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The Discreet Charm of the DS [#permalink]
02 Feb 2012, 03:15

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Expert's post

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I'm posting the next set of medium/hard DS questions. I'll post OA's with detailed explanations after some discussion. Please, post your solutions along with the answers. Good luck!

1. Bonnie can paint a stolen car in x hours, and Clyde can paint the same car in y hours. They start working simultaneously and independently at their respective constant rates at 9:45am. If both x and y are odd integers, is x=y? (1) x^2+y^2<12 (2) Bonnie and Clyde complete the painting of the car at 10:30am

4. How many numbers of 5 consecutive positive integers is divisible by 4? (1) The median of these numbers is odd (2) The average (arithmetic mean) of these numbers is a prime number

7. A certain fruit stand sold total of 76 oranges to 19 customers. How many of them bought only one orange? (1) None of the customers bought more than 4 oranges (2) The difference between the number of oranges bought by any two customers is even

10. The function f is defined for all positive integers a and b by the following rule: f(a,b)=(a+b)/GCF(a,b), where GCF(a,b) is the greatest common factor of a and b. If f(10,x)=11, what is the value of x? (1) x is a square of an integer (2) The sum of the distinct prime factors of x is a prime number.

Re: The Discreet Charm of the DS [#permalink]
12 Dec 2012, 18:36

Bunuel wrote:

SOLUTIONS:

1. Bonnie can paint a stolen car in x hours, and Clyde can paint the same car in y hours. They start working simultaneously and independently at their respective constant rates at 9:45am. If both x and y are odd integers, is x=y?

Bonnie and Clyde when working together complete the painting of the car ins \frac{xy}{x+y} hours (sum of the rates equal to the combined rate or reciprocal of total time: \frac{1}{x}+\frac{1}{y}=\frac{1}{T} --> T=\frac{xy}{x+y}). Now, if x=y then the total time would be: \frac{x^2}{2x}=\frac{x}{2}, since x is odd then this time would be odd/2: 0.5 hours, 1.5 hours, 2.5 hours, ....

(1) x^2+y^2<12 --> it's possible x and y to be odd and equal to each other if x=y=1 but it's also possible that x=1 and y=3 (or vise-versa). Not sufficient.

(2) Bonnie and Clyde complete the painting of the car at 10:30am --> they complete the job in 3/4 of an hour (45 minutes), since it's not odd/2 then x and y are not equal. Sufficient.

Answer: B.

i didnot understand this..... the questions says if they are working independently.... why are you considering combined rate? my analysis I ended up at the same answer though

Re: The Discreet Charm of the DS [#permalink]
13 Dec 2012, 02:24

Expert's post

Amateur wrote:

Bunuel wrote:

SOLUTIONS:

1. Bonnie can paint a stolen car in x hours, and Clyde can paint the same car in y hours. They start working simultaneously and independently at their respective constant rates at 9:45am. If both x and y are odd integers, is x=y?

Bonnie and Clyde when working together complete the painting of the car ins \frac{xy}{x+y} hours (sum of the rates equal to the combined rate or reciprocal of total time: \frac{1}{x}+\frac{1}{y}=\frac{1}{T} --> T=\frac{xy}{x+y}). Now, if x=y then the total time would be: \frac{x^2}{2x}=\frac{x}{2}, since x is odd then this time would be odd/2: 0.5 hours, 1.5 hours, 2.5 hours, ....

(1) x^2+y^2<12 --> it's possible x and y to be odd and equal to each other if x=y=1 but it's also possible that x=1 and y=3 (or vise-versa). Not sufficient.

(2) Bonnie and Clyde complete the painting of the car at 10:30am --> they complete the job in 3/4 of an hour (45 minutes), since it's not odd/2 then x and y are not equal. Sufficient.

Answer: B.

i didnot understand this..... the questions says if they are working independently.... why are you considering combined rate? my analysis I ended up at the same answer though

Because they are working simultaneously and independently to paint the same car. _________________

Re: The Discreet Charm of the DS [#permalink]
29 Dec 2012, 06:37

Bunuel wrote:

2. Is xy<=1/2?

(1) x^2+y^2=1. Recall that (x-y)^2\geq{0} (square of any number is more than or equal to zero) --> x^2-2xy+y^2\geq{0} --> since x^2+y^2=1 then: 1-2xy\geq{0} --> xy\leq{\frac{1}{2}}. Sufficient.

(2) x^2-y^2=0 --> |x|=|y|. Clearly insufficient.

Answer: A.

Dear Bunuel

Why haven't we used (x+y)^2 instead of (x-y)^2 in statement 1 ? Sorry if its a silly question _________________

Re: The Discreet Charm of the DS [#permalink]
11 Feb 2013, 02:38

Bunuel wrote:

SOLUTIONS:

1. Bonnie can paint a stolen car in x hours, and Clyde can paint the same car in y hours. They start working simultaneously and independently at their respective constant rates at 9:45am. If both x and y are odd integers, is x=y?

