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Math Expert V
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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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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! --> if $$n=2$$ then $$(n+2)!-n!=22$$ and for $$p=2$$ then answer will be YES but for $$p=11$$ the answer will be NO. Not sufficient.

(2) p is a factor of (n+2)!/n! --> $$\frac{(n+2)!}{n!}=(n+1)(n+2)$$ --> if $$n=2$$ then $$(n+1)(n+2)=12$$ and for $$p=2$$ then answer will be YES but for $$p=3$$ the answer will be NO. Not sufficient.

(1)+(2) $$(n+2)!-n!=n!((n+1)(n+2)-1)$$. Now, $$(n+1)(n+2)-1$$ and $$(n+1)(n+2)$$ are consecutive integers. Two consecutive integers are co-prime, which means that they don't share ANY common factor but 1. For example 20 and 21 are consecutive integers, thus only common factor they share is 1. So, as from (2) $$p$$ is a factor of $$(n+1)(n+2)$$ then it can not be a factor of $$(n+1)(n+2)-1$$, thus in order $$p$$ to be a factor of $$n!*((n+1)(n+2)-1)$$, from (1), then it should be a factor of the first multiple of this expression: $$n!$$. Sufficient.

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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.

from statement 1, we get diffrent remainders for different values of p NS
statement 2 : all primes greater than 3 give 1 as remainder hence sufficient but less than 3 has diff remainders

IMO C
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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.

let the 3 numbers be a<b<c. abc=c^2. therefore,c(c-ab)=0.. since a,b,c are positive int. c !=0.. therfore c-ab=0. Hence c=ab

from statement2 : ab =24
statement 1 NS

IMO B.
Pleas correct if wrong
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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.

I didnt solve this one .. but from statement 1 and II since i know the n and K both I can calculate the probability..IMo C
Please correct if wrong
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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.

statement I since a-b = 15. the third side c should be greater than 15. for convenience I set this as 15.1 and b= 0.1 therefore a = 15.1 and a+b+c = 30.2 >30 hence Suff
Statement II :NS

IMo A
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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.

IMO E statements together are not suff
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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) 13 patients of Vertigo Hospital have arachnophobia but not acrophobia. Clearly insufficient.

On solving statement I : 13 ppl have acrophobia hence, suff.

IMO A
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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.

IMO D.

each statement gives a linear eqn in 2 variables. since i just need the diffrenec between both and not the exact value for Each . I and II are individually sufficient Hence D
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Bunuel wrote:
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.

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!

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

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) 13 patients of Vertigo Hospital have arachnophobia but not acrophobia.

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.

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.

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.

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.

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.

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

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.

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

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.

1. D
Jules'invested amt - x
Jim's invested amt - y
Q is x-y?
a) x+0.12x-y-0.12y = 24 => x-y can be found
b) x+0.2x-y-0.2y = 40 => x-y can be found

2. C
a) n!(n+2)(n+1) - n! = pK cant say about (n+2)(n+1) being a multiple of p
b) it confirms the above missing element as (n+2)(n+1) is a multiple of p

3.C
Using both it comes to y^4 = 3x^4-29x^2 +64 => either x being odd or even 3x^4-29x^2 is always even hence y^4 is even and hence y is even.

4.A
let be n(acrophobia) and c be common to both
a) 58 = 45 + x - c + c => x can be determined
b) no of patients dont belong to both is not mentioned

5.E
No division of how many listen and how many dont is provided

6. A
a) a-b=15 => c>15 => a+b >15+(c-15) => a+b+c >30
b) not relevant

7. C
I think the prob will be 1/(K*(K-1)*..(K-(n-1))
so we need both k & n

8. C
I know that remainder is always 1 when n^2 is divided by 12 given n is a prime greater than 3. Got from plugging nos

9. B
I think we need to find the largest no as xyz=z^2 =>xy=z which is given in second statement. Not sure if I am missing a trap.

10. E
As it is not mentioned if there any more varieties of snakesother than cobras and vipers.

11. E

Not sure about this. I think each of the statements along with the premise gives that m > b, but by how much can not be concluded and hence cant say.

12. B
Again using both also I get yz =9 which is directly implied by statement b. Hence I go with B. Not sure of any trap.

13.E
not used to giving explanations hence taking a lot of time Originally posted by mithun2vrs on 22 Mar 2012, 10:01.
Last edited by mithun2vrs on 22 Mar 2012, 12:31, edited 1 time in total.
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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.

1)I set up a table of M/F and Listeners/Non-Listeners. Since 56% are male and there are 35 astronauts, reduce 56/100 where the demoninator is <= 35. The only solution is 14/25. Thus, 25 astronauts do NOT listen to Bach and 10 astronauts listen to Bach. Sufficient.

2)With the same table I already have set up, I can determine that either 10, 20, or 30 listen to Bach; therefore, not sufficient.

C would be a trap answer. Based on (1) our options of 20 and 30 from (2) are eliminated, for a total of 7 female listeners, 3 male listeners, 11 female non-listeners and 14 male non-listeners. However, this information is unneccessary since (1) is enough.

