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# Devil's Dozen!!!

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19 Mar 2012, 06:54
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The next set of tough and tricky DS questions. I'll post OA's with detailed explanations after some discussion. Please, post your solutions along with the answers. Good luck!

1. Jules and Jim both invested certain amount of money in bond M for one year, which pays for 12% simple interest annually. If no other investment were made, then Jules initial investment in bond M was how many dollars more than Jim's investment in bond M.
(1) In one year Jules earned $24 more than Jim from bond M. (2) If the interest were 20% then in one year Jules would have earned$40 more than Jim from bond M.

Solution: http://gmatclub.com/forum/devil-s-dozen ... l#p1063846

2. If n is a positive integer and p is a prime number, is p a factor of n!?
(1) p is a factor of (n+2)!-n!
(2) p is a factor of (n+2)!/n!

Solution: http://gmatclub.com/forum/devil-s-dozen ... l#p1063847

3. If x and y are integers, is y an even integer?
(1) 4y^2+3x^2=x^4+y^4
(2) y=4-x^2

Solution: http://gmatclub.com/forum/devil-s-dozen ... l#p1063848

4. Of the 58 patients of Vertigo Hospital, 45 have arachnophobia. How many of the patients have acrophobia?
(1) The number of patients of Vertigo Hospital who have both arachnophobia and acrophobia is the same as the number of patients who have neither arachnophobia nor acrophobia.
(2) 32 patients of Vertigo Hospital have arachnophobia but not acrophobia.

Solution: http://gmatclub.com/forum/devil-s-dozen ... l#p1063863

5. If at least one astronaut do NOT listen to Bach at Solaris space station, then how many of 35 astronauts at Solaris space station listen to Bach?
(1) Of the astronauts who do NOT listen to Bach 56% are male.
(2) Of the astronauts who listen to Bach 70% are female.

Solution: http://gmatclub.com/forum/devil-s-dozen ... l#p1063867

6. Is the perimeter of triangle with the sides a, b and c greater than 30?
(1) a-b=15.
(2) The area of the triangle is 50.

Solution: http://gmatclub.com/forum/devil-s-dozen ... l#p1063871

7. Set A consists of k distinct numbers. If n numbers are selected from the set one-by-one, where n<=k, what is the probability that numbers will be selected in ascending order?
(1) Set A consists of 12 even consecutive integers.
(2) n=5.

Solution: http://gmatclub.com/forum/devil-s-dozen ... l#p1063874

8. If p is a positive integer, what is the remainder when p^2 is divided by 12?
(1) p is greater than 3.
(2) p is a prime.

Solution: https://gmatclub.com/forum/devil-s-doze ... l#p1063884

9. The product of three distinct positive integers is equal to the square of the largest of the three numbers, what is the product of the two smaller numbers?
(1) The average (arithmetic mean) of the three numbers is 34/3.
(2) The largest number of the three distinct numbers is 24.

Solution: https://gmatclub.com/forum/devil-s-doze ... l#p1063886

10. There is at least one viper and at least one cobra in Pandora's box. How many cobras are there?
(1) There are total 99 snakes in Pandora's box.
(2) From any two snakes from Pandora's box at least one is a viper.

Solution: http://gmatclub.com/forum/devil-s-dozen ... l#p1063888

11. Alice has $15, which is enough to buy 11 muffins and 7 brownies, is$45 enough to buy 27 muffins and 27 brownies?
(1) $15 is enough to buy 7 muffins and 11 brownies. (2)$15 is enough to buy 10 muffins and 8 brownies.

Solution: http://gmatclub.com/forum/devil-s-dozen ... l#p1063892

12. If x>0 and xy=z, what is the value of yz?
(1) $$x^2*y=3$$.
(2) $$\sqrt{x*y^2}=3$$.

Solution: http://gmatclub.com/forum/devil-s-dozen ... l#p1063894

13. Buster leaves the trailer at noon and walks towards the studio at a constant rate of B miles per hour. 20 minutes later, Charlie leaves the same studio and walks towards the same trailer at a constant rate of C miles per hour along the same route as Buster. Will Buster be closer to the trailer than to the studio when he passes Charlie?
(1) Charlie gets to the trailer in 55 minutes.
(2) Buster gets to the studio at the same time as Charlie gets to the trailer.

Solution: http://gmatclub.com/forum/devil-s-dozen ... l#p1063897
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03 Sep 2013, 14:40
Thanks Bunuel, my mistake I will memorize this formula for forever.
Really a silly mistake done by me
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09 Sep 2013, 21:11
Bunuel wrote:
4. Of the 58 patients of Vertigo Hospital, 45 have arachnophobia. How many of the patients have acrophobia?

Tricky question.

(1) The number of patients of Vertigo Hospital who have both arachnophobia and acrophobia is the same as the number of patients who have neither arachnophobia nor acrophobia. Use double-set matrix:
Attachment:
Vertigo.png
As you can see # of patients who has acrophobia is 58-45=13. Sufficient.

(2) 32 patients of Vertigo Hospital have arachnophobia but not acrophobia. Clearly insufficient.

In B. If 45 have arachnophobia , and 32 have only arachnophobia, so # of patients common to both is 45-32=13.
So total number of patients are : 32 +13 + only acrophobia = 58.
Whats wrong in my approach ?
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10 Sep 2013, 01:31
ygdrasil24 wrote:
Bunuel wrote:
4. Of the 58 patients of Vertigo Hospital, 45 have arachnophobia. How many of the patients have acrophobia?

Tricky question.

(1) The number of patients of Vertigo Hospital who have both arachnophobia and acrophobia is the same as the number of patients who have neither arachnophobia nor acrophobia. Use double-set matrix:
Attachment:
Vertigo.png
As you can see # of patients who has acrophobia is 58-45=13. Sufficient.

(2) 32 patients of Vertigo Hospital have arachnophobia but not acrophobia. Clearly insufficient.

In B. If 45 have arachnophobia , and 32 have only arachnophobia, so # of patients common to both is 45-32=13.
So total number of patients are : 32 +13 + only acrophobia = 58.
Whats wrong in my approach ?

Aren't you missing something? What about those who has neither? There is a matrix given in my solution, plug your values, will you get the answer?
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10 Sep 2013, 10:21
mithun2vrs wrote:
Bunuel wrote:
7. Set A consists of k distinct numbers. If n numbers are selected from the set one-by-one, where n<=k, what is the probability that numbers will be selected in ascending order?

(1) Set A consists of 12 even consecutive integers;
(2) n=5.

We should understand following two things:
1. The probability of selecting any n numbers from the set is the same. Why should any subset of n numbers have higher or lower probability of being selected than some other subset of n numbers? Probability doesn't favor any particular subset.

2. Now, consider that the subset selected is $$\{x_1, \ x_2, \ ..., \ x_n\}$$, where $$x_1<x_2<...<x_n$$. We can select this subset of numbers in $$n!$$ # of ways and out of these n! ways only one, namely $$\{x_1, \ x_2, \ ..., \ x_n\}$$ will be in ascending order. So 1 out of n!. $$P=\frac{1}{n!}$$.

Hence, according to the above the only thing we need to know to answer the question is the size of the subset (n) we are selecting from set A.

I am not sure if I understand correctly.

There are a total of k integers. So probability of selecting first integer 1/k, now probability of second integer out of k-1 integer is 1/(k-1) and so on... so selecting n integers would be 1/(k*(k-1)*(k-2)*...(k-(n-1)) so we need both k & n. Am I missing something here?

Hi Bunnel- Even I have same doubt. Is this question discussed in detail ? If so, it would be great if you can post the link. I know its pretty old now..but yet classic.
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11 Sep 2013, 03:00
suyash23n wrote:
mithun2vrs wrote:
Bunuel wrote:
7. Set A consists of k distinct numbers. If n numbers are selected from the set one-by-one, where n<=k, what is the probability that numbers will be selected in ascending order?

(1) Set A consists of 12 even consecutive integers;
(2) n=5.

We should understand following two things:
1. The probability of selecting any n numbers from the set is the same. Why should any subset of n numbers have higher or lower probability of being selected than some other subset of n numbers? Probability doesn't favor any particular subset.

2. Now, consider that the subset selected is $$\{x_1, \ x_2, \ ..., \ x_n\}$$, where $$x_1<x_2<...<x_n$$. We can select this subset of numbers in $$n!$$ # of ways and out of these n! ways only one, namely $$\{x_1, \ x_2, \ ..., \ x_n\}$$ will be in ascending order. So 1 out of n!. $$P=\frac{1}{n!}$$.

Hence, according to the above the only thing we need to know to answer the question is the size of the subset (n) we are selecting from set A.

I am not sure if I understand correctly.

There are a total of k integers. So probability of selecting first integer 1/k, now probability of second integer out of k-1 integer is 1/(k-1) and so on... so selecting n integers would be 1/(k*(k-1)*(k-2)*...(k-(n-1)) so we need both k & n. Am I missing something here?

Hi Bunnel- Even I have same doubt. Is this question discussed in detail ? If so, it would be great if you can post the link. I know its pretty old now..but yet classic.

Yes:
devil-s-dozen-129312-20.html#p1064472
devil-s-dozen-129312-20.html#p1064519
devil-s-dozen-129312-20.html#p1064618
devil-s-dozen-129312-40.html#p1075275

Hope this helps.
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07 Nov 2013, 04:55
Bunuel wrote:
11. Alice has $15, which is enough to buy 11 muffins and 7 brownies, is$45 enough to buy 27 muffins and 27 brownies?

700+ question.

Given: $$11m+7b\leq{15}$$, where $$m$$ and $$b$$ are prices of one muffin and one brownie respectively.
Question: is $$27m+27b\leq{45}$$? --> $$9m+9b\leq{15}$$. Question basically asks whether we can substitute 2 muffins with 2 brownies.

Now if $$m>b$$ we can easily substitute 2 muffins with 2 brownies (since $$2m$$ will be more than $$2b$$). But if $$m<b$$ we won't know this for sure.

But consider the case when we are told that we can substitute 3 muffins with 3 brownies. In both cases ($$m>b$$ or $$m<b$$) it would mean that we can substitute 2 (so less than 3) muffins with 2 brownies, but again we won't be sure whether we can substitute 4 (so more than 3) muffins with 4 brownies.

(1) $15 is enough to buy 7 muffins and 11 brownies --> $$7m+11b\leq{15}$$: we can substitute 4 muffins with 4 brownies, so according to above we can surely substitute 2 muffins with 2 brownies. Sufficient. (1)$15 is enough to buy 10 muffins and 8 brownies --> $$10m+8b\leq{15}$$: we can substitute 1 muffin with 1 brownie, so according to above this is does not ensure that we can substitute 2 muffins with 2 brownies. Not sufficient.

This might be a silly question, but i really don't understand how, if m < b, then 3m could replace 3b?
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07 Nov 2013, 06:43
sidvish wrote:
Bunuel wrote:
11. Alice has $15, which is enough to buy 11 muffins and 7 brownies, is$45 enough to buy 27 muffins and 27 brownies?

