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Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | So you have 0, 1, 2, and 1 and 2 thirds will be right around there. We're going to include it. That right there is 5 thirds. And everything greater than or equal to that will be included in our solution set. Let's do another one. Let's say we have x over negative 3 is greater than negative 10 over 9. So we want to just isolate the x on the left-hand side. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | And everything greater than or equal to that will be included in our solution set. Let's do another one. Let's say we have x over negative 3 is greater than negative 10 over 9. So we want to just isolate the x on the left-hand side. So let's multiply both sides by negative 3. This is essentially the coefficient you could imagine is negative 1 over 3. So we want to multiply by the inverse, which would be negative 3. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | So we want to just isolate the x on the left-hand side. So let's multiply both sides by negative 3. This is essentially the coefficient you could imagine is negative 1 over 3. So we want to multiply by the inverse, which would be negative 3. So if you multiply both sides by negative 3, you get negative 3 times, this you could rewrite it as negative 1 third x. And on this side, you have negative 10 over 9 times negative 3. And the inequality will switch because we are multiplying or dividing by a negative number. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | So we want to multiply by the inverse, which would be negative 3. So if you multiply both sides by negative 3, you get negative 3 times, this you could rewrite it as negative 1 third x. And on this side, you have negative 10 over 9 times negative 3. And the inequality will switch because we are multiplying or dividing by a negative number. So the inequality will switch. It will go from greater than to less than. So the left-hand side of the equation just becomes an x. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | And the inequality will switch because we are multiplying or dividing by a negative number. So the inequality will switch. It will go from greater than to less than. So the left-hand side of the equation just becomes an x. That was the whole point. That cancels out with that. The negatives cancel out. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | So the left-hand side of the equation just becomes an x. That was the whole point. That cancels out with that. The negatives cancel out. x is less than. And then you have a negative times a negative. That will make it a positive. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | The negatives cancel out. x is less than. And then you have a negative times a negative. That will make it a positive. Then if you divide the numerator and the denominator by 3, you get a 1 and a 3. So x is less than 10 over 3. So if we were to write this in interval notation, the solution set, the upper bound will be 10 over 3. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | That will make it a positive. Then if you divide the numerator and the denominator by 3, you get a 1 and a 3. So x is less than 10 over 3. So if we were to write this in interval notation, the solution set, the upper bound will be 10 over 3. And it won't include 10 over 3. This isn't less than or equal to. So we're going to put a parentheses here. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | So if we were to write this in interval notation, the solution set, the upper bound will be 10 over 3. And it won't include 10 over 3. This isn't less than or equal to. So we're going to put a parentheses here. Notice here it included 5 thirds. We put a bracket. Here we're not including 10 thirds. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | So we're going to put a parentheses here. Notice here it included 5 thirds. We put a bracket. Here we're not including 10 thirds. We put a parentheses. And it'll go from 10 over 3 all the way down to negative infinity. And everything less than 10 over 3 is in our solution set. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | Here we're not including 10 thirds. We put a parentheses. And it'll go from 10 over 3 all the way down to negative infinity. And everything less than 10 over 3 is in our solution set. Let's draw that. Let's draw the solution set. So 10 over 3. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | And everything less than 10 over 3 is in our solution set. Let's draw that. Let's draw the solution set. So 10 over 3. So we might have 0, 1, 2, 3, 4. 10 over 3 is 3 and 1 third. So it might sit, let me do it in a different color. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | So 10 over 3. So we might have 0, 1, 2, 3, 4. 10 over 3 is 3 and 1 third. So it might sit, let me do it in a different color. It might be over here. We're not going to include that. It's less than 10 over 3. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | So it might sit, let me do it in a different color. It might be over here. We're not going to include that. It's less than 10 over 3. 10 over 3 is not in the solution set. That is 10 over 3 right there. And everything less than that, but not including 10 over 3, is in our solution set. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | It's less than 10 over 3. 10 over 3 is not in the solution set. That is 10 over 3 right there. And everything less than that, but not including 10 over 3, is in our solution set. Let's do one more. Let's do one more. Say we have x over negative 15 is less than 8. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | And everything less than that, but not including 10 over 3, is in our solution set. Let's do one more. Let's do one more. Say we have x over negative 15 is less than 8. So once again, let's multiply both sides of this equation by negative 15. So negative 15 times x over negative 15. Then you have an 8 times a negative 15. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | Say we have x over negative 15 is less than 8. So once again, let's multiply both sides of this equation by negative 15. So negative 15 times x over negative 15. Then you have an 8 times a negative 15. And when you multiply both sides of an inequality by a negative number, or divide both sides by a negative number, you swap the inequality. It's less than, you change it to greater than. And now the left-hand side just becomes an x, because these guys cancel out. