even say 0.3 repeating. Although I tend to see this more often. Maybe I'm just mistaken. But in general, this line on top of the decimal means that this number pattern repeats indefinitely. So 1/3 is equal to 0.33333 and it goes on forever. Another way of writing that is 0.33 repeating. Let's do a couple of, maybe a little bit harder, but they all follow the same pattern. Let me pick some weird numbers. Let me actually do an improper fraction. Let me say 17/9. So here, it's interesting. The numerator is bigger than the denominator. So actually we're going to get a number larger than 1. But let's work it out. So we take 9 and we divide it into 17. And let's add some trailing 0's for the decimal point here. So 9 goes into 17 one time. 1 times 9 is 9. 17 minus 9 is 8. Bring down a 0. 9 goes into 80-- well, we know that 9 times 9 is 81, so it has to go into it only eight times because it can't go into it nine times. 8 times 9 is 72. 80 minus 72 is 8. Bring down another 0. I think we see a pattern forming again. 9 goes into 80 eight times. 8 times 9 is 72. And clearly, I could keep doing this forever and we'd keep getting 8's. So we see 17 divided by 9 is equal to 1.88 where the 0.88 actually repeats forever. Or, if we actually wanted to round this we could say that that is also equal to 1.-- depending where we wanted to round it, what place. We could say roughly 1.89. Or we could round in a different place. I rounded in the 100's place. But this is actually the exact answer. 17/9 is equal to 1.88. I actually might do a separate module, but how would we write this as a mixed number? Well actually, I'm going to do that in a separate. I don't want to confuse you for now. Let's do a couple more problems. Let me do a real weird one. Let me do 17/93. What does that equal as a decimal? Well, we do the same thing. 93 goes into-- I make a really long line up here because I don't know how many decimal places we'll do. And remember, it's always the denominator being divided into the numerator. This used to confuse me a lot of times because you're often dividing a larger number into a smaller number. So 93 goes into 17 zero times. There's a decimal. 93 goes into 170? Goes into it one time. 1 times 93 is 93. 170 minus 93 is 77. Bring down the 0. 93 goes into 770? Let's see. It will go into it, I think, roughly eight times. 8 times 3 is 24. 8 times 9 is 72. Plus 2 is 74. And then we subtract. 10 and 6. It's equal to 26. Then we bring down another 0. 93 goes into 26-- about two times. 2 times 3 is 6. 18. This is 74. 0. So we could keep going. We could keep figuring out the decimal points. You could do this indefinitely. But if you wanted to at least get an approximation, you would say 17 goes into 93 0.-- or 17/93 is equal to 0.182 and then the decimals will keep going. And you can keep doing it if you want. If you actually saw this on exam they'd probably tell you to stop at some point. You know, round it to the nearest hundredths or thousandths place. And just so you know, let's try to convert it the other way, from decimals to fractions. Actually, this is, I think, you'll find a much easier thing to do. If I were to ask you what 0.035 is as a fraction? Well, all you do is you say, well, 0.035, we could write it this way-- we could write that's the same thing as 03-- well, I shouldn't write 035. That's the same thing as 35/1,000. And you're probably saying, Sal, how did you know it's 35/1000? Well because we went to 3-- this is the 10's place. Tenths not 10's. This is hundreths. This is the thousandths place. So we went to 3 decimals of significance. So this is 35 thousandths. If the decimal was let's say, if it was 0.030. There's a couple of ways we could say this. Well, we could say, oh well we got to 3-- we went to the thousandths Place. So this is the same thing as 30/1,000. or. We could have also said, well, 0.030 is the same thing as 0.03 because this 0 really doesn't add any value. If we have 0.03 then we're only going to the hundredths place. So this is the same thing as 3/100. So let me ask you, are these two the same? Well, yeah. Sure they are. If we divide both the numerator and the denominator of both of these expressions by 10 we get 3/100. Let's go back to this case. Are we done with this? Is 35/1,000-- I mean, it's right. That is a fraction. 35/1,000. But if we wanted to simplify it even more looks like we could divide both the numerator and the denominator by 5. And then, just to get it into simplest form, that equals 7/200. And if we wanted to convert 7/200 into a decimal using the technique we just did, so we would do 200 goes into 7 and figure it out. We should get 0.035. I'll leave that up to you as an exercise. Hopefully now you get at least an initial understanding of how to convert a fraction into a decimal and maybe vice versa. And if you don't, just do some of the practices. And I will also try to record another module on this or another presentation. Have fun with the exercises.