700+ question.

Given: $$11m+7b\leq{15}$$, where $$m$$ and $$b$$ are prices of one muffin and one brownie respectively.
Question: is $$27m+27b\leq{45}$$? --> $$9m+9b\leq{15}$$. Question basically asks whether we can substitute 2 muffins with 2 brownies.

Now if $$m>b$$ we can easily substitute 2 muffins with 2 brownies (since $$2m$$ will be more than $$2b$$). But if $$m<b$$ we won't know this for sure.

But consider the case when we are told that we can substitute 3 muffins with 3 brownies. In both cases ($$m>b$$ or $$m<b$$) it would mean that we can substitute 2 (so less than 3) muffins with 2 brownies, but again we won't be sure whether we can substitute 4 (so more than 3) muffins with 4 brownies.

(1) $15 is enough to buy 7 muffins and 11 brownies --> $$7m+11b\leq{15}$$: we can substitute 4 muffins with 4 brownies, so according to above we can surely substitute 2 muffins with 2 brownies. Sufficient. (1)$15 is enough to buy 10 muffins and 8 brownies --> $$10m+8b\leq{15}$$: we can substitute 1 muffin with 1 brownie, so according to above this is does not ensure that we can substitute 2 muffins with 2 brownies. Not sufficient.

This might be a silly question, but i really don't understand how, if m < b, then 3m could replace 3b?

The following post might help: devil-s-dozen-129312-80.html#p1246581
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23 May 2014, 00:29
Bunuel wrote:
5. If at least one astronaut do NOT listen to Bach at Solaris space station, then how many of 35 astronauts at Solaris space station listen to Bach?

Also very tricky.

(1) Of the astronauts who do NOT listen to Bach 56% are male --> if # of astronauts who do NOT listen to Bach is $$x$$ then $$0.56x$$ is # of males who do NOT listen to Bach. Notice that $$0.56x=\frac{14}{25}x$$ must be an integer. Hence x must be a multiple of 25: 25, 50, 75, ... But $$x$$ (# of astronauts who do NOT listen to Bach) must also be less than (or equal to) 35. So $$x$$ can only be 25, which makes # of astronauts who do listen to Bach equal to 35-25=10. Sufficient.

(2) Of the astronauts who listen to Bach 70% are female. Now, if we apply the same logic here we get that, if # of astronauts who listen to Bach is $$y$$ then $$0.7y$$ is # of females who listen to Bach: $$0.7y=\frac{7}{10}y$$ must be an integer. Hence it must be a multiple of 10, but in this case it can take more than 1 value: 10, 20, 30. So, this statement is not sufficient.

Hi Bunnel,

How we are getting 14/25

Thanks.
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23 May 2014, 01:15
PathFinder007 wrote:
Bunuel wrote:
5. If at least one astronaut do NOT listen to Bach at Solaris space station, then how many of 35 astronauts at Solaris space station listen to Bach?

Also very tricky.

(1) Of the astronauts who do NOT listen to Bach 56% are male --> if # of astronauts who do NOT listen to Bach is $$x$$ then $$0.56x$$ is # of males who do NOT listen to Bach. Notice that $$0.56x=\frac{14}{25}x$$ must be an integer. Hence x must be a multiple of 25: 25, 50, 75, ... But $$x$$ (# of astronauts who do NOT listen to Bach) must also be less than (or equal to) 35. So $$x$$ can only be 25, which makes # of astronauts who do listen to Bach equal to 35-25=10. Sufficient.

(2) Of the astronauts who listen to Bach 70% are female. Now, if we apply the same logic here we get that, if # of astronauts who listen to Bach is $$y$$ then $$0.7y$$ is # of females who listen to Bach: $$0.7y=\frac{7}{10}y$$ must be an integer. Hence it must be a multiple of 10, but in this case it can take more than 1 value: 10, 20, 30. So, this statement is not sufficient.

Hi Bunnel,

How we are getting 14/25

Thanks.

Hi Pathfinder,
Suppose x = no. of people who do not listen to Bach , 0< x<=35

56% of x = x*(56/100) = x*(14/25) ..... on taking out 4(the common factor) both from Numerator and Denominator

Now, this 56% must be an integer implies x should be a multiple of 25
and x<=35 as given above
----------------> x =25, hence sufficient.
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23 May 2014, 04:54
Bunuel wrote:
11. Alice has $15, which is enough to buy 11 muffins and 7 brownies, is$45 enough to buy 27 muffins and 27 brownies?

700+ question.

Given: $$11m+7b\leq{15}$$, where $$m$$ and $$b$$ are prices of one muffin and one brownie respectively.
Question: is $$27m+27b\leq{45}$$? --> $$9m+9b\leq{15}$$. Question basically asks whether we can substitute 2 muffins with 2 brownies.

Now if $$m>b$$ we can easily substitute 2 muffins with 2 brownies (since $$2m$$ will be more than $$2b$$). But if $$m<b$$ we won't know this for sure.

But consider the case when we are told that we can substitute 3 muffins with 3 brownies. In both cases ($$m>b$$ or $$m<b$$) it would mean that we can substitute 2 (so less than 3) muffins with 2 brownies, but again we won't be sure whether we can substitute 4 (so more than 3) muffins with 4 brownies.

(1) $15 is enough to buy 7 muffins and 11 brownies --> $$7m+11b\leq{15}$$: we can substitute 4 muffins with 4 brownies, so according to above we can surely substitute 2 muffins with 2 brownies. Sufficient. (1)$15 is enough to buy 10 muffins and 8 brownies --> $$10m+8b\leq{15}$$: we can substitute 1 muffin with 1 brownie, so according to above this is does not ensure that we can substitute 2 muffins with 2 brownies. Not sufficient.

Hi Bunnel,

I just want to know why statement 2 is not sufficient. following is my logic

from question 11m+7b =15

st1- 7m+11b = 15

subtract st2 from 1

4m-4b = 0
m=b

so in question we can say 18b = 15
b=15/18=5/6. using this we can get the price for 27m and 27b

now same i can get from st2 then why this is not sufficient?

Thanks.
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23 May 2014, 05:14
PathFinder007 wrote:
Bunuel wrote:
11. Alice has $15, which is enough to buy 11 muffins and 7 brownies, is$45 enough to buy 27 muffins and 27 brownies?

700+ question.

Given: $$11m+7b\leq{15}$$, where $$m$$ and $$b$$ are prices of one muffin and one brownie respectively.
Question: is $$27m+27b\leq{45}$$? --> $$9m+9b\leq{15}$$. Question basically asks whether we can substitute 2 muffins with 2 brownies.

Now if $$m>b$$ we can easily substitute 2 muffins with 2 brownies (since $$2m$$ will be more than $$2b$$). But if $$m<b$$ we won't know this for sure.

But consider the case when we are told that we can substitute 3 muffins with 3 brownies. In both cases ($$m>b$$ or $$m<b$$) it would mean that we can substitute 2 (so less than 3) muffins with 2 brownies, but again we won't be sure whether we can substitute 4 (so more than 3) muffins with 4 brownies.

(1) $15 is enough to buy 7 muffins and 11 brownies --> $$7m+11b\leq{15}$$: we can substitute 4 muffins with 4 brownies, so according to above we can surely substitute 2 muffins with 2 brownies. Sufficient. (1)$15 is enough to buy 10 muffins and 8 brownies --> $$10m+8b\leq{15}$$: we can substitute 1 muffin with 1 brownie, so according to above this is does not ensure that we can substitute 2 muffins with 2 brownies. Not sufficient.

Hi Bunnel,

I just want to know why statement 2 is not sufficient. following is my logic

from question 11m+7b =15

st1- 7m+11b = 15

subtract st2 from 1

4m-4b = 0
m=b

so in question we can say 18b = 15
b=15/18=5/6. using this we can get the price for 27m and 27b

now same i can get from st2 then why this is not sufficient?

Thanks.

There is a huge difference between $$11m+7b\leq{15}$$ and $$11m+7={15}$$. "Enough" should be translated as $$\leq$$ only.
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02 Jul 2014, 22:20
Bunuel wrote:
priyavenugopal wrote:
Sure. Thanks. I just tried it in the vry usual algebraic eqn solving way.

Given,
From qn stem - 11m + 7b = 15 (i took the 'enough' word as 'equal to'..as the 'enough' keyword can be rephrased as '<=', i thought that,considering the max limit '=' would do in such price lmt case..not sure, whthr thrs any trap thr..)
From (1) stmt - 7m + 11b = 15
From (2) stmt - 10m + 8b = 15
Solving qn stem n stmt 1 eqns, got m and b as 5/6 (didnt get an int val thou..not sure whether thrs a trap)
Solving qn stem n stmt 2 eqns, got m and b as 5/6

Having m and b values, we can find whthr 45$is enough to buy 27m + 27b or not. (i.e. applying m n b in eqn 27m + 27b = 45 => 2 * 27 * 5/6 = 45). So, arrived at D. Not even to analyze the rest of it, I must say that there is a huge difference between $$11m+7b\leq{15}$$ and $$11m+7={15}$$. You can not just write = sing instead of <= just because it's more convenient and "enough" should be translated as <= only. Bunuel the statement says 15$ is enough, so it implies <= 15$, consider max situation that it takes 15$ to buy 10m and 8b, which implies 45$is sufficient . Can we not imply that for any amount less than 15$ , the solution will hold?.

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03 Jul 2014, 05:57
ayushee01 wrote:
Bunuel wrote:
priyavenugopal wrote:
Sure. Thanks. I just tried it in the vry usual algebraic eqn solving way.

Given,
From qn stem - 11m + 7b = 15 (i took the 'enough' word as 'equal to'..as the 'enough' keyword can be rephrased as '<=', i thought that,considering the max limit '=' would do in such price lmt case..not sure, whthr thrs any trap thr..)
From (1) stmt - 7m + 11b = 15
From (2) stmt - 10m + 8b = 15
Solving qn stem n stmt 1 eqns, got m and b as 5/6 (didnt get an int val thou..not sure whether thrs a trap)
Solving qn stem n stmt 2 eqns, got m and b as 5/6

Having m and b values, we can find whthr 45$is enough to buy 27m + 27b or not. (i.e. applying m n b in eqn 27m + 27b = 45 => 2 * 27 * 5/6 = 45). So, arrived at D. Not even to analyze the rest of it, I must say that there is a huge difference between $$11m+7b\leq{15}$$ and $$11m+7={15}$$. You can not just write = sing instead of <= just because it's more convenient and "enough" should be translated as <= only. Bunuel the statement says 15$ is enough, so it implies <= 15$, consider max situation that it takes 15$ to buy 10m and 8b, which implies 45$is sufficient . Can we not imply that for any amount less than 15$ , the solution will hold?.