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | Then you have an 8 times a negative 15. And when you multiply both sides of an inequality by a negative number, or divide both sides by a negative number, you swap the inequality. It's less than, you change it to greater than. And now the left-hand side just becomes an x, because these guys cancel out. x is greater than 8 times 15 is 80 plus 40 is 120. So negative 120. Is that right? |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | And now the left-hand side just becomes an x, because these guys cancel out. x is greater than 8 times 15 is 80 plus 40 is 120. So negative 120. Is that right? 80 plus 40. Negative 120. Or we could write the solution set as starting at negative 120. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | Is that right? 80 plus 40. Negative 120. Or we could write the solution set as starting at negative 120. But we're not including negative 120. We don't have an equal sign here. Going all the way up to infinity. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | Or we could write the solution set as starting at negative 120. But we're not including negative 120. We don't have an equal sign here. Going all the way up to infinity. And if we were to graph it, we draw the number line here. I'll do a real quick one. Let's say that that is negative 120. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | Going all the way up to infinity. And if we were to graph it, we draw the number line here. I'll do a real quick one. Let's say that that is negative 120. Maybe 0 sitting up here. This would be negative 121. This would be negative 119. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | Let's say that that is negative 120. Maybe 0 sitting up here. This would be negative 121. This would be negative 119. We are not going to include negative 120, because we don't have an equal sign there. It's going to be everything greater than negative 120. All of these things that I'm shading in green would satisfy the inequality. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | This would be negative 119. We are not going to include negative 120, because we don't have an equal sign there. It's going to be everything greater than negative 120. All of these things that I'm shading in green would satisfy the inequality. And you could even try it out. Does 0 work? 0 over 15? |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | All of these things that I'm shading in green would satisfy the inequality. And you could even try it out. Does 0 work? 0 over 15? Yeah, that's 0. That's definitely less than 8. I mean, that doesn't prove it to you, but you could try any of these numbers, and they should work. |
Percent from fraction models.mp3 | So we're told the square below represents one whole, so this entire square is a whole, and then they ask us what percent is represented by the shaded area. So why don't you pause this video and see if you can figure that out. So let's see, the whole is divided into one, two, three, four, five, six, seven, eight, nine, 10 equal sections, of which one, two, three, four, five, six, seven are actually filled in, that's the shaded area. So one way to think about it is, 7 tenths are shaded in. But how do we express this fraction as a percent? They're asking for a percent. Well remember, per cent, it literally means per hundred. |
Percent from fraction models.mp3 | So one way to think about it is, 7 tenths are shaded in. But how do we express this fraction as a percent? They're asking for a percent. Well remember, per cent, it literally means per hundred. Cent, same root as the word hundred, you see it in cents or century. And so, can we write this as per hundred instead of per 10? Well, seven per 10 is the same thing as 70 per hundred, or 70% And how did I go from 7 tenths to 70 over 100? |
Percent from fraction models.mp3 | Well remember, per cent, it literally means per hundred. Cent, same root as the word hundred, you see it in cents or century. And so, can we write this as per hundred instead of per 10? Well, seven per 10 is the same thing as 70 per hundred, or 70% And how did I go from 7 tenths to 70 over 100? Well, I just multiply both the numerator and the denominator by 10. And once you do more and more percents, you'll get a hang of it, you say, oh, 7 tenths, that's the same thing as 70 per 100, which is 70%. Let's do another example. |
Percent from fraction models.mp3 | Well, seven per 10 is the same thing as 70 per hundred, or 70% And how did I go from 7 tenths to 70 over 100? Well, I just multiply both the numerator and the denominator by 10. And once you do more and more percents, you'll get a hang of it, you say, oh, 7 tenths, that's the same thing as 70 per 100, which is 70%. Let's do another example. Here we're told 100% is shown on the following tape diagram. So just this amount right over here is 100%. And then they ask us, what percent is represented by the entire tape diagram? |
Percent from fraction models.mp3 | Let's do another example. Here we're told 100% is shown on the following tape diagram. So just this amount right over here is 100%. And then they ask us, what percent is represented by the entire tape diagram? So by this entire thing right over here. Pause this video and see if you can answer that. Well, one way to think about 100%, 100% is equivalent to a whole. |
Percent from fraction models.mp3 | And then they ask us, what percent is represented by the entire tape diagram? So by this entire thing right over here. Pause this video and see if you can answer that. Well, one way to think about 100%, 100% is equivalent to a whole. And now we have three times as much of that for the entire tape diagram. So you could view this as three wholes, or you could say that's 100%, we have another 100% right over here, and then we have another 100% right over here. So the whole tape diagram, that would be 300%. |
Percent from fraction models.mp3 | Well, one way to think about 100%, 100% is equivalent to a whole. And now we have three times as much of that for the entire tape diagram. So you could view this as three wholes, or you could say that's 100%, we have another 100% right over here, and then we have another 100% right over here. So the whole tape diagram, that would be 300%. Let's do another example. This is strangely fun. And I'll see, it says, the large rectangle below represents one whole. |
Percent from fraction models.mp3 | So the whole tape diagram, that would be 300%. Let's do another example. This is strangely fun. And I'll see, it says, the large rectangle below represents one whole. So that's this whole thing is one whole. What percentage is represented by the shaded area? So pause the video and see if you can figure that out again. |
Percent from fraction models.mp3 | And I'll see, it says, the large rectangle below represents one whole. So that's this whole thing is one whole. What percentage is represented by the shaded area? So pause the video and see if you can figure that out again. So let's just express it as a fraction first. So we have a total of one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 squares. So out of those 20 squares, we see that six of them are actually shaded in. |
