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24-29, pgs. 155-156. Created by Sal Khan.
Want to join the conversation?
- What are vertices?(4 votes)
- how do i solve this problem (5th grade) which number below has a 3 that has a value 1/10 as much as the 3 in 9,532.07 a.8,447.36 b.7.143.14 c.3,748.51 d.5,307.06(2 votes)
- Can anyone explain #29 better? If he factors out 32 as 16 * 16, then 8(16 * 16), then cancels out one of the 16s, why does he still have the initial 16 at the start of the equation (16 * 20)? That makes very little sense to me.
And how does he end up at 16(20 + 16)? Where did the unaccounted for large number go that would have been the result of 8 * 32?(2 votes)
- The question is √(16∙20 + 8∙32), so I will explain what he is doing, then try to answer your questions:
So, you could just use PEMDAS and grind through all that math, OR you can factor out a common factor and use distribution to do the heavy lifting in just a few seconds. Remember that distribution is that handy math property so that
4(m + n) = 4m + 4n
but also, 4m + 4n can be factored to 4 (m + n)
It is a great time-saver in mental math, and also an important basis for algebra.
In his case, he is using 16
If I rewrite this original problem as
√(16∙a + b∙32), it is easier to see that both 16 and 32 contain a juicy factor equal to 16. You can use distribution to take out 16 from both those terms:
√(16∙a + b∙32) = √(16(a + 2b))
Now we make use of the fact that √XY = √X∙√Y and the first part here is √16
Let's see what is left when 16 is factored out of each term of the (16∙20 + 8∙32)
a = 20
b = 8
when we separate out the 16, we have (a + 2b)
that means we have 20 + 2∙8 which equals 20 + 16 = 36
So √16 ∙ √36 is the result, which we can recognize as 4 ∙ 6 = 24, the answer
I actually was more ambitious than Sal. I saw that √(16∙20 + 8∙32) can be rewritten as
8∙8∙4) =√(64∙5 + 64∙4) = √64(5 + 4) =√64 ∙ √(5 + 4) = √64 ∙ √9 = 8∙3 = 24
The grinding way is √(16∙20 + 8∙32) = √(320 + 256) = √576
Not a lot of people recognize that square root without a calculator, but it is the exact same answer of 24 (That is √576 = 24)
Now, what he says about the 32 is that 32 = 16 times 2, and he wrote 16∙2 not 16²
In that row, he had 8∙16∙2 which equals 16∙16, which he wrote as his next row. There is NO extra 8 times 16∙16
He probably should have spaced the numbers better.(0 votes)
- in question 24 , i cant understand why is the answer not 22 .. cant get it(1 vote)
- The total length of the rope according to the question stem is 40 feet. Also given, is that the rope can be cut into two segments, one of which is 18 feet longer than the other so the two segments can be represented as x (shorter segment) and x+18 (longer segment). From there, you can set up the equation: shorter segment + longer segment = 40. Alternatively, another way of writing this is x + x+18 = 40 or 2x +18 = 40. Solving for x, you get 2x = 22 (after subtracting 18 from both sides) which simplifies to x = 11 feet. Hope this helps.(2 votes)
- how do you divide fractions(1 vote)
- if X/Y is a fraction and A/B is another fraction .
Then (X/Y)/(A/B) = (X x B)/(Y x A) .
For Example , (2/3)/(4/5) = (2 x 5)/(3 x 4) = 10/12 .
You can further simplify them by dividing the numerator and denominator by common factors.
In case of 10 and 12 , 2 is a common factor . so you divide 10 by 2 , you get 5 , similarly , you divide 12 by 2 , you get 6 . so 10/12 = 5/6 .(1 vote)
- A board measureing 30 5/6 is to be cut into 3 equal places. How long will each piece be?