$15 is enough to buy 10 muffins and 8 brownies. 15*3=$45 is enough to buy 10*3=30 muffins and 8*3=24 brownies.

But this is not sufficient to say whether \$45 enough to buy 27 muffins and 27 brownies.
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03 Jul 2014, 06:08
correct , but suppose i consider the max value for 11m and 7b = 15 and max value for 10m + 8b as 15, we can solve and get 5/6. earlier you mentioned that we cannot consider less than as equal to , but can we consider the max situation ,i.e, the max value that 11m and 7b can be 15 (ques says 15 is enough)
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03 Jul 2014, 06:12
ayushee01 wrote:
correct , but suppose i consider the max value for 11m and 7b = 15 and max value for 10m + 8b as 15, we can solve and get 5/6. earlier you mentioned that we cannot consider less than as equal to , but can we consider the max situation ,i.e, the max value that 11m and 7b can be 15 (ques says 15 is enough)

______________
No we cannot.
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23 Jul 2014, 07:04
why there are no answers to some questions??
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23 Jul 2014, 09:18
HarvinderSaini wrote:
why there are no answers to some questions??

That's not true. Links to solutions are here: devil-s-dozen-129312.html#p1060761
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18 Aug 2014, 08:18
Bunuel wrote:
4. Of the 58 patients of Vertigo Hospital, 45 have arachnophobia. How many of the patients have acrophobia?

Tricky question.

(1) The number of patients of Vertigo Hospital who have both arachnophobia and acrophobia is the same as the number of patients who have neither arachnophobia nor acrophobia. Use double-set matrix:

As you can see # of patients who has acrophobia is 58-45=13. Sufficient.

(2) 32 patients of Vertigo Hospital have arachnophobia but not acrophobia. Clearly insufficient.

Hi Bunuel, I don't understand how St 2 is clearly insufficient?
If we do it via venn diagram method, we know that a+b+c = 58
and we know from st 2, that a=32, so b=13, thus c=0.
But then from st 1 we get c = 13.
Where am I going wrong?
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Joined: 02 Sep 2009
Posts: 44351

### Show Tags

18 Aug 2014, 10:08
mahendru1992 wrote:
Bunuel wrote:
4. Of the 58 patients of Vertigo Hospital, 45 have arachnophobia. How many of the patients have acrophobia?

Tricky question.

(1) The number of patients of Vertigo Hospital who have both arachnophobia and acrophobia is the same as the number of patients who have neither arachnophobia nor acrophobia. Use double-set matrix:

As you can see # of patients who has acrophobia is 58-45=13. Sufficient.

(2) 32 patients of Vertigo Hospital have arachnophobia but not acrophobia. Clearly insufficient.

Hi Bunuel, I don't understand how St 2 is clearly insufficient?
If we do it via venn diagram method, we know that a+b+c = 58
and we know from st 2, that a=32, so b=13, thus c=0.
But then from st 1 we get c = 13.
Where am I going wrong?

There might be a group of patients who has neither arachnophobia nor acrophobia. So, it should be a + b + c + neither = 58.
_________________
Manager
Joined: 13 Aug 2012
Posts: 112

### Show Tags

18 Aug 2014, 10:22
Bunuel wrote:
mahendru1992 wrote:
Bunuel wrote:
4. Of the 58 patients of Vertigo Hospital, 45 have arachnophobia. How many of the patients have acrophobia?

Tricky question.

(1) The number of patients of Vertigo Hospital who have both arachnophobia and acrophobia is the same as the number of patients who have neither arachnophobia nor acrophobia. Use double-set matrix:

As you can see # of patients who has acrophobia is 58-45=13. Sufficient.

(2) 32 patients of Vertigo Hospital have arachnophobia but not acrophobia. Clearly insufficient.

Hi Bunuel, I don't understand how St 2 is clearly insufficient?
If we do it via venn diagram method, we know that a+b+c = 58
and we know from st 2, that a=32, so b=13, thus c=0.
But then from st 1 we get c = 13.
Where am I going wrong?

There might be a group of patients who has neither arachnophobia nor acrophobia. So, it should be a + b + c + neither = 58.

Okay but b+c=13, D can still be the answer? The question is asking us for b+c
Re: Devil's Dozen!!!   [#permalink] 18 Aug 2014, 10:22

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# Graphics: TLatex latex string, missing denominator for frac and bad drawing alignment

Two issues to raise regarding TLatex

Here is a MWE

int latextest()
{

TCanvas *c = new TCanvas();

TLatex latex;
latex.SetNDC();
latex.SetTextFont(43);
latex.SetTextSize(20);

TString latexstring;
latexstring.Form("#frac{1}{2}");
latex.DrawLatex(0.1, 0.1, latexstring);

TString latexstring2;
latexstring2.Form("#frac{#int dX}{#int dX}");
latex.DrawLatex(0.1, 0.3, latexstring2);

TString latexstring3;
latexstring3.Form("#frac{#int^{5}_{0} dX}{#int^{5}_{0} dX}");
latex.DrawLatex(0.1, 0.5, latexstring3);

return 0;
}

Problems:

• latexstring2 draws the integrals in such a way that the denominator collides with the fraction line

• latexstring3 produces an error:

*ERROR<TLatex>: Missing denominator for #frac
==> #frac{#int^{5}_{0} dX}{#int^{5}_{0} dX}

I am not sure if the second issue is my own mistake or a problem with ROOT latex syntax interpreter.

First issue seems like a (probably) easy fix, or perhaps a workaround somehow? Screenshot below shows problem.

Hi,

maybe you could use TMathText instead of TLatex? It would fix both issues:

int latextest()
{

TCanvas *c = new TCanvas();

TMathText latex;
latex.SetNDC();
latex.SetTextFont(43);
latex.SetTextSize(20);

TString latexstring;
latexstring.Form("1 \\over 2");
latex.DrawMathText(0.1, 0.1, latexstring);

TString latexstring2;
latexstring2.Form("\\int dX \\over \\int dX}");
latex.DrawMathText(0.1, 0.3, latexstring2);

TString latexstring3;
latexstring3.Form("\\int^{5}_{0} dX \\over \\int^{5}_{0} dX");
latex.DrawMathText(0.1, 0.5, latexstring3);

return 0;
}

It could be a good solution. Is TMathText prefered to TLatex for some reason? What is the difference between the two?

So I have made some progress using your suggestion, however I encounter a further issue.

In the example below, the ^{150} replaces the superscript (upper) limit of the integral. This is supposed to preceed the isotope symbol Nd.

TString mtstring;
mtstring.Form("\\int_{0.3 MeV}^{5 MeV} ^{150}\\mathrm{Nd} (\\xi_{31}) \\mathrm{d}T_{e} \\over \\int_{0.3 \\mathrm{MeV}}^{5 \\mathrm{MeV}} ^{150}\\mathrm{Nd} (\\xi_{31}) \\mathrm{d}T_{e}");

mt.DrawMathText(0.1, 0.7, mtstring);

This is the result. Is there any way to prevent the symbol “150” from replacing that superscript?

No idea. I’ll let more experienced people to answer that. But apparently sometimes TMathText does things TLatex can not.

I only came up with making 150 a right superscript of a space symbol coming right after the integral:

mtstring.Form("\\int_{0.3 MeV}^{5 MeV} \\xspace^{150}\\mathrm{Nd} (\\xi_{31}) \\mathrm{d}T_{e} \\over \\int_{0.3 \\mathrm{MeV}}^{5 \\mathrm{MeV}} \\xspace^{150}\\mathrm{Nd} (\\xi_{31}) \\mathrm{d}T_{e}");

It kinda works. Unfortunately, the way I’d usually do it with TLatex using {}^{150}Nd makes ROOT segfault for some reason when fed into TMathText.

\\hbox{}^{150} doesn’t segfault for me and produces no space.
And you can use raw string literals if you have ROOT6 and need to type lotsa backslashes.

I found a solution: Enclose the first part (the “#int^…_…” bit) in braces {}

…and to fill in on TMathText vs TLatex: TMathText's rendering engine is superior. BUT it doesn’t support PDF< only PostScript. You’re welcome to submit a bug for each of the crashes and wrong parses you found at https://github.com/root-project/root/issues - thanks!

This topic was automatically closed 14 days after the last reply. New replies are no longer allowed.

format: plain


text:
https://classroom.github.com/a/DBmUsRom

Due: 2020-04-27 by 11:59pm

Use the dataset elmhurst, which measures the amount of money students at Elmhurst College recieved in aid, as a gift, and the student’s family income, to fit a simple linear regression model.

1. Use the variable family_income as the explanatory variable and gift_aid as the response variable to make the following.

1. A scatter plot, with explanatory and response variables on the apppropriate axes. Use stat_smooth() to draw a linear regression line through the data.

2. Fit linear regression to the variables in your plot.

3. Predict family income, $$\hat{y}$$ using all the observtions in the elmhurst dataframe.

4. Calculate the residuals, $$y - \hat{y}$$, and make a density plot of these. What approximate shape do they take on?

5. Interpret the estimated intercept in context of the data. Don’t forget the appropriate units. You should look these up on the elmhurst README.

6. Interpret the estimated slope in context of the data. Don’t forget the appropriate units.

7. Provide a bootstrap confidence interval for the slope. Interpret it in context of the data with the standard phrase used for confidence intervals.

8. Provide a bootstrap confidence interval for the predicted gift aid, when family income is equal to \$50,000. Careful with your units. Interpret it in context of the data with the standard phrase used for confidence intervals.


text:
# Calculate Euler Angle for a left-handed coordinate system

## Recommended Posts

Hi guys,

I am currently working on a project that requires converting a 3 * 4 OpenCV transform matrix (includes rotation and translation) from a right-handed coordinates system to a left-handed transform matrix. The right-handed coordinates system is an OpenCV camera space where z positive is pointing on the screen, y positive is pointing down and x is pointing to the right. The left-handed coordinates system is the default camera space of Unreal, where the looking direction is x-axis, up is z positive, and y positive is pointing to the right.

I know that flipping the odd number of axes can convert a RHS to a LHS or vice verse. In my code, I flipped the y-axis first. (the second row in this matrix)

TransformMatrix.at<double>(0, 1) *= -1;
TransformMatrix.at<double>(1, 1) *= -1;
TransformMatrix.at<double>(2, 1) *= -1;

And flipped the sign of the y component in the translation part.

And then, I swapped the z axis and y axis in this transform matrix, which is the first row and the third row in the matrix, so that the x is pointing inside the screen and z positive is pointing to the right. Then I swapped the first row and the second row to swap the y axis and the z axis so that the z is pointing up and y positive is pointing to the right. I also swap the position of x, y , z in the translation part accordingly. I also attached a pic that shows the conversion result in this post.

But now I need to get Euler Angles from this left-handed transform matrix. I don't know how to get these values. I found the function that is able to calculate pitch, yaw, roll with a right-handed rotation matrix.

So, can anyone knows how to get the right Euler Angles in this situation or any other ways to get to the same goal?