Percent from fraction models.mp3 | So pause the video and see if you can figure that out again. So let's just express it as a fraction first. So we have a total of one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 squares. So out of those 20 squares, we see that six of them are actually shaded in. So 6 20ths, can we write that as per 100? Well, let's see. If I were to go from 20 to 100, I multiply by five. |
Percent from fraction models.mp3 | So out of those 20 squares, we see that six of them are actually shaded in. So 6 20ths, can we write that as per 100? Well, let's see. If I were to go from 20 to 100, I multiply by five. And so if I multiply the numerator by five, I'll get the same value. Six times five is 30. So six per 20 is the same thing as 30 per 100, which is the same thing as 30%, which literally means per 100. |
Percent from fraction models.mp3 | If I were to go from 20 to 100, I multiply by five. And so if I multiply the numerator by five, I'll get the same value. Six times five is 30. So six per 20 is the same thing as 30 per 100, which is the same thing as 30%, which literally means per 100. So this is 30%. Let's do one last example. Here we are told, which large rectangle below represents one whole? |
Percent from fraction models.mp3 | So six per 20 is the same thing as 30 per 100, which is the same thing as 30%, which literally means per 100. So this is 30%. Let's do one last example. Here we are told, which large rectangle below represents one whole? So this is a whole, and then this whole thing right over here is another whole. What percentage is represented by the shaded area? Again, pause the video. |
Percent from fraction models.mp3 | Here we are told, which large rectangle below represents one whole? So this is a whole, and then this whole thing right over here is another whole. What percentage is represented by the shaded area? Again, pause the video. See if you can answer that. So this one, we have shaded in a whole, so that is 100%. And then over here, we have shaded in one, two, three, four fifths of the whole. |
Percent from fraction models.mp3 | Again, pause the video. See if you can answer that. So this one, we have shaded in a whole, so that is 100%. And then over here, we have shaded in one, two, three, four fifths of the whole. So four fifths, if I were to express it as per 100, what would it be? Five times 20 is 100, so four times 20 is 80. So four fifths, or 80 hundredths, is filled out here, or you could say 80 per 100, which is the same thing as 80%. |
Percent from fraction models.mp3 | And then over here, we have shaded in one, two, three, four fifths of the whole. So four fifths, if I were to express it as per 100, what would it be? Five times 20 is 100, so four times 20 is 80. So four fifths, or 80 hundredths, is filled out here, or you could say 80 per 100, which is the same thing as 80%. So this right over here is 80%. So what percent is represented by the shaded area? Well, we have 100%, and then we have 80%, so we have 180%. |
Percent from fraction models.mp3 | So four fifths, or 80 hundredths, is filled out here, or you could say 80 per 100, which is the same thing as 80%. So this right over here is 80%. So what percent is represented by the shaded area? Well, we have 100%, and then we have 80%, so we have 180%. It's more than a whole. If you have a percentage that is larger than 100%, you're talking about something that is more than a whole, and then we see that. We have a whole right over here, and then we have 80% more than that. |
Proportion word problem (example 2) 7th grade Khan Academy.mp3 | So a big clue is the same pace. I have to remove a hair from my tongue, alright. A big clue is the same pace. That means that the hot dogs, hot dogs per minute, is going to be constant, is always going to be the same. Always the same. Because this is essentially the pace. Her hot dogs per minute are going to stay the same. |
Proportion word problem (example 2) 7th grade Khan Academy.mp3 | That means that the hot dogs, hot dogs per minute, is going to be constant, is always going to be the same. Always the same. Because this is essentially the pace. Her hot dogs per minute are going to stay the same. It's going to stay at the same pace. So it tells us that she can eat 21 hot dogs in 66 minutes. So her hot dogs per minute, at least up here, is 21 hot dogs in 66 minutes. |
Proportion word problem (example 2) 7th grade Khan Academy.mp3 | Her hot dogs per minute are going to stay the same. It's going to stay at the same pace. So it tells us that she can eat 21 hot dogs in 66 minutes. So her hot dogs per minute, at least up here, is 21 hot dogs in 66 minutes. So 21 hot dogs in 66 minutes. Well if her pace is always going to be the same, well it's going to take her, this ratio over here, is going to be the ratio between 35 hot dogs and however long it takes her to eat 35 hot dogs. So once again, hot dogs per minute has to be a constant because it's going to be the same pace. |
Proportion word problem (example 2) 7th grade Khan Academy.mp3 | So her hot dogs per minute, at least up here, is 21 hot dogs in 66 minutes. So 21 hot dogs in 66 minutes. Well if her pace is always going to be the same, well it's going to take her, this ratio over here, is going to be the ratio between 35 hot dogs and however long it takes her to eat 35 hot dogs. So once again, hot dogs per minute has to be a constant because it's going to be the same pace. Hot dogs per minute. If 21 hot dogs take 66 minutes, 35 hot dogs take M minutes, these two ratios are going to be the same. We're dealing with a proportional relationship that's going to be happening at the same rate. |
Proportion word problem (example 2) 7th grade Khan Academy.mp3 | So once again, hot dogs per minute has to be a constant because it's going to be the same pace. Hot dogs per minute. If 21 hot dogs take 66 minutes, 35 hot dogs take M minutes, these two ratios are going to be the same. We're dealing with a proportional relationship that's going to be happening at the same rate. And then we're left with a situation where we just have to solve for M. And there's a bunch of different ways that you could tackle this. The easiest way that I can think of doing it is, I don't like this M sitting here in the denominator, so let's multiply both sides by M. Let me do that in a different color. So I multiply that side by M, and this side by M. And so what do we get? |