Edit: I wanted to add that I came to these conclusions because it's stated "at least one astronaut do NOT listen to Bach." I think that oversight is why many people picked E?
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Thanks for welcoming, Bunuel..awaiting ur OAs n explanations for some of those tricky qns in the list.

@dyaffe55 - Yeah, thts rgt, i overlooked the given data for qn #5. Thanks for the explanation, it makes more sense.

There are 2 more wrong answers frm me other than this in the qn list, to my knwledge till thn..sry if thr is any more..
9. the answer is B (and not D, i mistyped the answer choice in my prev post)
10. guess its B (and not E..came across a similar qn post frm Bunuel..guess i should be more alert on the 'atleast' keyword usage in the qn stems, hereafter.)
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SOLUTIONS:

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.

Question: $$x-y=?$$

(1) In one year Jules earned $24 more than Jim from bond M. $$0.12x-0.12y=24$$ --> $$0.12(x-y)=24$$ --> $$x-y=200$$. Sufficient. (2) If the interest were 20% then in one year Jules would have earned$40 more than Jim from bond M. Basically the same type of information as above: $$0.2x-0.2y=40$$ --> $$0.2(x-y)=40$$ --> $$x-y=200$$. Sufficient.

Important note when two quantities are increased (decreased) by the same percent their difference also increase (decrease) by the same percent. For example if you increase 100 and 150 by 20% to 120 and 180 respectively, then their difference will also increase by the same 20% from 50 to 60.
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3. If x and y are integers, is y an even integer?

(1) 4y^2+3x^2=x^4+y^4 --> rearrange: $$3x^2-x^4=y^4-4y^2$$ --> $$x^2(3-x^2)=y^2(y^2-4)$$. Notice that LHS is even for any value of $$x$$: if $$x$$ is odd then $$3-x^2=odd-odd=even$$ and if $$x$$ is even then the product is naturally even. So, $$y^2(y^2-4)$$ is also even, but in order it to be even $$y$$ must be even, since if $$y$$ is odd then $$y^2(y^2-4)=odd*(odd-even)=odd*odd=odd$$. Sufficient.

(2) y=4-x^2 --> if $$x=odd$$ then $$y=even-odd=odd$$ but if $$x=even$$ then $$y=even-even=even$$. Not sufficient.

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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 [ 7.34 KiB | Viewed 84265 times ]
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.

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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.

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6. Is the perimeter of triangle with the sides a, b and c greater than 30?

700+ question.

(1) a-b=15. Must know for the GMAT: the length of any side of a triangle must be larger than the positive difference of the other two sides, but smaller than the sum of the other two sides. So, a+b>c>15 --> a+b+c>30. Sufficient.

(2) The area of the triangle is 50. For a given perimeter equilateral triangle has the largest area. Now, if the perimeter were equal to 30 then it would have the largest area if it were equilateral. Let's find what this area would be: $$Area_{equilateral}=s^2*\frac{\sqrt{3}}{4}=(\frac{30}{3})^2*\frac{\sqrt{3}}{4}=25*\sqrt{3}<50$$. Since even equilateral triangle with perimeter of 30 can not produce the area of 50, then the perimeter must be more that 30. Sufficient.

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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.

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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.

(1) p is greater than 3. Clearly insufficient: different values of $$p$$ will give different values of the remainder.
(2) p is a prime. Also insufficient: if $$p=2$$ then the remainder is 4 but if $$p=3$$ then the remainder is 9.

(1)+(2) You can proceed with number plugging and try several prime numbers greater than 3 to see that the remainder will always be 1 (for example try $$p=5$$, $$p=7$$, $$p=11$$).

If you want to double-check this with algebra you should apply the following property of the prime number: any prime number greater than 3 can be expressed either as $$p=6n+1$$ or $$p=6n-1$$.

If $$p=6n+1$$ then $$p^2=36n^2+12n+1$$ which gives remainder 1 when divided by 12;

If $$p=6n-1$$ then $$p^2=36n^2-12n+1$$ which also gives remainder 1 when divided by 12.

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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?

700 question.

Let the three integers be $$a$$, $$b$$, and $$c$$, where $$0<a<b<c$$. Given: $$abc=c^2$$ --> $$ab=c$$. Question: $$ab=c=?$$

(1) The average (arithmetic mean) of the three numbers is 34/3 --> $$a+b+c=34$$ --> $$a+b+ab=34$$ --> $$(a+1)(b+1)=35$$. Now, since $$a$$ and $$b$$ are integers, then $$a+1=5$$ and $$b+1=7$$. $$a=4$$ and $$b=6$$ --> $$ab=24$$. Sufficient. (Notice that $$a+1=1$$ and $$b+1=35$$ is not possible since in this case $$a=0$$ and we are told that all integers are positive).

(2) The largest number of the three distinct numbers is 24. Directly give the value of c. Sufficient.

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