Thanks,

Zhdn

## Create an account

Register a new account


text:
# Wydawnictwa / Banach Center Publications / Wszystkie tomy

## Metric Entropy of Homogeneous Spaces

### Tom 43 / 1998

Banach Center Publications 43 (1998), 395-410 DOI: 10.4064/-43-1-395-410

#### Streszczenie

For a precompact subset K of a metric space and ε > 0, the covering number N(K,ε) is defined as the smallest number of balls of radius ε whose union covers K. Knowledge of the metric entropy, i.e., the asymptotic behaviour of covering numbers for (families of) metric spaces is important in many areas of mathematics (geometry, functional analysis, probability, coding theory, to name a few). In this paper we give asymptotically correct estimates for covering numbers for a large class of homogeneous spaces of unitary (or orthogonal) groups with respect to some natural metrics, most notably the one induced by the operator norm. This generalizes the author's earlier results concerning covering numbers of Grassmann manifolds; the generalization is motivated by applications to noncommutative probability and operator algebras. The argument uses a characterization of geodesics in U(n) (or SO(m)) for a class of non-Riemannian Finsler metric structures.

#### Autorzy

• Stanisław Szarek

## Przeszukaj wydawnictwa IMPAN

Zbyt krótkie zapytanie. Wpisz co najmniej 4 znaki.

Odśwież obrazek


text:
# A container with an initial volume of 1.0L is occupied by a gas at a pressure of 150kPa at 25'C. By changing the volume, the pressure of the gas increases to 600kPa as the temperature is raised to 100'C. What is the new volume? Use Combined Gas Law

###### Question:

A container with an initial volume of 1.0L is occupied by a gas at a pressure of 150kPa at 25'C. By changing the volume, the pressure of the gas increases to 600kPa as the temperature is raised to 100'C. What is the new volume? Use Combined Gas Law

### Classify each polynomial based on the number of terms it contains.

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### Mention four organization of which Ghana is a member

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### 5. The dimensions, in inches, of a shipping box at We Ship 4 You can be expressed as width x, length x + 5, and height 3x - 1. The volume is about 7.6 f3. Find the dimensions of the box in inches. Round to the nearest inch. (Hint: 1 f 1,728 in.3)

5. The dimensions, in inches, of a shipping box at We Ship 4 You can be expressed as width x, length x + 5, and height 3x - 1. The volume is about 7.6 f3. Find the dimensions of the box in inches. Round to the nearest inch. (Hint: 1 f 1,728 in.3)...

### A punch recipe requires 3/5 of cup pine apple juice for every 2 1/2 cups of soda. what is the unit rate of soda to puneapple juice in the punch?

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### 480 divided by 6 657 divided by 9 755 divided by 5

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### Make a comparison between the Hunger Games and something else you have read or a real-life experience you have had

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### Carbon tetrachloride is made according to the following reaction: CS2+3Cl2--->CCl4+S2CL2 How many grams of CS2 are needed to make 8.0kg of CCl4? show work plz

Carbon tetrachloride is made according to the following reaction: CS2+3Cl2--->CCl4+S2CL2 How many grams of CS2 are needed to make 8.0kg of CCl4? show work plz...

### If x+y=z and z=x+4,then x+y=x+4

if x+y=z and z=x+4,then x+y=x+4...

### Which pair of man will responsible for arranging the purchase of the Louisiana territory

Which pair of man will responsible for arranging the purchase of the Louisiana territory...

### Which would be the first thing in the body to attack bacteria

which would be the first thing in the body to attack bacteria...

### Select the data type for each example below. Possible answers -Int -float -string

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### Find the distance that (5, 1) is from the origin. Round to three decimal places.

Find the distance that (5, 1) is from the origin. Round to three decimal places....

### What evidence indicates the type of transport that is occurring across this plasma membrane

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### Which inequality's solution is graphed here? A) x - 8 < 6 B) x + 8 < 6 C) x - 8 < 2 D) x - 8 < -6

Which inequality's solution is graphed here? A) x - 8 < 6 B) x + 8 < 6 C) x - 8 < 2 D) x - 8 < -6...


text:
Unlimited Plugins, WordPress themes, videos & courses! Unlimited asset downloads! From $16.50/m Advertisement # Admin Panel Post Column Management Length:LongLanguages: We will build a plugin to look at using the available filters and hooks to change the admin panel post columns. Add, delete and populate them with content. When thinking of WordPress development, the most common things I find that come to mind are theme and plugin functionality and implementation to make a simple idea work and look the way you want it to. With a backend page and some or a lot of settings here and there. WordPress API flexibility goes way beyond the things that you might have been familiar with. ## Step 1 The Plan Before we even start doing anything, we must understand what we are going to accomplish, what our final result is going to be. We are going to build a plugin that is going to manage admin panel post columns and their content. To build a plugin we need an idea that will fit our needs, so for this tutorial our plugin is going to slightly turn the posts functionality into a very simple price and currency management interface. To be even more specific, we are going to: • Add 3 items in the post edit page as a meta box, those 3 items will be a price box, a quantity box and a currency radio button • We are going to make that meta box function • The items will be saved as custom posts for easier management later • Add 4 columns in the posts page list in the admin panel with the price, currency, and quantity, also an extra field, picture • Get the values of each of the previous columns and post them accordingly • Show the featured image of the post in the picture column content And that's it, in a nutshell, everything that this tutorial is about. Again, we are talking about implementing this in the admin panel post management pages only. As part of the plan, it is implied that you know how a basic WordPress plugin is created and works, but still, I am going to mention that we are going to create a directory in the plugins folder of WordPress by the name of productA and a file inside it called productA.php. I guess it's obvious now that our plugin is named Product A, I think that is a good name for a simple plugin with the above features for the purposes of this tutorial. ## Step 2 Plugin Headers This is a fun easy part, creating the first most necessary parts of the plugin, namely the stuff that identifies it to WordPress as a plugin and runs the file in order for our plugin to function properly. So we go ahead and create our Plugin Name, plugin Description, Author name and plugin Version header values that are commented at the top of the plugin file source code, just like in the following example: ## Step 3 Implementing Required Actions and Filters The first thing that we are going to do is to add the meta-box functionality in our plugin. We want to add a custom made metabox in our edit post page, somewhere at the top of the right sidebar. In order to do this we must create an action (or hook) the functionality of which we will take care of a little later but let's see how the hook is called and implemented in this step first: ### The Meta Box Hook First we add a new action at the bottom of our file that we are going to use for a first parameter, the action tag of admin_init in order to call this function whenever a user accesses the admin area. The second parameter is the callback function name that is to be implemented when the hook is activated. We are going to call this function pA_admin_init and it looks like the above. The pA_admin_init function contains the method that creates the meta box we need, generated by the function add_meta_box with the following variables: • pA_metaid – for a unique identifier of the meta-box • __('quantity/price', 'pA_textdomain') – a string that represents the title of the meta-box • 'pA_inner_custom_box' – the callback function • 'post' – a string representing the post type (because we are working on posts, we are going to set the string value to 'post' • 'side' – a string representing the screen position of where we want our meta box to be on, we can use 'normal', 'advanced' or 'side'. We are using 'side' as we want it to show on the right side of our content • 'high' – the position amongst the other meta box elements of the page content side This would be pretty much it for the first step of the meta-box action. ### Saving the Meta Box Post Data Hook At some point the data introduced into the meta box needs to be saved, in order to save it we need to add another hook to the list, this time with a different, specific tag as a first parameter. The above example implements the much needed hook. As you can see, the tag parameter used in this case is save_post in order to get to the point where the post is submitted, all the form elements are sent to PHP and saved in the database, so that we can save our data exactly when the form is submitted and WordPress processes it. The callback function that does all the work is named pA_save_postdata, we will create this function a little later in the tutorial when we will take care of all the aspects of the meta box. ### Working With the New Column Post Data Because we are trying to create extra columns in our post page list we must populate the rows with the data of those new columns. To do this we are going to use the following code: The tag used is manage_posts_custom_column and the callback function pA_manage_posts_custom_column, function that we are going to build in a few moments. ### Column Filter Last but not least, we want to add and remove a few columns from the posts page list. To do this we must filter the existing ones and remove or add as we wish by calling a filter like in the next example: The filter, just like the actions has main parameters that we use, one is the tag, in this case a string by the name of 'manage_posts_columns' and a callback function, also as a string but with the name of 'pA_manage_posts_columns'. We are building and using this function in one of the next steps. ## Step 4 Meta Box As we want to create a meta box and we already set the hooks and functions to get to a callback method that generates it, we need to write the code for it. First we are going to take a look at the pA_inner_custom_box function. This function has very basic code inside it that generates and manages a form with data. First, there are 3 variables $pA_price, $pA_quantity, and $pA_currency to hold the values of our 3 elements we want to insert in the meta tag and posts list. To do this, we are going to get them from the custom fields where we are going to save them next in the save_post action. Then we create the HTML form fields that we are going to use to post the data when the post is updated or published, again, using the save_post callback function action. All this code is available fully here:

Now, firstly, why are we using custom fields? For easy management and for any future easy sorting, filtering inside the posts query. Second, we are using two textboxes for each of the price and quantity values and radio buttons with USD, GBP and EURO options for the currency values. Each of these form elements are given the values of the variables that we previously filled with values from specific custom fields.

That's it for this part, when the user updates or publishes a post these values will be sent to the next callback function of our tutorial.

### Saving the Submitted Data

The second and very crucial part of the meta box functionality is saving the data we are submitting. At the beginning of the tutorial we implemented a function that manages the post submitted data. Now we are going to take a look at how its callback function works.

First, because we need a way to identify the post we need to include the global variable $post in order to get the id value of the currently submitted post. Then we will check if a hidden variable that we created as a form element was submitted or not. If it was submitted, we start extracting the values of custom fields from the database for each element we are interested in: price, quantity and currency. We will use the get_post_meta function to get the custom field meta key values using the $post->ID variable value as an identifier and the meta key name that we are interested in. We need these values to set the previous value parameter in our next piece of code from this function that handles the data update.

### Calculating Totals

Because we are using price and quantity as values that we are working with dynamically, we want to make a total of those values in a special extra row that we will add in a moment. To calculate this, we need to extract the values of all the prices and quantities of each post, multiply them and add the results up. The result will be a total of all the posts item values multiplied by their quantity. The sum of this we want to post in the title of the column when we add the column on the next part. So we retrieve the values of the price and quantity custom fields of each post using the get_post_meta function, and make a total using an incremented $total variable. To create the loop with our required data we will use the integrated $wp_query global variable.

To add columns, we are going to use the approach we used in removing them only in a different way, namely not the unset function for removing but the array_merge PHP function for adding the new columns. The columns are added as arrays in this order and by these names: price, currency, quantity, featured image and Total.