Proportion word problem (example 2) 7th grade Khan Academy.mp3 | We're dealing with a proportional relationship that's going to be happening at the same rate. And then we're left with a situation where we just have to solve for M. And there's a bunch of different ways that you could tackle this. The easiest way that I can think of doing it is, I don't like this M sitting here in the denominator, so let's multiply both sides by M. Let me do that in a different color. So I multiply that side by M, and this side by M. And so what do we get? On the left-hand side, we have 21 over 66 M, 21 over 66 times M, times M, is equal to, well, you divide by M and multiply by M, those are going to cancel out and you're just going to have 35. So now you just have to solve for M, and the best way I can think of doing that is multiply both sides times the reciprocal, both sides times the reciprocal of the coefficient on the M. So let's multiply both sides by, let's multiply both sides by 66 over 21. Once again, I've just swapped the numerator and the denominator here to get the reciprocal, but I can't just do it to one side of an equation, I have to do it to both sides, otherwise it's not going to be equal anymore. |
Proportion word problem (example 2) 7th grade Khan Academy.mp3 | So I multiply that side by M, and this side by M. And so what do we get? On the left-hand side, we have 21 over 66 M, 21 over 66 times M, times M, is equal to, well, you divide by M and multiply by M, those are going to cancel out and you're just going to have 35. So now you just have to solve for M, and the best way I can think of doing that is multiply both sides times the reciprocal, both sides times the reciprocal of the coefficient on the M. So let's multiply both sides by, let's multiply both sides by 66 over 21. Once again, I've just swapped the numerator and the denominator here to get the reciprocal, but I can't just do it to one side of an equation, I have to do it to both sides, otherwise it's not going to be equal anymore. So times 66 over 21. This is just going to be one, you multiply something times its reciprocal, you're just going to end up with one. So you're going to be left with M, is equal to, now 35 times 66 divided by 21. |
Proportion word problem (example 2) 7th grade Khan Academy.mp3 | Once again, I've just swapped the numerator and the denominator here to get the reciprocal, but I can't just do it to one side of an equation, I have to do it to both sides, otherwise it's not going to be equal anymore. So times 66 over 21. This is just going to be one, you multiply something times its reciprocal, you're just going to end up with one. So you're going to be left with M, is equal to, now 35 times 66 divided by 21. Well, 35 is the same thing as, 35 is five times seven, and 21 is three times seven. So you're multiplying by seven up here, and here you have a seven in the denominator, you're dividing by seven, so they're going to cancel out. So this is going to simplify to five times 66 over three, and then we could simplify it even more, because 66 is the same thing as three times 22. |
Proportion word problem (example 2) 7th grade Khan Academy.mp3 | So you're going to be left with M, is equal to, now 35 times 66 divided by 21. Well, 35 is the same thing as, 35 is five times seven, and 21 is three times seven. So you're multiplying by seven up here, and here you have a seven in the denominator, you're dividing by seven, so they're going to cancel out. So this is going to simplify to five times 66 over three, and then we could simplify it even more, because 66 is the same thing as three times 22. Three times 22. And so you have a three in the numerator, you're multiplying by three, and a three in the denominator, dividing by three, three divided by three is one. So you're left with five times 22, which is 110. |
Proportion word problem (example 2) 7th grade Khan Academy.mp3 | So this is going to simplify to five times 66 over three, and then we could simplify it even more, because 66 is the same thing as three times 22. Three times 22. And so you have a three in the numerator, you're multiplying by three, and a three in the denominator, dividing by three, three divided by three is one. So you're left with five times 22, which is 110. So it would take her M minutes to eat 35 hot dogs at the same pace. Now, when some of you might have tackled it, you might have had a different equation set up here. Instead of thinking of hot dogs per minute, you might have thought about minutes per hot dog. |
Proportion word problem (example 2) 7th grade Khan Academy.mp3 | So you're left with five times 22, which is 110. So it would take her M minutes to eat 35 hot dogs at the same pace. Now, when some of you might have tackled it, you might have had a different equation set up here. Instead of thinking of hot dogs per minute, you might have thought about minutes per hot dog. And so in that situation, if you thought in terms of minutes per hot dog, you might have said, okay, look, it took Micah 66 minutes to eat 21 hot dogs, and it's going to take her M minutes to eat 35 hot dogs, and if it's the same pace, then these two rates are going to be equal. They have to be the same pace. And so then you can solve for M, and actually this one's easier to solve for M. You just multiply both sides by 35. |
Proportion word problem (example 2) 7th grade Khan Academy.mp3 | Instead of thinking of hot dogs per minute, you might have thought about minutes per hot dog. And so in that situation, if you thought in terms of minutes per hot dog, you might have said, okay, look, it took Micah 66 minutes to eat 21 hot dogs, and it's going to take her M minutes to eat 35 hot dogs, and if it's the same pace, then these two rates are going to be equal. They have to be the same pace. And so then you can solve for M, and actually this one's easier to solve for M. You just multiply both sides by 35. Multiply both sides by 35, and you're left with, on the right-hand side, you're left with just an M, and on the left-hand side, same idea. You're taking 35. You have 35 times 66, 21st, which we already figured out is 110. |
Multiplying three numbers in scientific notation (example) Pre-Algebra Khan Academy.mp3 | So this is 1.45 times 10 to the 8th power times, and I could just write the parentheses again like this, but I'm just gonna write it as another multiplication, times 9.2 times 10 to the negative 12th and then times, times 3.01 times 10 to the negative 5th. All this meant when I wrote these parentheses times next to each other, I'm just going to multiply this expression times this expression times this expression. And since everything is involved in multiplication, it actually doesn't matter what order I multiply in. And so with that in mind, I can swap the order here. This is going to be the same thing as 1.45. That's that right there times 9.2 times 9.2 times 3.01 times 3.01 times 10 to the 8th. Let me do that in that purple color. |