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Rewriting tricky fractions to decimals (video) | Khan Academy

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Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

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Adding and Subtracting - Negative Numbers

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Adding Negatives = Bigger Negative Number!

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Adding Negatives = Bigger Negative Number!

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Find the sum of negative 15 plus negative 46 plus negative 29. To do this, let's just first visualize what each of these numbers look like. So I'm going to draw a number line for each of them. Negative 15 might look something like this. So if this is 0 and let's say that this is negative 15, I could represent negative 15 as-- if I'd like to-- an arrow that points from 0 to negative 15. The length of the arrow is the absolute value of this. It's the distance from 0. So the length here is 15. And the negative says that we are pointing to the left. So the absolute value is 15. That's the length of that arrow. Let's do the same thing for negative 46. So once again, let me draw my number line. 0 is going to be right over there. And negative 46 is going to be someplace over here. Negative 46. Notice, same exact idea. The distance between negative 46 and 0-- or another way to think about that is the absolute value of negative 46-- this distance right over here is going to be 46. And its direction is to the left. That is why we get to the number negative 46. The negative really tells you whether you're to the left or to the right of 0, but the absolute value says how far are you to the left or the right of 0. And then, finally, let's do the same thing for negative 29. So once again, let me draw my number line. I want to use the yellow again. So draw my number line. Let's say this is 0. Then negative 29 is going to be roughly over here. I'm estimating it. And once again, negative 29 is exactly 29 away from 0. So this length right over here is 29, and it is to the left. That is why it is negative 29. If it was positive 29, it would be 29 to the right of 0. So we've represented all of these numbers. And you can see what their absolute values are like. And now let's think about what happens if you add them. So one way to think about adding these numbers is if you added these arrows, if you put this arrow on top of this arrow or to the left of this arrow, if you started where this arrow leaves off-- and then you put this green arrow, and then you put the orange arrow. So let's do that. Let's draw that. It's going to be a longer arrow now. So we're going to start at 0. First, we have the negative 15. So we're going to move 15 spaces to the left to get us to negative 15. Then we're going to go 46 spaces to the left to get us to negative 15 plus negative 46. So let me draw that. So then we're going to go-- and we're going to figure out what that number is in a second. So then we're going to go 46 to the left. That's about that far. Really just this arrow, I'm now placing it. I'm starting off where the negative 15 left off, and then I'm putting that arrow after that. And we don't know yet what number this gets us to. We're going to have to do a little math here. And that's actually the point of this problem. But we know what the length of this arrow is. The length of this arrow is 46. It is 46 to the left, and we know the length of this magenta arrow is 15. And then, finally, we have this orange arrow. The orange arrow, which we know has a length of 29. Although, it's 29 to the left. It looks like that. It has a length of 29. So when all is said and done, where are we? Where do we end up on our number line? Well, the total length that we are to the left is going to be 15 plus 46 plus 29. But it's going to be to the left, so it's going to be negative. So what we can really view this as, since all of these have the same sign, is we could say, look, this is the same thing as the absolute value of negative 15, which is 15, plus the absolute value of negative 46, which is 46, plus the absolute value of negative 29, which is 29, but then take the negative value of it. So let's just add that. 15 plus 46 plus 29 is equal to, let's see-- 5 plus 6 is 11. 11 plus 9 is 20. Carry the 2. 2 plus 1 is 3. 3 plus 4 is 7. 7 plus 2 is 9. So you get 90. So this entire length here is 90. The entire length, if you add up the arrows, you get a 90. But it is not 90 to the right. So if it was 90 to the right, these would all be positives. And then we would just get a positive 90. But this is 90 to the left. So when you add these guys up, you don't get to positive 90. You get to negative 90. So one way to think about it is these are all the same sign. So this is going to be equal to the negative of the absolute value of negative 15 plus the absolute value of negative 46 plus the absolute value of negative 29. And the reason why we're writing this here-- this might look fancy, but this is really just the length of this purple arrow. This absolute value of negative 46, that's really just the length of the green arrow. 46. And this is just the length of the orange arrow. So this is 15 plus 46 plus 29, gave us 90, but it is to the left. So that is why it is negative 90.

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Adding negative numbers example (video) | Khan Academy

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Find the sum of negative 15 plus negative 46 plus negative 29. To do this, let's just first visualize what each of these numbers look like. So I'm going to draw a number line for each of them. Negative 15 might look something like this.

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Subtracting a negative = adding a positive

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Find out why subtracting a negative number is the same as adding the absolute value of that number.

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Adding and subtracting negative numbers | Pre-Algebra | Khan Academy

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Add and subtract negative fractions with unlike denominators.