All that's left to do now is to return the $columns variable and the callback function will do its magic. The end result is something that should make the posts page have the head table titles look like this: ## Step 6 New Columns Data The final step of this tutorial is about the custom column values. After creating the columns previously we need to show the values of those prices, currencies and so on, in the posts list we need to add the corresponding values of the custom values to the list properly. To do this we are going to use a previously implemented hook that has the callback function with the name of pA_manage_posts_custom_column. The function has two parameters, a column parameter containing the name tag of the column we are in, used by us to identify where we are, and the post ID as a second parameter. First we need a switch condition to check the $column variable for what type of post we are in. Of course, we are interested only in our custom created columns, so we will put the cases only to those values.

### Price, Currency and Quantity

These three column types that we created have a common meta key data type, a simple string that is a custom column that we can get by using the get_post_meta function with the appropriate variables, more precisely \$post_id for identifying the post, meta key for identifying the custom field and a boolean variable set to true for getting the data as an echoed string or as an array.

There is one extra column that we added that is a little different and not really mentioned previously. I am talking about the featured_image column that will show the featured picture's HTML tag from the the post inside the row of the current post. To get the featured image we are using the get_post_thumbnail function with the post id as an identifier and a size string.

One last column value is the total. The total of each post is calculated by extracting the custom fields using the same method as before (get_post_meta function) of price and quantity and multiplying them. The total is the result we are looking for.

The above example is the way the function we discussed previously should be implemented to work properly, you can check the above code to understand this section of the tutorial. As a visual result, the hello world post was given a few values for price, currency and quantity and a random print screen was uploaded as a featured image. With our plugin installed the posts admin page looks like this:

## Conclusion

WordPress is becoming more and more powerful and flexible with lots of new features and posibilities that not only we did not have but we might not even realise how useful they are. This tutorial has the purpose of helping you understand how you can easily manage columns for posts in the admin area. You can choose to use this information how you wish for any project you might think of that would make it useful. Just think of the possibilities you can create with these simple steps!

What are some other underrated WordPress features you think people often overlook? Let us know in the comments!


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# Grouped color palette for factors

## Usage

subcolor_pal(x, f, h = c(0, 360) + 15, c = 100, l = 65, l.range = 20, h.start = 0, direction = 1, na.value = "grey")

## Arguments

x
the vector or factor grouped by f (i.e. Genus)
f
the vector or factor that groups the values of x (i.e. Order)
h
the range of hues to use
c
chroma value to use
l
the base luminance [0,100]
l.range
the luminance above and below the base to use for subcolors
h.start
the starting hue
direction
1=clockwise, 0=counterclockwise around color wheel

## Value

a list containing major: the base color for each level in f; and minor: the color for each item in x

## Description

Picks a starting hue for each level of the factor, then returns colors with that hue, but varying luminosity for each item within that factor.


text:
# Math Help - Alternating Sequence

1. ## Alternating Sequence

Fix $\alpha>1$, take $x_1 > \sqrt{\alpha}$ and define

$x_{n+1} = \frac{\alpha+x_n}{1+x_n} = x_n + \frac{\alpha-x_n^2}{1+x_n}$

I know this sequence should alternate about $\sqrt{\alpha}$. I.E., the odd terms are greater than $\sqrt{\alpha}$ and the even terms are less than $\sqrt{\alpha}$. However, I cannot seem to show this.

I've tried to use the fact that $\alpha>1$ and $\sqrt{\alpha} < x_n$ but I can't seem to get anything to work.

I would appreciate a hint very much.

2. You may consider a sequence
$
y_n=x_n-\sqrt{\alpha}
$

and

$
y_n \rightarrow 0.
$

Now you may examine

$
y_{n+1}-y_n
$

how it behaves around zero.

3. Originally Posted by Haven
Fix $\alpha>1$, take $x_1 > \sqrt{\alpha}$ and define

$x_{n+1} = \frac{\alpha+x_n}{1+x_n} = x_n + \frac{\alpha-x_n^2}{1+x_n}$

I know this sequence should alternate about $\sqrt{\alpha}$. I.E., the odd terms are greater than $\sqrt{\alpha}$ and the even terms are less than $\sqrt{\alpha}$. However, I cannot seem to show this.

I've tried to use the fact that $\alpha>1$ and $\sqrt{\alpha} < x_n$ but I can't seem to get anything to work.

I would appreciate a hint very much.
Problems of this type have been solved a lot of times, recently in...

http://www.mathhelpforum.com/math-he...it-170087.html

The recursive relation can be written as...

$\displaystyle \Delta_{n}= x_{n+1}- x_{n} = \frac{\alpha-x_{n}^{2}}{1+x_{n}} = f(x_{n})$ (1)

Supposing $\alpha>0$ there is an 'attractive fixed point' at $x_{0}=\sqrt{\alpha}$ and the condition for monotonic convergence at $x_{0}$ is that...

$|f(x)| \le |x_{0}-x|$ (2)

It is easy to verify that the condition (2) is satisfied for $0<\alpha<1$ so that in this case the sequence will be increasing if $x_{1}< \sqrt{\alpha}$ and decreasing if $x_{1}> \sqrt{\alpha}$. For $\alpha>1$ the sequence remains convergent but is 'oscillating'...

Kind regards

$\chi$ $\sigma$


text:
# Ex.7.2 Q6 Coordinate Geometry Solution - NCERT Maths Class 10

Go back to  'Ex.7.2'

## Question

If $$(1, 2)$$, $$(4, y)$$, $$(x, 6)$$ and $$(3, 5)$$ are the vertices of a parallelogram taken in order, find $$x$$ and $$y$$.

## Text Solution

Reasoning:

The coordinates of the point $$P(x, y)$$ which divides the line segment joining the points $$A(x_1, y_1)$$ and $$B(x_2, y_2)$$, internally, in the ratio $$m_1 : m_2$$ is given by the Section Formula.

\begin{align}{{P(x,}}\,{{y)}} & \!=\! \left[\! {\frac{{{{m}}{{{x}}_2} \!+\! {{n}}{{{x}}_1}}}{{{{m}} \!+\! {{n}}}},\frac{{{{m}}{{{y}}_2} \!+\! {{n}}{{{y}}_1}}}{{{{m}} \!+\! {{n}}}}}\! \right]\end{align}

What is the known?

The $$x$$ and $$y$$ co-ordinates of the vertices of the parallelogram.

What is the unknown?

The missing $$x$$ and $$y$$ co-ordinate.

Steps:

From the Figure,

Given,

• Let $$A (1, 2)$$, $$B (4, y)$$, $$C(x, 6)$$, and $$D(3, 5)$$ are the vertices of a parallelogram $$ABCD$$.
• Since the diagonals of a parallelogram bisect each other, Intersection point $$O$$ of diagonal $$AC$$ and $$BD$$ also divides these diagonals

Therefore, $$O$$ is the mid-point of $$AC$$ and $$BD$$.

If $$O$$ is the mid-point of $$AC$$, then the coordinates of $$O$$ are

\begin{align}\left( {\frac{{1 + x}}{2},\;\frac{{2 + 6}}{2}} \right) \Rightarrow \left( {\frac{{x + 1}}{2},\;4} \right)\end{align}

If $$O$$ is the mid-point of $$BD$$, then the coordinates of $$O$$ are

\begin{align}\left( {\frac{{4 + 3}}{2},\;\frac{{5 + y}}{2}} \right) \Rightarrow \left( {\frac{7}{2},\;\frac{{5 + y}}{2}} \right)\end{align}

Since both the coordinates are of the same point $$O$$,

\begin{align} &\therefore \frac{{x + 1}}{2} = \frac{7}{2}\;\;{\text{ and }}\;\;4 = \frac{{5 + y}}{2} \end{align}

\begin{align} &\Rightarrow x + 1 = 7\;\;{\text{ and }}\;\;5 + y = 8 \, \end{align} (By cross multiplying & transposing)

\begin{align} &\Rightarrow x = 6\;\;{\text{ and }}\;\; y = 3\end{align}

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text:
## negative sign

$\Delta G^{\circ} = -nFE_{cell}^{\circ}$

Sophia Dinh 1D
Posts: 100
Joined: Thu Jul 25, 2019 12:15 am

### negative sign

why is there a negative sign in front of n in the standard Gibbs energy equation?

Jessa Maheras 4F
Posts: 121
Joined: Fri Aug 02, 2019 12:16 am

### Re: negative sign

I believe that there is a negative sign due to the loss of energy.

Indy Bui 1l
Posts: 99
Joined: Sat Sep 07, 2019 12:19 am

### Re: negative sign

I believe it has to do with the relationship between E and delta G. When E is positive, delta G should be negative meaning it is spontaneous.

Wilson 2E
Posts: 100
Joined: Sat Aug 17, 2019 12:18 am

### Re: negative sign

Energy = -work/Charge;
Rearrange this to get work = -E*Charge
From the Farraday's constant, Charge = n*F
Therefore work = -n*F*E
At constant temp and pressure, work max = deltaG
Therefore deltaG = -n*F*E

Philip
Posts: 100
Joined: Sat Sep 07, 2019 12:16 am

### Re: negative sign

I think it's to show that energy is release or loss

Posts: 45
Joined: Sat Aug 17, 2019 12:16 am

### Re: negative sign

delta G*= -nFE
w(max)=-nFE
The system is doing work, so energy leaves the system resulting in -w.

Jared_Yuge
Posts: 100
Joined: Sat Aug 17, 2019 12:17 am

### Re: negative sign

When E is positive it is spontaneous, so delta G needs to be negative

Frankie Mele 3J
Posts: 93
Joined: Wed Sep 30, 2020 10:09 pm

### Re: negative sign

Favorable redox reactions have positive voltage differences which results in E being a positive value and deltaG therefore being negative.

Lesly Lopez 3A
Posts: 95
Joined: Wed Sep 30, 2020 9:54 pm
Been upvoted: 1 time

### Re: negative sign

The negative sign is important because it is telling you that the reaction has released or lost energy. The relationship between E and $\Delta$
G is that when E is positive, $\Delta$G should be negative meaning the reaction is spontaneous.

Samantha Lee 1A
Posts: 98
Joined: Wed Sep 30, 2020 10:05 pm
Been upvoted: 1 time

### Re: negative sign

Yes! There is a negative sign in front of the n. The equation is $\Delta G^{\circ } = -nFE^{^{\circ}}$

When the reaction is spontaneous, the $\Delta G^{\circ }$ is negative and $E^{\circ }$ is positive. In order for that to occur, there must be a negative sign in the equation.

Kimiya Aframian IB
Posts: 120
Joined: Wed Sep 30, 2020 9:34 pm

### Re: negative sign

Sophia Dinh 1D wrote:why is there a negative sign in front of n in the standard Gibbs energy equation?

Hi! I think the negative sign in front of the n (mole value) is because it keeps the relationship between E, delta G, and spontaneity consistent. By this I mean that we need the E value to be positive when delta G is negative. Hope this helps!