Express the answer using the same format as the orinal board length as a whole number and
Fraction. Additionally express the answer as a number rounded to two decimal points.(1 vote)
- suppose you can send 2 text messages per minute, how many text messages you can send in 25 days(1 vote)
- lets convert this into a mathematics equation.
The texting rate is 2 text/ minute.
60 minutes/ 1 hour is equivalent in time.
24 hours/ 1 day is equivalent in time.
The number of days is 25 so we will leave it as 1/25 days
now to multiple
(2 texts/ 1 minute)*(60 minutes/ 1 hour)*(24 hours/ 1 day)*(1 day/25 days)
Everything crosses out aside from 72,000 texts/ 25 days. This means in 25 days you could text 72,000 times.(1 vote)
We're on problem 24. A rope 40 feet long is cut into two pieces. Let's draw that. Let's say that's where I cut them. If one piece is 18 feet longer than the other, what is the length in feet of the shorter piece? So let's say this is the shorter piece. This is the longer piece. x plus 18. And they also tell us that both pieces combined-- it's a 40-foot long rope, so if I add these together it should be equal to 40. So x plus x, plus 18 is equal to 40. x plus x is equal to 2x, plus 18 is equal to 40. And then I get 2x is equal to-- subtract 18 from both sides, that's what? 22. x is equal to 11. Next problem. Let me switch colors. 25. The Earth travels around the sun at a speed of approximately 18.5 miles per second. This approximate speed is how many miles per hour? So how many seconds are there in a mile? So you have 60 seconds per minute. And then you have times 60 minutes per hour. So you get 60 times 60 is 3,600. And the units work out, too. Minutes in the denominator. Minutes in the numerator. So you get 3,600 seconds per hour. So 18.5 miles per second, you're going to go 3,600 times as far in an hour. So let's multiply that times 3,600 seconds per hour. And you'll get whatever this number is. So let's multiply it out. 18.5. I'm going to multiply it times 36 and then add two 0's later. 6 times 5 is 30. 6 times 8 is 48, plus 3 is 51. 6 times 1 is 6, plus 5 is 11. 3 times 5 is 15. 3 times 8 is 24, plus 1 is 25. 3 times 1 is 3, plus 2 is 5. And so you get a 0, a 6, a 6, and a 6. And then I have one number behind the decimal point. So 18.5 times 36 is 666. But I'm multiplying it not by 36 but by 3,600. So 18.5 times 3,600 is going to be equal to this times 100. So 66,600. And the units work out. Second in the denominator. Second in the numerator. Miles per hour. And that's choice D. And they didn't even let you approximate it, because choice C is really close. You really have to, as far as I can tell, go through the multiplication. Next problem. 26. If the quotient a/b is positive-- that's what they're telling us-- which of the following must be true? So if a/b is positive, that essentially tells us that a and b have to have the same sign, right? The only way you can get a positive number when you divide two numbers is if they're both positive, or they're both negative. If they were different signs-- if one was positive and one was negative, or one was negative and one was positive-- then this would be a negative number. So this tells us that they're the same sign. So statement A, choice A, tells us a is greater than 0. No, this doesn't have to be true. They both could be less than 0. B tells us that b is greater than 0. Once again, this isn't necessarily true. They both could be less than 0 and this would still be true. Choice C. ab is greater than 0. Well, think about it. This implies that both a and b have to have the same sign. If a and b have the same sign when I multiply them, I'm still going to get a number bigger than 0, right? A positive times a positive is a positive. A negative times a negative is a positive. So it's choice C. Problem 27. And I saw this coming, so I drew this ahead of time. Actually, no I didn't. Well, I thought I had drawn it ahead of time. Let me actually open it up. I thought I had drawn it. Oh, well, it must have gotten erased. Well, let's just do it in real time, then. So it says, the dots on the graph above indicate the weights and fuel efficiency ratings for 20 cars. How many of the cars weigh more than 2,500 pounds-- so weight is more than 2,500-- and also get more than 25 miles per gallon? So fuel efficiency is more than 22 miles per gallon. And notice, they said more than 22, not 22 miles or more. And they said more than 