Multiplying three numbers in scientific notation (example) Pre-Algebra Khan Academy.mp3 | And so with that in mind, I can swap the order here. This is going to be the same thing as 1.45. That's that right there times 9.2 times 9.2 times 3.01 times 3.01 times 10 to the 8th. Let me do that in that purple color. Times 10 to the 8th, so times 10 to the 8th times 10 to the negative 12th power, 10 to the negative 12th power times 10 to the negative 5th power times 10 to the negative 5th power. And this is useful because now I have all of my powers of 10 right over here. I can put parentheses around that. |
Multiplying three numbers in scientific notation (example) Pre-Algebra Khan Academy.mp3 | Let me do that in that purple color. Times 10 to the 8th, so times 10 to the 8th times 10 to the negative 12th power, 10 to the negative 12th power times 10 to the negative 5th power times 10 to the negative 5th power. And this is useful because now I have all of my powers of 10 right over here. I can put parentheses around that. And I have all of my non-powers of 10 right over there. And so I can simplify it if I have the same base 10 or right over here, so I can add the exponents. This is going to be 10 to the eight minus 12 minus five power, minus five power. |
Multiplying three numbers in scientific notation (example) Pre-Algebra Khan Academy.mp3 | I can put parentheses around that. And I have all of my non-powers of 10 right over there. And so I can simplify it if I have the same base 10 or right over here, so I can add the exponents. This is going to be 10 to the eight minus 12 minus five power, minus five power. And then all of this on the left-hand side, let me get a calculator out. I have 1.45, you could do it by hand, but this is a little bit faster and less likely to make a careless mistake, times 9.2 times 3.01, which is equal to 40.1534. So this is equal to 40.1534. |
Multiplying three numbers in scientific notation (example) Pre-Algebra Khan Academy.mp3 | This is going to be 10 to the eight minus 12 minus five power, minus five power. And then all of this on the left-hand side, let me get a calculator out. I have 1.45, you could do it by hand, but this is a little bit faster and less likely to make a careless mistake, times 9.2 times 3.01, which is equal to 40.1534. So this is equal to 40.1534. And of course, this is going to be multiplied times 10 to this thing. And so if you simplify this exponent, you get 40.1534 times 10 to the eight minus 12 is negative four minus five is negative nine, 10 to the negative nine power. Now, you might be tempted to say that this is already in scientific notation because I have some number here times some power of 10, but this is not quite official scientific notation. |
Multiplying three numbers in scientific notation (example) Pre-Algebra Khan Academy.mp3 | So this is equal to 40.1534. And of course, this is going to be multiplied times 10 to this thing. And so if you simplify this exponent, you get 40.1534 times 10 to the eight minus 12 is negative four minus five is negative nine, 10 to the negative nine power. Now, you might be tempted to say that this is already in scientific notation because I have some number here times some power of 10, but this is not quite official scientific notation. And that's because in order for it to be in scientific notation, this number right over here has to be greater than or equal to one and less than 10. And this is obviously not less than 10. Essentially for it to be in scientific notation, you want a non-zero digit right over here, and then you want your decimal and then the rest of everything else. |
Multiplying three numbers in scientific notation (example) Pre-Algebra Khan Academy.mp3 | Now, you might be tempted to say that this is already in scientific notation because I have some number here times some power of 10, but this is not quite official scientific notation. And that's because in order for it to be in scientific notation, this number right over here has to be greater than or equal to one and less than 10. And this is obviously not less than 10. Essentially for it to be in scientific notation, you want a non-zero digit right over here, and then you want your decimal and then the rest of everything else. So here, and you want a non-zero single digit over here. Here we obviously have, here we have two digits. This is larger than 10, or this is greater than or equal to 10. |
Multiplying three numbers in scientific notation (example) Pre-Algebra Khan Academy.mp3 | Essentially for it to be in scientific notation, you want a non-zero digit right over here, and then you want your decimal and then the rest of everything else. So here, and you want a non-zero single digit over here. Here we obviously have, here we have two digits. This is larger than 10, or this is greater than or equal to 10. You want this thing to be less than 10 and greater than or equal to one. So the best way to do that is to write this thing right over here in scientific notation. This is the same thing as 4.01534 times 10. |
Multiplying three numbers in scientific notation (example) Pre-Algebra Khan Academy.mp3 | This is larger than 10, or this is greater than or equal to 10. You want this thing to be less than 10 and greater than or equal to one. So the best way to do that is to write this thing right over here in scientific notation. This is the same thing as 4.01534 times 10. And one way to think about it is to go from 40 to four, we had to move this decimal over to the left. Moving a decimal over to the left to go from 40 to four, you're dividing by 10. So you have to multiply by 10 so it all equals out. |
Multiplying three numbers in scientific notation (example) Pre-Algebra Khan Academy.mp3 | This is the same thing as 4.01534 times 10. And one way to think about it is to go from 40 to four, we had to move this decimal over to the left. Moving a decimal over to the left to go from 40 to four, you're dividing by 10. So you have to multiply by 10 so it all equals out. Divide by 10 and then multiply by 10. Or another way to write it, or another way to think about it is 4.0 and all this stuff times 10 is going to be 40.1534. And so you're going to have four, all of this times 10 to the first power, that's the same thing as 10, times this thing, times 10 to the negative ninth power. |