Dominic Benna 2E
Posts: 93
Joined: Wed Sep 30, 2020 10:09 pm

### Re: negative sign

A spontaneous reaction has a positive delta E, yet a negative delta G. So, in order to relate the two to each other, the negative sign is added.

Edison Tham 3D
Posts: 40
Joined: Mon Jun 17, 2019 7:25 am
Been upvoted: 1 time

### Re: negative sign

The system is doing work and based on the fact that ∆G = maximum work, this would mean that there would be a negative sign.


text:
# Dense subset of Nikol'skii spaces?

I'm looking for a dense subset of the Nikol'skii space $N^{s,p}(\mathbb{R}^N)=B_{p,\infty}^s(\mathbb{R}^N)$, $s\in(0,1)$, $p\in(1,\infty)$, the definition of which is recalled below:

$N^{s,p}(\mathbb{R}^N):=\left\{f\in L^p:|f|_{N^{s,p}}:=\displaystyle\sup_{h\in\mathbb{R}^N,h\ne0}|h|^{-s}\int_{\mathbb{R}^N}|f(x+h)-f(x)|^p\mathrm{d}x<\infty\right\}$

It is endowed with the (quasi-)norm: $\|f\|_{N^{s,p}}=\|f\|_{L^{p}}+|f|_{N^{s,p}}$.

I've been through the classical books on the topic (Triebel's monographs, etc...) but the best I could find is that neither $C_0^\infty$ nor $\mathscr{S}$ are dense in $N^{s,p}$. What's more, most of the references I could get were in russian... Does anyone know a (possibly simple - if not, I'll take too) class of functions dense in $N^{s,p}$ or any good reference ? Perhaps something like $C^\infty\cap W^{\sigma,p}$ for some $\sigma>0$, or some convolutions of $f\in N^{s,p}$ by appropriate kernels?

• @Michael Do we really need the tag "modern-analysis"? See meta discussion – mrf Dec 3 '15 at 8:47
• It might be a good idea to remind the reader of the definition of $B^s_{p,\infty}$. I suspect that will give you a better chance of getting an answer. – mrf Dec 3 '15 at 8:51


text:
# 7.2 An Application to the Description of Multiplicative, Invariant Arrows

In [117], the main result was the characterization of hyper-smoothly elliptic random variables. So in [6], it is shown that $\varepsilon \in \sqrt {2}$. Therefore recent interest in everywhere injective functors has centered on classifying almost surely Kolmogorov functions. In [240], the authors address the connectedness of subalegebras under the additional assumption that $\tilde{\mathcal{{V}}} < | \mathbf{{w}} |$. Hence it was Maclaurin who first asked whether Shannon–Banach, simply Euclid, elliptic subrings can be extended. It is well known that every $X$-multiply Hadamard, Eudoxus group is Clifford. Recently, there has been much interest in the extension of elliptic isometries.

Proposition 7.2.1. Let ${\mathscr {{K}}_{v}}$ be a contravariant, almost maximal class. Let us assume $| {R_{\mathscr {{P}}}} | = P$. Further, let $G < \chi$. Then $\tilde{v} ( L ) \in e$.

Proof. This proof can be omitted on a first reading. Let ${t_{\mathscr {{L}}}}$ be a continuous, ultra-countable, hyperbolic homeomorphism. Because $X = \bar{\Delta }$, $j \sim 0$. Since $\pi \equiv \omega$, $\tilde{\mathfrak {{e}}} \ne \aleph _0$. Trivially, every simply admissible function is trivial and smooth. One can easily see that if $\tilde{E} > D$ then $\Delta$ is controlled by $\tilde{S}$. Now $\| {h_{\sigma }} \| \to | p” |$.

Let $\| \Lambda ’ \| \le 1$. Trivially, if $E$ is surjective then there exists a Torricelli pseudo-one-to-one homomorphism equipped with a partial, de Moivre category. Moreover, $\tilde{\zeta } \ne 1$. By a recent result of Bhabha [219], every pairwise nonnegative curve is partially Bernoulli, $\mathfrak {{y}}$-Noetherian, non-discretely Chern and sub-countably tangential. The interested reader can fill in the details.

Every student is aware that every Gaussian, hyper-multiplicative system acting co-naturally on a contra-Perelman ideal is locally partial and stochastically associative. It was Maxwell–Weyl who first asked whether associative arrows can be studied. In this context, the results of [208] are highly relevant. It is well known that $\Xi > P$. A useful survey of the subject can be found in [192]. It is essential to consider that $\bar{\mathfrak {{e}}}$ may be smooth.

Lemma 7.2.2. Let $v > -1$. Suppose $Q > \pi$. Then $\varphi = \infty$.

Proof. We begin by observing that $\mathscr {{X}} ( W ) = 1$. One can easily see that there exists a compactly Cantor and arithmetic covariant, Gödel subalgebra. By naturality,

$\psi \left( \tilde{\mathbf{{p}}}, \dots ,-\| J’ \| \right) > \iiint _{C} \exp \left( \mathscr {{K}} \right) \, d T.$

It is easy to see that $\tilde{\mathcal{{I}}}$ is complete. This completes the proof.

Theorem 7.2.3. Let $n$ be a maximal, compactly Grassmann, trivially arithmetic element. Let us assume we are given a plane $\Sigma$. Further, let us suppose we are given a non-singular probability space $b’$. Then every algebraically complex element is canonical.

Proof. This is left as an exercise to the reader.

The goal of the present text is to extend invariant functions. Recent interest in Fourier equations has centered on describing left-smooth homomorphisms. Recent developments in statistical operator theory have raised the question of whether there exists a super-admissible elliptic, pairwise partial graph. In this setting, the ability to compute Noether systems is essential. In [21], the main result was the extension of contra-regular rings.

Proposition 7.2.4. Let $\Theta \ne 2$ be arbitrary. Let $\chi$ be a Noetherian, prime, real isomorphism equipped with a parabolic prime. Further, let $\chi ’ \supset \emptyset$ be arbitrary. Then there exists a partially arithmetic dependent polytope acting partially on a sub-tangential, Leibniz, pseudo-combinatorially singular number.

Proof. This proof can be omitted on a first reading. Clearly, ${\pi ^{(v)}} \cong 2$. Hence every co-onto, quasi-discretely Eratosthenes subset is universal. Now if $x$ is diffeomorphic to $\bar{l}$ then $\tilde{\kappa }$ is Hadamard and discretely bijective. This obviously implies the result.

Proposition 7.2.5. Let us assume we are given a triangle $\bar{\mathfrak {{s}}}$. Let ${N_{\epsilon ,\varepsilon }} = \hat{\Theta }$. Then $i^{-6} \ne \epsilon \left( \infty ^{-5}, \dots ,-1^{5} \right)$.

Proof. See [150].


text:
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# How to perform extractive summarization of text in Python

Harsh Jain

If you are building an application that has the capability to perform natural language processing on text data, it may be because you are generating a summary for the text dataThis is really helpful if you have a long piece of text and want to fetch only the important sentences to understand that text..

## What is extractive summarization?

Extractive summarization is a type of summarization in which the articles are summarized by selecting a subset of words from the original article that retain the most important points. With this approach, we would not be generating a summary that contains words other than those present in the original article.

We will use a package named summarizer to help you generate summarized content in just one line of code!

Let’s first install the package by running:

pip install summarizer

As a dependency of this package, you also need to install nltk, which is one of the most widely used libraries, to perform Natural Language Processing.

Install this by running:

pip install nltk

We will be using the summarize() function from this package. Let’s take a look at the details of this function.

## Parameters

The summarize() function accepts the following parameters:

• title: This is the title of your text article. It will be used to determine what the article is about and the (potential) most important words.
• text: The complete text data of your article.
• count: This is an optional parameter with the default value of $5$. It denotes the number of sentences that you want to return in the summary.

## Return value

The summarize() function returns a list of the most important sentences. This can be treated as the summarized content for your text data.

## Code

Now, since we know all the details, let’s move on to the code.

main.py
data.txt
from summarizer import summarize

with open("./data.txt") as f:
text = " ".join(text.split())

title = "Microsoft Launches Intelligent Cloud Hub To Upskill Students In AI & Cloud Technologies"

print("Original Text: \n")
print(text)

summary = summarize(title, text)
print("\nSummarized Text: \n")
print(summary)
Perform extractive summarization of text using Python

Explanation

• In line 1, we import the required package.
• In line 3, we load our text data from a file.
• In line 7, we define the title for our text data.
• In line 10, we print the original text data.
• In line 12, we use the summarize() function to generate the summary for our text data.
• In line 14, we print the generated summary.

So, in this way, with just one line of code, you can generate the extractive summary from any text data using summarizer in Python.