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Adding and Subtracting Fractions with Unlike Denominators

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Adding and Subtracting Fractions with Unlike Denominators

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- [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.

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Adding fractions with unlike denominators (video) | Khan Academy

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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."

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Adding and Subtracting with Good Visuals

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Adding and Subtracting with Good Visuals

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How to Multiply and Divide Fractions in Algebra For Dummies

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Dividing fractions: 3/5 ÷ 1/2 (video) | Khan Academy

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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 deno…

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Multiplying Positive and Negative Fractions

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Multiplying Positive and Negative Fractions

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Multiplying positive and negative fractions

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Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

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Video transcript

divide simplify the answer and write as a mixed number and we have two and one-fourth divided by one and three-fourths so the first thing we want to do since both of these are mixed numbers is to convert them both into improper fractions so let's start with two and one-fourth so we're still going to have four in the denominator but instead of two and one-fourth remember two is the same thing as eight forts so we have eight forts and then we have another 1/4 that gives us nine fourths or another way to come up with this 9 you take 4 you take 4 times 2 which is 8 plus 1 that gives you 9 and then the 1 and 3/4 same process you're going to have 4 in the denominator and then the numerator is going to be 4 times 1 which is 4 plus 3 which is 7 so this is the exact same problem here 2 and 1/4 divided by 1 and 3/4 is the same thing as 9 fourths divided by 7/4 and we saw in several videos already that dividing by a fraction is the same thing as multiplying by its reciprocal so this is equivalent to this is equivalent to so these are all equivalent this is equivalent to 9/4 x times the reciprocal of this we're taking changing the division operation to a multiplication and we're taking the reciprocal of the 7/4 the reciprocal of 7/4 you swap the numerator denominator or the top number in the bottom number and you get 4 over 7 now we could just multiply these we could just say this is 9 times 4 which would be 36 over 4 times 7 which is 28 and then try to put it in lowest terms or we could do it right now because it'd be simpler we have a 4 in the numerator we have a 4 in the denominator that will eventually be in the denominator so let's divide our eventual numerators on our debt ventral denominators both by 4 so you divide this 4 by 4 you get 1 this 4 by 4 you get 1 so now when you multiply it you get 9 times 1 which is 9 over 1 times 7 which is 7 so we have our answer but it's writ right now it's an improper fraction they want us to write it as a mixed number and to figure out as a mixed number we can do it in our now I think we've seen this enough times we say how many times does 7 how many times does 7 go into 9 what goes into it exactly one time but when you take 7 into 9 one time what do you have left over we're going to have 2 left over right 7 times 1 is 7 and you're gonna have 2 left over you need 2 more to get to 9 so you're going to have 2 left over so this is one and two sevens and we're done

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Dividing mixed numbers (video) | Khan Academy

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Improper Fractions and Mixed Numbers by Shmoop

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Write 7/4 as a mixed number. So right now it's an improper fraction. 7 is larger than 4. Let's write it is a mixed number. So first I'm just going to show you a fairly straightforward way of doing it and then we're going to think a little bit about what it actually means. So to figure out what 7/4 represents as a mixed number, let me write it in different colors. So this is going to be equal to-- the easiest way I do it is you say, well, you divide 4 it 7. If we're dealing with fourths, 4 goes into 7 a total of one time. Let me do this in another color. 1 times 4 is 4. And then what is our remainder? 7 minus 4 is 3. So if we wanted to write this in plain-- well, let me just do the problem, and then we'll think about what it means in a second. So you see that 4 goes into 7 one time, so you have one whole here, and then how much do you have left over? Well, you have 3 left over, and that comes from right over there. That is the remainder when you divide 4 into 7. 3 left over, but it's 3 of your 4, or 3/4 left over. So that's the way we just converted it from an improper fraction to a mixed number. Now, it might seem a little bit like voodoo what I just did. I divided 4 into 7, it goes one time, and then the remainder is 3, so I got 1 and 3/4. But why does that make sense? Why does that actually makes sense? So let's draw fourths. Let's draw literally 7 fourths and maybe it'll become clear. So let's do a little square as a fourth. So let's say I have a square like that, and that is 1/4. Now, let's think about what seven of those mean, so let me copy and paste that. Copy and then paste it. So here I have 2 one-fourths, or you could see I have 2/4. Now I have 3 one-fourths. Now, I have 4 one-fourths. Now this is a whole, right? I have 4 one-fourths. This is a whole. So let me start on another whole. So now I have 5. Now I have 6 one-fourths, and now I have 7 one-fourths. Now, what does this look like? So all I did is I rewrote 7/4, or 7 one-fourths. I just kind of drew it for you. Now, what does this represent? Well, I have 4 fourths here, so this is 4/4. This right here is 3/4. Notice, 7/4 is 4/4 with 3/4 left over. So let me write it this way. 7/4 is 4/4 with 3/4 left over. Now what is 4/4? 4/4 is one whole. So you have one whole with 3/4 left over, so you end up with 1 and 3/4. So that is the 3/4 part and that is your one whole. Hopefully that makes sense and hopefully you understand why it connects. Because you say, well, how many wholes do you have? When you're dividing the 4 into the 7 and getting the one, you're essentially saying how many wholes? So the number of wholes, or you can imagine, the number of whole pies. And then how many pieces do we have left over? Well, we have 3 pieces and each piece is 1/4, so we have 3/4 left over. So we have one whole pie and three pieces, which are each a fourth left over.