Multiplying three numbers in scientific notation (example) Pre-Algebra Khan Academy.mp3 | So you have to multiply by 10 so it all equals out. Divide by 10 and then multiply by 10. Or another way to write it, or another way to think about it is 4.0 and all this stuff times 10 is going to be 40.1534. And so you're going to have four, all of this times 10 to the first power, that's the same thing as 10, times this thing, times 10 to the negative ninth power. And then once again, powers of 10, so it's 10 to the first times 10 to the negative nine is going to be 10 to the negative eighth power. 10 to the negative eighth power and we still have this 4.01534 times 10 to the negative eight and now we have written it in scientific, now we have written it in scientific notation. Now they wanted us to express it in both decimal and scientific notation. |
Multiplying three numbers in scientific notation (example) Pre-Algebra Khan Academy.mp3 | And so you're going to have four, all of this times 10 to the first power, that's the same thing as 10, times this thing, times 10 to the negative ninth power. And then once again, powers of 10, so it's 10 to the first times 10 to the negative nine is going to be 10 to the negative eighth power. 10 to the negative eighth power and we still have this 4.01534 times 10 to the negative eight and now we have written it in scientific, now we have written it in scientific notation. Now they wanted us to express it in both decimal and scientific notation. And when they're asking us to write it in decimal notation, they essentially want us to multiply this out, expand this out. And so the way to think about it, write these digits out. So I have 4.01534. |
Multiplying three numbers in scientific notation (example) Pre-Algebra Khan Academy.mp3 | Now they wanted us to express it in both decimal and scientific notation. And when they're asking us to write it in decimal notation, they essentially want us to multiply this out, expand this out. And so the way to think about it, write these digits out. So I have 4.01534. And if I'm just looking at this number, I start with the decimal right over here. Now every time I divide by 10, or if I multiply by 10 to the negative one, I'm moving this over to the left one spot. So 10 to the negative one, if I multiply by 10 to the negative one, that's the same thing as dividing by 10. |
Multiplying three numbers in scientific notation (example) Pre-Algebra Khan Academy.mp3 | So I have 4.01534. And if I'm just looking at this number, I start with the decimal right over here. Now every time I divide by 10, or if I multiply by 10 to the negative one, I'm moving this over to the left one spot. So 10 to the negative one, if I multiply by 10 to the negative one, that's the same thing as dividing by 10. And so I'm moving the decimal over to the left one. Here, I'm multiplying by 10 to the negative eight. Or you could say I'm dividing by 10 to the eighth power. |
Multiplying three numbers in scientific notation (example) Pre-Algebra Khan Academy.mp3 | So 10 to the negative one, if I multiply by 10 to the negative one, that's the same thing as dividing by 10. And so I'm moving the decimal over to the left one. Here, I'm multiplying by 10 to the negative eight. Or you could say I'm dividing by 10 to the eighth power. So I'm going to want to move the decimal to the left eight times. So move a decimal to left eight times. And one way to remember it, look, this is a very, very, very, very small number. |
Multiplying three numbers in scientific notation (example) Pre-Algebra Khan Academy.mp3 | Or you could say I'm dividing by 10 to the eighth power. So I'm going to want to move the decimal to the left eight times. So move a decimal to left eight times. And one way to remember it, look, this is a very, very, very, very small number. If I multiply this, I should get a smaller number. So I should be moving the decimal to the left. If this was a positive eight, then this would be a very large number. |
Multiplying three numbers in scientific notation (example) Pre-Algebra Khan Academy.mp3 | And one way to remember it, look, this is a very, very, very, very small number. If I multiply this, I should get a smaller number. So I should be moving the decimal to the left. If this was a positive eight, then this would be a very large number. And so if I multiply by a large power of 10, I'm going to be moving the decimal to the right. So this whole thing should evaluate to being smaller than 4.01534. So I move the decimal eight times to the left, eight times to the left. |
Multiplying three numbers in scientific notation (example) Pre-Algebra Khan Academy.mp3 | If this was a positive eight, then this would be a very large number. And so if I multiply by a large power of 10, I'm going to be moving the decimal to the right. So this whole thing should evaluate to being smaller than 4.01534. So I move the decimal eight times to the left, eight times to the left. I move it one time to the left to get it right over here. And then the next seven times, I'm just going to add zeros. So one, two, three, four, five, six, seven zeros. |
Multiplying three numbers in scientific notation (example) Pre-Algebra Khan Academy.mp3 | So I move the decimal eight times to the left, eight times to the left. I move it one time to the left to get it right over here. And then the next seven times, I'm just going to add zeros. So one, two, three, four, five, six, seven zeros. And I'll put a zero in front of the decimal just to clarify it. So now I notice if you include this digit right over here, I have a total of eight digits. I have an eight, I have eight, sorry, I have seven zeros and this digit gives us eight. |
Multiplying three numbers in scientific notation (example) Pre-Algebra Khan Academy.mp3 | So one, two, three, four, five, six, seven zeros. And I'll put a zero in front of the decimal just to clarify it. So now I notice if you include this digit right over here, I have a total of eight digits. I have an eight, I have eight, sorry, I have seven zeros and this digit gives us eight. So again, one, two, three, four, five, six, seven, eight. The best way to think about it is I started with the decimal right here. I moved once, twice, three, four, five, six, seven, eight times. |
Multiplying three numbers in scientific notation (example) Pre-Algebra Khan Academy.mp3 | I have an eight, I have eight, sorry, I have seven zeros and this digit gives us eight. So again, one, two, three, four, five, six, seven, eight. The best way to think about it is I started with the decimal right here. I moved once, twice, three, four, five, six, seven, eight times. That's what multiplying times 10 to the negative eight did for us. And I get this number right over here. And when you see a number like this, you start to appreciate why we write things in scientific notation. |
Multiplying three numbers in scientific notation (example) Pre-Algebra Khan Academy.mp3 | I moved once, twice, three, four, five, six, seven, eight times. That's what multiplying times 10 to the negative eight did for us. And I get this number right over here. And when you see a number like this, you start to appreciate why we write things in scientific notation. This is much easier to, it takes less space to write. And you immediately know roughly how big this number is. This is much harder to write. |