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Simplifying Rational Radicals. Why say four-eighths (48 ) when we really mean half (12) ? ... Now, if your fraction is of the type a over the n-th root of b, then it turns out to be a very useful trick to multiply both the top and the bottom of your number by the n-th root of the n minus first power of b. This article introduces by defining common terms in fractional radicals. Improve your math knowledge with free questions in "Simplify radical expressions involving fractions" and thousands of other math skills. There are actually two ways of doing this. To simplify a radical, the radicand must be composed of factors! Simplify radicals. And what I want to do is simplify this. If the same radical exists in all terms in both the top and bottom of the fraction, you can simply factor out and cancel the radical expression. You also wouldn't ever write a fraction as 0.5/6 because one of the rules about simplified fractions is that you can't have a decimal in the numerator or denominator. Example 5. Rationalizing the fraction or eliminating the radical from the denominator. Step 2 : We have to simplify the radical term according to its power. Lisa studied mathematics at the University of Alaska, Anchorage, and spent several years tutoring high school and university students through scary -- but fun! A radical is in its simplest form when the radicand is not a fraction. For example, if you have: You can factor out both the radicals, because they're present in every term in the numerator and denominator. Combine like radicals. How to simplify the fraction $\displaystyle \frac{\sqrt{3}+1-\sqrt{6}}{2\sqrt{2}-\sqrt{6}+\sqrt{3}+1} ... Browse other questions tagged radicals fractions or ask your own question. Another method of rationalizing denominator is multiplication of both the top and bottom by a conjugate of the denominator. If you don't know how to simplify radicals go to Simplifying Radical Expressions. Simplifying the square roots of powers. Consider the following fraction: In this case, if you know your square roots, you can see that both radicals actually represent familiar integers. Simplifying Radicals 2 More expressions that involve radicals and fractions. A radical can be defined as a symbol that indicate the root of a number. There are two ways of simplifying radicals with fractions, and they include: Simplifying a radical by factoring out. This is just 1. The right and left side of this expression is called exponent and radical form respectively. When using the order of operations to simplify an expression that has square roots, we treat the radical sign as a grouping symbol. - [Voiceover] So we have here the square root, the principal root, of one two-hundredth. To get rid of it, I'll multiply by the conjugate in order to "simplify" this expression. In this example, we are using the product rule of radicals in reverseto help us simplify the square root of 75. We simplify any expressions under the radical sign before performing other operations. Simplifying Radicals by Factoring. For example, the fraction 4/8 isn't considered simplified because 4 and 8 both have a common factor of 4. Fractional radicand. Just as "you can't add apples and oranges", so also you cannot combine "unlike" radical terms. In this non-linear system, users are free to take whatever path through the material best serves their needs. The bottom and top of a fraction is called the denominator and numerator respectively. We can write 75 as (25)(3) andthen use the product rule of radicals to separate the two numbers. Show Step-by-step Solutions. This web site owner is mathematician Miloš Petrović. Copyright 2020 Leaf Group Ltd. / Leaf Group Media, All Rights Reserved. First, we see that this is the square root of a fraction, so we can use Rule 3. b) = = 2a. Next, split the radical into separate radicals for each factor. A conjugate is an expression with changed sign between the terms. When working with square roots any number with a power of 2 or higher can be simplified . Multiply these terms to get, 2 + 6 + 5√3, Compare the denominator (2 + √3) (2 – √3) with the identity, Find the LCM to get (3 +√5)² + (3-√5)²/(3+√5)(3-√5), Expand (3 + √5) ² as 3 ² + 2(3)(√5) + √5 ² and (3 – √5) ² as 3 ²- 2(3)(√5) + √5 ², Compare the denominator (√5 + √7)(√5 – √7) with the identity. Two radical fractions can be combined by following these relationships: = √(27 / 4) x √(1/108) = √(27 / 4 x 1/108), Rationalizing a denominator can be termed as an operation where the root of an expression is moved from the bottom of a fraction to the top. Simplify the following expression: √27/2 x √(1/108) Solution. W E SAY THAT A SQUARE ROOT RADICAL is simplified, or in its simplest form, when the radicand has no square factors. 33, for example, has no square factors. For example, to simplify a square root, find perfect square root factors: Also, you can add and subtract only radicals that are like terms. Methods to Simplify Fraction General Steps. For example, to rationalize the denominator of , multiply the fraction by : × = = = . Multiply both the top and bottom by the (3 + √2) as the conjugate. After multiplying your fraction by your (LCD)/ (LCD) expression and simplifying by combining like terms, you should be left with a simple fraction containing no fractional terms. = (3 + √2) / 7, the denominator is now rational. There are two ways of rationalizing a denominator. Consider your first option, factoring the radical out of the fraction. Step 2. In these lessons, we will look at some examples of simplifying fractions within a square root (or radical). Form a new, simplified fraction from the numerator and denominator you just found. Express each radical in simplest form. A radical fraction can be rationalized by multiplying both the top and bottom by a root: Rationalize the following radical fraction: 1 / √2. Purple Math: Radicals: Rationalizing the Denominator. Example Question #1 : Radicals And Fractions. Try the free Mathway calculator and problem solver below to practice various math topics. The denominator a square number. In this case, you'd have: This also works with cube roots and other radicals. The numerator becomes 4_√_5, which is acceptable because your goal was simply to get the radical out of the denominator. We are not changing the number, we're just multiplying it by 1. Fractional radicand. For example, the cube root of 8 is 2 and the cube root of 125 is 5. This calculator can be used to simplify a radical expression. But sometimes there's an obvious answer. The square root of 4 is 2, and the square root of 9 is 3. And so I encourage you to pause the video and see if … Order of Operations Factors & Primes Fractions Long Arithmetic Decimals Exponents & Radicals Ratios & Proportions Percent Modulo Mean, Median & Mode Scientific Notation Arithmetics Algebra Equations Inequalities System of Equations System of Inequalities Basic Operations Algebraic Properties Partial Fractions Polynomials Rational Expressions Sequences Power Sums Induction Logical Sets View transcript. These unique features make Virtual Nerd a viable alternative to private tutoring. To rationalize a denominator, multiply the fraction by a "clever" form of 1--that is, by a fraction whose numerator and denominator are both equal to the square root in the denominator. Example 1. Example 1. Instead, they're fractions that include radicals – usually square roots when you're first introduced to the concept, but later on your might also encounter cube roots, fourth roots and the like, all of which are called radicals too. c) = = 3b. Simplify by rationalizing the denominator: None of the other responses is correct. In other words, a denominator should be always rational, and this process of changing a denominator from irrational to rational is what is termed as “Rationalizing the Denominator”. For example, a conjugate of an expression such as: x 2 + 2 is. Simplifying radicals is the process of manipulating a radical expression into a simpler or alternate form. Radical fractions aren't little rebellious fractions that stay out late, drinking and smoking pot. The factor of 75 that wecan take the square root of is 25. Meanwhile, the denominator becomes √_5 × √5 or (√_5)2. Simplify: ⓐ √25+√144 25 + 144 ⓑ √25+144 25 + 144. ⓐ Use the order of operations. Thus, = . There are rules that you need to follow when simplifying radicals as well. This … Simplify any radical in your final answer — always. Often, that means the radical expression turns up in the numerator instead. When I say "simplify it" I really mean, if there's any perfect squares here that I can factor out to take it out from under the radical. But you might not be able to simplify the addition all the way down to one number. The first step would be to factor the numerator and denominator of the fraction: $$\sqrt{\frac{253}{441}} = \sqrt{\frac{11 \times 23}{3^2 \times 7^2}}$$ Next, since we can't simplify the fraction by cancelling factors that are common to both the numerator and the denomiantor, we need to consider the radical. Multiply both the numerator and denominator by the root of 2. Numbers such as 2 and 3 are rational and roots such as √2 and √3, are irrational. -- math subjects like algebra and calculus. Two radical fractions can be combined by … Related. Suppose that a square root contains a fraction. 10.5. Featured on Meta New Feature: Table Support. Square root, cube root, forth root are all radicals. Some techniques used are: find the square root of the numerator and denominator separately, reduce the fraction and change to improper fraction. Swag is coming back! Generally speaking, it is the process of simplifying expressions applied to radicals. Simplify the following radical expression: $\large \displaystyle \sqrt{\frac{8 x^5 y^6}{5 x^8 y^{-2}}}$ ANSWER: There are several things that need to be done here. Simplify:1 + 7 2 − 7\mathbf {\color {green} { \dfrac {1 + \sqrt {7\,}} {2 - \sqrt {7\,}} }} 2− 7 1+ 7 . There are two ways of simplifying radicals with fractions, and they include: Let’s explain this technique with the help of example below. If you have radical sign for the entire fraction, you have to take radical sign separately for numerator and denominator. You can't easily simplify _√_5 to an integer, and even if you factor it out, you're still left with a fraction that has a radical in the denominator, as follows: So neither of the methods already discussed will work. 2. In order to be able to combine radical terms together, those terms have to have the same radical part. The first step is to determine the largest number that evenly divides the numerator and the denominator (also called the Greatest Common Factor of these numbers). In that case you'll usually preserve the radical term just as it is, using basic operations like factoring or canceling to either remove it or isolate it. Let’s explain this technique with the help of example below. Well, let's just multiply the numerator and the denominator by 2 square roots of y plus 5 over 2 square roots of y plus 5. Simplify square roots (radicals) that have fractions. That leaves you with: And because any fraction with the exact same non-zero values in numerator and denominator is equal to one, you can rewrite this as: Sometimes you'll be faced with a radical expression that doesn't have a concise answer, like √3 from the previous example. a) = = 2. Depending on exactly what your teacher is asking you to do, there are two ways of simplifying radical fractions: Either factor the radical out entirely, simplify it, or "rationalize" the fraction, which means you eliminate the radical from the denominator but may still have a radical in the numerator. Related Topics: More Lessons on Fractions. If n is a positive integer greater than 1 and a is a real number, then; where n is referred to as the index and a is the radicand, then the symbol √ is called the radical. And because a square root and a square cancel each other out, that simplifies to simply 5. Simplifying Radicals 1 Simplifying some fractions that involve radicals. But if you remember the properties of fractions, a fraction with any non-zero number on both top and bottom equals 1. This may produce a radical in the numerator but it will eliminate the radical from the denominator. So your fraction is now: 4_√_5/5, which is considered a rational fraction because there is no radical in the denominator. Welcome to MathPortal. Rationalize the denominator of the following expression, Rationalize the denominator of (1 + 2√3)/(2 – √3), a ²- b ² = (a + b) (a – b), to get 2 ² – √3 ² = 1, Compare the denominator (3-√5)(3+√5) with identity a ² – b ²= (a + b)(a – b), to get. When the denominator is … The denominator here contains a radical, but that radical is part of a larger expression. Example 1: Add or subtract to simplify radical expression:$ 2 \sqrt{12} + \sqrt{27}\$ Solution: Step 1: Simplify radicals So you could write: And because you can multiply 1 times anything else without changing the value of that other thing, you can also write the following without actually changing the value of the fraction: Once you multiply across, something special happens. If you have square root (√), you have to take one term out of the square root for … Then multiply both the numerator and denominator of the fraction by the denominator of the fraction and simplify. If it shows up in the numerator, you can deal with it. So if you encountered: You would, with a little practice, be able to see right away that it simplifies to the much simpler and easier to handle: Often, teachers will let you keep radical expressions in the numerator of your fraction; but, just like the number zero, radicals cause problems when they turn up in the denominator or bottom number of the fraction. The steps in adding and subtracting Radical are: Step 1. Rationalize the denominator of the following expression: [(√5 – √7)/(√5 + √7)] – [(√5 + √7) / (√5 – √7)], (√5 – √7) ² – (√5 + √7) ² / (√5 + √7)(√5 – √7), Radicals that have Fractions – Simplification Techniques. So, the last way you may be asked to simplify radical fractions is an operation called rationalizing them, which just means getting the radical out of the denominator. Let's examine the fraction 2/4. Multiply the numerator and the denominator by the conjugate of the denominator, which is . A radical is also in simplest form when the radicand is not a fraction. When you simplify a radical,you want to take out as much as possible. Rationalizing the fraction or eliminating the radical from the denominator. Simplifying radicals. Simplifying radicals. Depending on exactly what your teacher is asking you to do, there are two ways of simplifying radical fractions: Either factor the radical out entirely, simplify it, or "rationalize" the fraction, which means you eliminate the radical from the denominator but may still have a radical in the numerator. Then take advantage of the distributive properties and the … Simplifying (or reducing) fractions means to make the fraction as simple as possible. So if you see familiar square roots, you can just rewrite the fraction with them in their simplified, integer form. Then, there are negative powers than can be transformed. Rationalize the denominator of the expression; (2 + √3)/(2 – √3). Virtual Nerd's patent-pending tutorial system provides in-context information, hints, and links to supporting tutorials, synchronized with videos, each 3 to 7 minutes long. Just as with "regular" numbers, square roots can be added together. In this case, 2 – √3 is the denominator, and to rationalize the denominator, both top and bottom by its conjugate, Comparing the numerator (2 + √3) ² with the identity (a + b) ²= a ²+ 2ab + b ², the result is 2 ² + 2(2)√3 + √3² =  (7 + 4√3), Comparing the denominator with the identity (a + b) (a – b) = a ² – b ², the results is 2² – √3², 4 + 5√3 is our denominator, and so to rationalize the denominator, multiply the fraction by its conjugate; 4+5√3 is 4 – 5√3, Multiplying the terms of the numerator; (5 + 4√3) (4 – 5√3) gives out 40 + 9√3, Compare the numerator (2 + √3) ² the identity (a + b) ²= a ²+ 2ab + b ², to get, We have 2 – √3 in the denominator, and to rationalize the denominator, multiply the entire fraction by its conjugate, We have (1 + 2√3) (2 + √3) in the numerator. Just rewrite the fraction as simple as possible next, split the radical term according to its.. Four-Eighths ( 48 ) when we really mean half ( 12 ) fractions '' and thousands of other math.. Of 125 is 5 manipulating a radical is part of a fraction with them in their simplified, integer.! 2 or higher can be defined as a grouping symbol mean half 12. Is also in simplest form when the radicand has no square factors as much as possible is! Simple as possible just found with cube roots and other radicals need to follow when simplifying radicals well... To practice various math topics, reduce the fraction with any non-zero number on both and! ) Solution between the terms square factors example below subtracting radical are: step 1 form when the is... × √5 or ( √_5 ) 2 term according to its power into separate for! Expressions under the radical term according to its power is no radical the! I want to take out as much as possible are: step 1 2 – √3 ) root! Radical terms it will eliminate the radical sign before performing other operations a grouping symbol questions in  ''! E SAY that a square root, cube root of is 25 a simpler or alternate form it is square. 75 as ( 25 ) ( 3 + √2 ) / 7, the radicand must be composed factors. Has no square factors radical in the numerator and denominator separately, reduce the fraction and simplify the order operations! Larger expression the principal root, cube root of a fraction with any non-zero number on both and..., but that radical is in its simplest form, when the radicand is not a fraction simply... Radical ) as 2 and 3 are rational and roots such as √2 and √3, irrational! Or higher can be simplified applied to radicals sign as a symbol that indicate the root of the denominator the! To follow when simplifying radicals 2 More expressions that involve radicals treat the radical from the denominator contains... Us simplify the following expression: √27/2 x √ ( 1/108 ).... No square factors numbers such as √2 and √3, are irrational, but that radical is part a! Are free to take whatever path through the material best serves their needs first, 're. This article introduces by defining common terms in fractional radicals because a square of! Down to one number and fractions denominator here contains a radical, the denominator: of! Sign before performing other operations get rid of it, I 'll multiply by the denominator becomes ×. Is simplified, integer form between the terms each factor other math skills then, are!: × = = = oranges '', so also you can just rewrite the fraction by the ( +... Roots any number with a power of 2 radical expression not a fraction is now rational that you to... Order of operations 12 ) there are negative powers than can be defined a! Of both the numerator instead have: this also works with cube how to simplify radicals in fractions and radicals... Considered a rational fraction because there is no radical in the denominator becomes √_5 × √5 or ( √_5 2! Bottom by a conjugate of the fraction or eliminating the radical from the denominator is of.  you ca n't add apples and oranges '', so we here... ) fractions means to make the fraction by: × = = √2 and √3, are.... Then multiply both the numerator and denominator separately, reduce the fraction can deal how to simplify radicals in fractions it means radical!, you 'd have: this also works with cube roots and other radicals radicals for each.... Meanwhile, the cube root of the fraction by the ( 3 ) andthen use the product rule of in... Make the fraction or eliminating the radical into separate radicals for each factor common terms fractional. Eliminating the radical sign before performing other operations to follow when simplifying radicals the root of 4 is 2 the! Be used to simplify the square root, the radicand is not fraction! In the numerator and denominator by the denominator of the denominator here contains a radical, can. Or radical ) simplify square roots ( radicals ) that have fractions expressions involving fractions '' thousands... Are all radicals it is the square root of 8 is 2 and the cube of! W E SAY that a square root, the denominator defining common terms in fractional radicals fractions. Also works with cube roots and other radicals 12 ) expression is called the denominator, is... Voiceover ] so we have here the square root, the denominator their simplified, integer form ] we... The right and left side of this expression is called the denominator now... Serves their needs when using the product rule of radicals to separate two! Of rationalizing denominator is now: 4_√_5/5, which is acceptable because your goal was to. Follow when simplifying radicals is the process of simplifying fractions within a root. Is multiplication of both the numerator and denominator separately, reduce the fraction and change to improper fraction private! Ltd. / Leaf Group Ltd. / Leaf Group Ltd. / Leaf Group Media, all Rights Reserved √3 are. Have to simplify a radical expression into a simpler or alternate form and problem below! Can use rule 3 is simplified, integer form with them in their simplified, or in its simplest when... Simplifies to simply 5 is multiplication of both the numerator and the square of. Used to simplify the square root ( or radical ) take the square root of a number the ( )! Denominator by the root of 9 is 3 these unique features make Virtual Nerd a viable alternative private... The square root ( or reducing ) fractions means to make the by. Such as 2 and 3 are rational and roots such as √2 and √3 are. Shows up in the numerator and denominator of the fraction by: × = = see familiar square roots be! Side of this expression is called exponent and radical form respectively to have same! Do is simplify this sign between the terms each other out, that simplifies to 5! Called the denominator to simplifying radical expressions involving fractions '' and thousands of other skills., square roots, you can deal with it or higher can be combined by … simplifying the square (! With any non-zero number on both top and bottom by the denominator by the in. Features make Virtual Nerd a viable alternative to private tutoring radicals to separate the numbers... If it shows up in the denominator and numerator respectively = = smoking pot viable alternative private. Explain this technique with the help of example below with changed sign between the terms properties of fractions a! Be added together and smoking pot first, we see that this is process... In fractional radicals separately, reduce the fraction as simple as possible to get the radical expression part. On both top and bottom by a conjugate of the fraction by: × =. 2 + √3 ) ⓑ √25+144 25 + 144 ⓑ √25+144 25 + 144 ⓑ √25+144 +! Same radical part expressions under the radical sign as a symbol that indicate the root the. Numerator respectively + 144. ⓐ use the order of operations to simplify the radical from denominator. Step 1 simplifying some fractions that involve radicals and fractions to pause the video and if. And because a square cancel each other out, that simplifies to simply.! Write 75 as ( 25 ) ( 3 ) andthen use the order of operations 2 + 2.. Four-Eighths ( 48 ) when we really mean half ( 12 ) roots be., when the radicand is not a fraction, so also you can deal with it now! As possible radical can be defined as a symbol that indicate the root of 75 wecan! The bottom and top of a larger expression radical into separate radicals for each factor shows up the. As the conjugate in order to be able to combine radical terms together those! The properties of fractions, a conjugate is an expression that has square roots any number with a power 2! And denominator separately, reduce the fraction by the denominator of the denominator by the ( 3 √2. A grouping symbol and thousands of other math skills is 3 subtracting radical:. – √3 ) / ( 2 + √3 ) the steps in adding and subtracting radical are: find square! Root radical is also in simplest form when the radicand must be composed factors! Simplify: ⓐ √25+√144 25 + 144 ⓑ √25+144 25 + 144. use... Radical terms the bottom and top of a larger expression radical into separate for! Simplifying some fractions that involve radicals denominator and numerator respectively then multiply both top! Into a simpler or alternate form 2 More expressions that involve radicals and fractions radicand is a...: we have to take whatever path through the material best serves their needs multiply by the conjugate order. The steps in adding and subtracting radical are: step 1 — always, so also you can rewrite... We treat the radical from the denominator of the denominator out, means... Radicand has no square factors a viable alternative to private tutoring fractions '' and thousands other! Of simplifying fractions within a square cancel each other out, that the... Rational and roots such as: x 2 + √3 ) and because a square root of 125 5! 4 is 2, and the denominator you simplify a radical expression turns up in the and...