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Writing improper fractions as mixed numbers (video) | Khan Academy

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Write 7/4 as a mixed number. So right now it's an improper fraction. 7 is larger than 4. Let's write it is a mixed number. So first I'm just going to show you a fairly straightforward way of doing it and then we're going to think a little bit about what it actually means.

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Video transcript

rewrite the expression four times than in parentheses we have eight plus three using the distributive law of multiplication over addition then simplify the expression so let's just try to solve this or evaluate this expression and then we'll talk a little bit about the distributive law of multiplication over addition usually just called the distributive law so we have 4 times 8 4 times 8 plus 8 plus 3 8 plus 3 now there's two ways to do it normally when you have parentheses your inclination is well let me just evaluate what's in the parentheses first and then worry about what's outside of the parentheses and we can do that fairly easily here we can evaluate what 8 plus 3 is 8 plus 3 is 11 so if we do that let me do that in this direction so if we do that we get 4 times we get 4 times and in parentheses we have an 1183 is 11 and then this is going to be equal to well 4 times 11 is just 44 so you can evaluate it that way but they want us to use the distributive law of multiplication we did not use the distributive law just now we just evaluated the expression maybe use the parentheses first then multiplied by 4 and the distributive law we multiply by 4 first and it's called the distributive law because you distribute the 4 and we're going to think about what that means so in the distributive law what this will become it'll become 4 times 8 plus 4 times 3 and we're going to think about why that is in a second so this is going to be equal to 4 times 8 4 times 8 plus 4 times 4 times 3 4 times 3 a lot of people's first instinct is just to multiply the 4 times the 8 but know you have to distribute the 4 you have to multiply it times times the 8 and times the 3 this is right here this is the distributive property in action right here distributive property in action and then when you evaluate I'm going to show you in kind of a visual way why this word but then when you evaluate it four times eight this in a different color 4 times 8 is 32 and then so we have 32 plus 4 times 3 4 times 3 is 12 and 32 plus 12 is equal to 44 that is also equal to 44 so you can get it either way but when they want us to use the distributive law you would multiply it you distribute the 4 first now let's think about why that happens let's let's visualize just what 8 plus 3 is so let me draw 8 of something so 1 2 3 4 5 6 let me draw it this well 6 7 8 right 1 2 3 4 5 6 7 8 and then we're going to add to that we're going to add to that 3 of something of them maybe the same thing 1 2 3 so you could imagine this is what we have inside of the parentheses we have 8 circles plus 3 circles now when we're multiplying this whole thing this whole thing times 4 what does that mean well that means we're just going to add this to itself 4 times so let me do that with the copy and paste copy and paste let me copy and then let me paste there you go that's 2 that's 1 2 3 and then we have 4 and we're going to add them all together so this is literally what 4 times right we have it let me go back to the drawing tool we have it 1 2 3 4 times this expression which is 8 plus 3 now what is this thing over here what is this thing over here what is this thing over here if you were to count all of this stuff you would get 44 but what is this thing over here well that's 8 add it to itself 4 times you could imagine you're adding all of these so what's 8 add it to itself 4 times that is 4 times 8 so this is 4 times 8 and what is this over here in the orange we have 1 2 3 4 times well each time we have three so it's got four times this right here this right here is four times three so you see why the distributive property works if you do four times eight plus three you have to multiply what when you I guess you could imagine duplicate the thing four times both the eight and the three is getting duplicated four times or it's being added to itself four times and that's why we distribute the four

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Distributive property over addition (video) | Khan Academy

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We've learned about order of operations and combining like terms. Let's layer the distributive property on top of this.