Multiplying three numbers in scientific notation (example) Pre-Algebra Khan Academy.mp3 | And when you see a number like this, you start to appreciate why we write things in scientific notation. This is much easier to, it takes less space to write. And you immediately know roughly how big this number is. This is much harder to write. You might even forget a zero when you write it or you might add a zero. And now the person has to sit and count the zeros to figure out essentially how large or get a rough sense of how large this thing is. It's one, two, three, four, five, six, seven zeros and you have this digit right here. |
Multiplying binomials intro Mathematics II High School Math Khan Academy.mp3 | Let's see if we can figure out the product of x minus four and x plus seven. And we want to write that product in standard quadratic form, which is just a fancy way of saying a form where you have some coefficient on the second degree term, ax squared, plus some coefficient b on the first degree term, plus the constant term. So this right over here would be standard quadratic form. So that's the form that we want to express this product in. And I encourage you to pause the video and try to work through it on your own. All right, now let's work through this. And the key when we're multiplying two binomials like this, or actually when we're multiplying any polynomials, is just to remember the distributive property that we all by this point know quite well. |
Multiplying binomials intro Mathematics II High School Math Khan Academy.mp3 | So that's the form that we want to express this product in. And I encourage you to pause the video and try to work through it on your own. All right, now let's work through this. And the key when we're multiplying two binomials like this, or actually when we're multiplying any polynomials, is just to remember the distributive property that we all by this point know quite well. So what we could view this as is we can distribute this x minus four, this entire expression, over the x and the seven. So we could say that this is the same thing as x minus four times x, plus x minus four times seven. So let's write that. |
Multiplying binomials intro Mathematics II High School Math Khan Academy.mp3 | And the key when we're multiplying two binomials like this, or actually when we're multiplying any polynomials, is just to remember the distributive property that we all by this point know quite well. So what we could view this as is we can distribute this x minus four, this entire expression, over the x and the seven. So we could say that this is the same thing as x minus four times x, plus x minus four times seven. So let's write that. So x minus four times x, or we could write this as x times x minus four. That's distributing the, or multiplying the x minus four times x. That's right there. |
Multiplying binomials intro Mathematics II High School Math Khan Academy.mp3 | So let's write that. So x minus four times x, or we could write this as x times x minus four. That's distributing the, or multiplying the x minus four times x. That's right there. Plus seven times x minus four. Times x minus four. Notice, all we did is distribute the x minus four. |
Multiplying binomials intro Mathematics II High School Math Khan Academy.mp3 | That's right there. Plus seven times x minus four. Times x minus four. Notice, all we did is distribute the x minus four. We took this whole thing and we multiplied it by each term over here. We multiplied x by x minus four, and we multiplied seven by x minus four. Now we see that we have these, I guess you'd call them two separate terms, and to simplify each of them, or to multiply them out, we just have to distribute in this first, we have to distribute this blue x, and over here we have to distribute this blue seven. |
Multiplying binomials intro Mathematics II High School Math Khan Academy.mp3 | Notice, all we did is distribute the x minus four. We took this whole thing and we multiplied it by each term over here. We multiplied x by x minus four, and we multiplied seven by x minus four. Now we see that we have these, I guess you'd call them two separate terms, and to simplify each of them, or to multiply them out, we just have to distribute in this first, we have to distribute this blue x, and over here we have to distribute this blue seven. So let's do that. So here we could say x times x is going to be x squared. X times, we have a negative here, so we could say negative four is going to be negative four x. |
Multiplying binomials intro Mathematics II High School Math Khan Academy.mp3 | Now we see that we have these, I guess you'd call them two separate terms, and to simplify each of them, or to multiply them out, we just have to distribute in this first, we have to distribute this blue x, and over here we have to distribute this blue seven. So let's do that. So here we could say x times x is going to be x squared. X times, we have a negative here, so we could say negative four is going to be negative four x. And just like that, we get x squared minus four x. And then over here, we have seven times x, so that's going to be plus seven x. And then we have seven times the negative four, which is negative 28. |
Multiplying binomials intro Mathematics II High School Math Khan Academy.mp3 | X times, we have a negative here, so we could say negative four is going to be negative four x. And just like that, we get x squared minus four x. And then over here, we have seven times x, so that's going to be plus seven x. And then we have seven times the negative four, which is negative 28. And we are almost done. We can simplify it a little bit more. We have two first degree terms here. |
Multiplying binomials intro Mathematics II High School Math Khan Academy.mp3 | And then we have seven times the negative four, which is negative 28. And we are almost done. We can simplify it a little bit more. We have two first degree terms here. If I have negative four x's, and to that I add seven x's, what is that going to be? Well, those two terms together, these two terms together are going to be negative four plus seven x's. Negative four plus seven. |
Multiplying binomials intro Mathematics II High School Math Khan Academy.mp3 | We have two first degree terms here. If I have negative four x's, and to that I add seven x's, what is that going to be? Well, those two terms together, these two terms together are going to be negative four plus seven x's. Negative four plus seven. Negative four plus seven x's. So all I'm doing here, I'm making it very clear that I'm adding these two coefficients, and then we have all the other terms. We have the x squared, x squared plus this, and then we have the minus 28. |