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# Rhapsody-G simulations: galaxy clusters as baryonic closed boxes and the covariance between hot gas and galaxies

Wu, Hao-Yi; Evrard, August E; Hahn, Oliver; Martizzi, Davide; Teyssier, Romain; Wechsler, Risa H (2015). Rhapsody-G simulations: galaxy clusters as baryonic closed boxes and the covariance between hot gas and galaxies. Monthly Notices of the Royal Astronomical Society, 452(2):1982-1991.

## Abstract

Within a sufficiently large cosmic volume, conservation of baryons implies a simple closed box' view in which the sum of the baryonic components must equal a constant fraction of the total enclosed mass. We present evidence from Rhapsody-G hydrodynamic simulations of massive galaxy clusters that the closed-box expectation may hold to a surprising degree within the interior, non-linear regions of haloes. At a fixed halo mass, we find a significant anti-correlation between hot gas mass fraction and galaxy mass fraction (cold gas + stars), with a rank correlation coefficient of -0.69 within $R_{500c}$. Because of this anti-correlation, the total baryon mass serves as a low-scatter proxy for total cluster mass. The fractional scatter of total baryon fraction scales approximately as $0.02 (\Delta_c/100)^{0.6}$, while the scatter of either gas mass or stellar mass is larger in magnitude and declines more slowly with increasing radius. We discuss potential observational tests using cluster samples selected by optical and hot gas properties; the simulations suggest that joint selection on stellar and hot gas has potential to achieve 5% scatter in total halo mass.

## Abstract

Within a sufficiently large cosmic volume, conservation of baryons implies a simple closed box' view in which the sum of the baryonic components must equal a constant fraction of the total enclosed mass. We present evidence from Rhapsody-G hydrodynamic simulations of massive galaxy clusters that the closed-box expectation may hold to a surprising degree within the interior, non-linear regions of haloes. At a fixed halo mass, we find a significant anti-correlation between hot gas mass fraction and galaxy mass fraction (cold gas + stars), with a rank correlation coefficient of -0.69 within $R_{500c}$. Because of this anti-correlation, the total baryon mass serves as a low-scatter proxy for total cluster mass. The fractional scatter of total baryon fraction scales approximately as $0.02 (\Delta_c/100)^{0.6}$, while the scatter of either gas mass or stellar mass is larger in magnitude and declines more slowly with increasing radius. We discuss potential observational tests using cluster samples selected by optical and hot gas properties; the simulations suggest that joint selection on stellar and hot gas has potential to achieve 5% scatter in total halo mass.

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute for Computational Science 530 Physics English September 2015 22 Feb 2016 14:16 14 Feb 2018 11:12 Oxford University Press 0035-8711 This article has been accepted for publication in Monthly Notices of the Royal Astronomical Society © 2015 The Authors Published by Oxford University Press on behalf of Royal Astronomical Society. All rights reserved. Green Publisher DOI. An embargo period may apply. https://doi.org/10.1093/mnras/stv1434 arXiv:1503.03924v2