2,500, not 2,500 pounds or more. Let me just draw the part of the graph that seems relevant. So if that's the weight axis-- so they have a couple at 25, 26, 27, 28, 29, 30. And then, we want to know above 22. So let me actually draw all of it. So that's 25. I'm only drawing part of the graph because I don't want to go through the exercise. Because I actually already did it and I can't find it, all of a sudden. But I don't want to stop recording this video. So 20. This is 22. This is 24. They don't label the 22 on their drawing. This is 26. This is 28. And so they want to know everything that weighs more than 2,500 pounds. So everything to the right of that. Because all of these tend to be on integer numbers. None of them are in between. And they say, and get more than 22 miles per gallon. So let me draw all of the ones that apply there. So let's see, at 2,600 pounds, there's two, at 24 and 26 miles per gallon. At 2,600 there's two points right there. At 2,700 there are three points. 1, 2, but the other one falls on the 22 miles per gallon, but we're talking about greater than that. And then-- let's see, that's 26, 27-- 28 has one down here and one up here. When I say 28, I mean 2,800 pounds. I'm just trying to draw it for you. And then the rest of them. There's two here and there's one here. But all we care about is the ones in this. There's a bunch more over here. I could draw them real fast, if you want me to, just so you can visualize the graph. So it looks something like that. And then there's one more here. But all we care about is this range, and there's 1, 2, 3, 4, 5. Five of those. So the answer is B. A lot of work for a fairly straightforward problem. Let me get some clean space. Problem 28. How many minutes does it take John to type y words if he types at a rate of x words per minute? Now, this is a fairly straightforward thing, but it can get confusing. Do I divide x by y? Y by x? I think a lot of times this becomes a lot simpler if you just pick numbers. So if we just phrased it, how long does it take him to type 100 words if he can type, I don't know, 50 words per minute? Now your brain would say, oh, if I can do 50 words per minute, it's going to take me 2 minutes to do 100 words. And how did you get 2 minutes? You said 100 divided by 50. So if you need to type y words and you can type at x words per minute, the answer is y divided by x. Let's see, that's not one of the choices. But I think they've made a mistake, because this is a simple enough problem that I have conviction. And if you look at the choices, choice A and choice B, at least in the book I have, they wrote x/y, so I think that's a typo. One of those probably should have read y/x. So I'm going to stick by my answer. I think this was another mistake in the book. y/x. You might want to check the answer key, just to make sure, but I have no doubt in my conviction on this one. Let me see, do I have time for the next one? Well, sure. Problem 29. I'm running out of space. The square root-- this is just to make sure you know how to do square roots-- of 16 times 20, in order of operations, plus 8 times 32. So there's a bunch of ways we could think about this, but the easiest way, instead of doing all of the multiplication, let's see if we can factor 16 out of both of these numbers. You could multiple them out and try to figure out the square root, but it'll take you a long time. So if we rewrite this as 16 times 20 plus, what? This could also be written if we took-- 32 is 16 times 2, right? So 8 times 32 is the same thing as 8 times 16 times 2, which is the same thing as 16 times 16. So we could have rewritten this as 8 times 32 is 16, times 16. So if we're factoring a 16 out, that's 16 times 20, plus 16. If you were to just multiply this out it would be 16 times 20, plus 16 times 16, which is the same thing as 8 times 32. And I'm doing all of this. You could just do the math yourself, but it takes a lot of time if you don't have a calculator. Well now, you can say, well, this is the same thing as the square root of 16 times the square root of 20 plus 16. Well, this is plus or minus 4, although I don't think they're going to make us worry about the negative square root. So let's just say it's 4. Looking at the choices--yeah, everything 's positive. So it's 4 times. And then, what's this? The square root of 36. So that equals 4 times 6, which is equal to 24. And that's choice B. Se you in the next video.