Multiplying binomials intro Mathematics II High School Math Khan Academy.mp3 | Negative four plus seven. Negative four plus seven x's. So all I'm doing here, I'm making it very clear that I'm adding these two coefficients, and then we have all the other terms. We have the x squared, x squared plus this, and then we have the minus 28. And we're at the home stretch. This would simplify to x squared. Now negative four plus seven is three, so this is going to be plus three x. |
Multiplying binomials intro Mathematics II High School Math Khan Academy.mp3 | We have the x squared, x squared plus this, and then we have the minus 28. And we're at the home stretch. This would simplify to x squared. Now negative four plus seven is three, so this is going to be plus three x. That's what these two middle terms simplify to, to three x, and then we have minus 28. Minus 28. And just like that, we are done. |
Multiplying binomials intro Mathematics II High School Math Khan Academy.mp3 | Now negative four plus seven is three, so this is going to be plus three x. That's what these two middle terms simplify to, to three x, and then we have minus 28. Minus 28. And just like that, we are done. And a fun thing to think about, and notice it's in the same form. If we were to compare, a is one, b is three, and c is negative 28. But it's interesting here to look at the pattern. |
Multiplying binomials intro Mathematics II High School Math Khan Academy.mp3 | And just like that, we are done. And a fun thing to think about, and notice it's in the same form. If we were to compare, a is one, b is three, and c is negative 28. But it's interesting here to look at the pattern. When we multiply these two binomials, especially these two binomials, where the coefficient on the x term was a one. Notice, we have x times x, that's what actually forms the x squared term over here. We have negative four, let me do this in a new color. |
Multiplying binomials intro Mathematics II High School Math Khan Academy.mp3 | But it's interesting here to look at the pattern. When we multiply these two binomials, especially these two binomials, where the coefficient on the x term was a one. Notice, we have x times x, that's what actually forms the x squared term over here. We have negative four, let me do this in a new color. We have negative four times, that's not a new color. We have negative four times seven, which is going to be negative 28. And then how did we get this middle term? |
Multiplying binomials intro Mathematics II High School Math Khan Academy.mp3 | We have negative four, let me do this in a new color. We have negative four times, that's not a new color. We have negative four times seven, which is going to be negative 28. And then how did we get this middle term? How did we get this three x? Well, you had the negative four x plus the seven x, or you had the negative four plus the seven times x. You had the negative four plus the seven times x. |
Multiplying binomials intro Mathematics II High School Math Khan Academy.mp3 | And then how did we get this middle term? How did we get this three x? Well, you had the negative four x plus the seven x, or you had the negative four plus the seven times x. You had the negative four plus the seven times x. So hopefully you see a little bit of a pattern here. If you're multiplying two binomials where the coefficients on the x term are both one, it's going to be x squared, and then the last term, the constant term, is going to be the product of these two constants, negative four and seven. And then the first degree term, right over here, its coefficient is going to be the sum of these two constants, negative four and seven. |
Multiplying binomials intro Mathematics II High School Math Khan Academy.mp3 | You had the negative four plus the seven times x. So hopefully you see a little bit of a pattern here. If you're multiplying two binomials where the coefficients on the x term are both one, it's going to be x squared, and then the last term, the constant term, is going to be the product of these two constants, negative four and seven. And then the first degree term, right over here, its coefficient is going to be the sum of these two constants, negative four and seven. Now this might, you could view this pattern, if you practice it, as just something that'll help you multiply binomials a little bit faster. But it's super important that you realize where this came from. This came from nothing more than applying a distributive property twice. |
Percent word problem example 1 Ratios, rates, and percentages 6th grade Khan Academy.mp3 | This is for guavas. And it's only today. So I say, you know what, let me go buy a bunch of guavas. So I go and I buy 6 guavas. So I buy 6 guavas. It ends up when I go to the register, and we're assuming no tax for it's a grocery and I live in a state where they don't tax groceries. So for the 6 guavas, they charge me, I get the 30% off, they charge me $12.60. |
Percent word problem example 1 Ratios, rates, and percentages 6th grade Khan Academy.mp3 | So I go and I buy 6 guavas. So I buy 6 guavas. It ends up when I go to the register, and we're assuming no tax for it's a grocery and I live in a state where they don't tax groceries. So for the 6 guavas, they charge me, I get the 30% off, they charge me $12.60. So this is the 30% off sale price on 6 guavas. I go home and then my wife tells me, you know, Sal, can you go get 2 more guavas tomorrow? I say sure. |
Percent word problem example 1 Ratios, rates, and percentages 6th grade Khan Academy.mp3 | So for the 6 guavas, they charge me, I get the 30% off, they charge me $12.60. So this is the 30% off sale price on 6 guavas. I go home and then my wife tells me, you know, Sal, can you go get 2 more guavas tomorrow? I say sure. So the next day I go and I want to buy 2 more guavas. So 2 guavas. But now the sale is off. |
Percent word problem example 1 Ratios, rates, and percentages 6th grade Khan Academy.mp3 | I say sure. So the next day I go and I want to buy 2 more guavas. So 2 guavas. But now the sale is off. There's no more 30%. That was only that first day that I bought the 6. So how much are those 2 guavas going to cost me? |
Percent word problem example 1 Ratios, rates, and percentages 6th grade Khan Academy.mp3 | But now the sale is off. There's no more 30%. That was only that first day that I bought the 6. So how much are those 2 guavas going to cost me? How much are those 2 guavas going to cost at full price? At full price. So a good place to start is to think about how much would those 6 guavas cost us at full price? |