Bonnie and Clyde when working together complete the painting of the car ins \frac{xy}{x+y} hours (sum of the rates equal to the combined rate or reciprocal of total time: \frac{1}{x}+\frac{1}{y}=\frac{1}{T} --> T=\frac{xy}{x+y}). Now, if x=y then the total time would be: \frac{x^2}{2x}=\frac{x}{2}, since x is odd then this time would be odd/2: 0.5 hours, 1.5 hours, 2.5 hours, ....

(1) x^2+y^2<12 --> it's possible x and y to be odd and equal to each other if x=y=1 but it's also possible that x=1 and y=3 (or vise-versa). Not sufficient.

(2) Bonnie and Clyde complete the painting of the car at 10:30am --> they complete the job in 3/4 of an hour (45 minutes), since it's not odd/2 then x and y are not equal. Sufficient.

Answer: B.

Hi,

Just a question, Only if we get a answer which is "odd/2" then x & y are considered to be equal . rite?

Re: The Discreet Charm of the DS [#permalink]
11 Feb 2013, 04:45

Expert's post

FTG wrote:

Bunuel wrote:

SOLUTIONS:

1. Bonnie can paint a stolen car in x hours, and Clyde can paint the same car in y hours. They start working simultaneously and independently at their respective constant rates at 9:45am. If both x and y are odd integers, is x=y?

Bonnie and Clyde when working together complete the painting of the car ins \frac{xy}{x+y} hours (sum of the rates equal to the combined rate or reciprocal of total time: \frac{1}{x}+\frac{1}{y}=\frac{1}{T} --> T=\frac{xy}{x+y}). Now, if x=y then the total time would be: \frac{x^2}{2x}=\frac{x}{2}, since x is odd then this time would be odd/2: 0.5 hours, 1.5 hours, 2.5 hours, ....

(1) x^2+y^2<12 --> it's possible x and y to be odd and equal to each other if x=y=1 but it's also possible that x=1 and y=3 (or vise-versa). Not sufficient.

(2) Bonnie and Clyde complete the painting of the car at 10:30am --> they complete the job in 3/4 of an hour (45 minutes), since it's not odd/2 then x and y are not equal. Sufficient.

Answer: B.

Hi,

Just a question, Only if we get a answer which is "odd/2" then x & y are considered to be equal . rite?

Re: The Discreet Charm of the DS [#permalink]
14 Feb 2013, 00:24

Bunuel wrote:

SOLUTIONS:

1. Bonnie can paint a stolen car in x hours, and Clyde can paint the same car in y hours. They start working simultaneously and independently at their respective constant rates at 9:45am. If both x and y are odd integers, is x=y?

Bonnie and Clyde when working together complete the painting of the car ins \frac{xy}{x+y} hours (sum of the rates equal to the combined rate or reciprocal of total time: \frac{1}{x}+\frac{1}{y}=\frac{1}{T} --> T=\frac{xy}{x+y}). Now, if x=y then the total time would be: \frac{x^2}{2x}=\frac{x}{2}, since x is odd then this time would be odd/2: 0.5 hours, 1.5 hours, 2.5 hours, ....

(1) x^2+y^2<12 --> it's possible x and y to be odd and equal to each other if x=y=1 but it's also possible that x=1 and y=3 (or vise-versa). Not sufficient.

(2) Bonnie and Clyde complete the painting of the car at 10:30am --> they complete the job in 3/4 of an hour (45 minutes), since it's not odd/2 then x and y are not equal. Sufficient.

Answer: B.

Bunuel, I did not understand the reasoning behind statement 2 being sufficient. Line marked in red is my doubt. what does it mean ?

Bunuel, Here's how I did . I don't know why it is wrong. Please advise :

6a=3b=7c

=> 12a=6b=14c

From 1) ac=6b => ac=12a => c=12.. a can be cancelled because it is not inequality. .

so c =12..

however,

ac-12a=0 --> a(c-12)=0 so a=0 OR c=12 .. Is this the reason why my soln is wrong?

Never reduce an equation by a variable (or expression with a variable), if you are not certain that the variable (or the expression with a variable) doesn't equal to zero. We can not divide by zero.

So, if you divide (reduce) ac=12a by a, you assume, with no ground for it, that a does not equal to zero thus exclude a possible solution (notice that both a=0 AND c-12=0 satisfy the equation).

Re: The Discreet Charm of the DS [#permalink]
14 Feb 2013, 01:16

Expert's post

thinktank wrote:

Bunuel wrote:

SOLUTIONS:

1. Bonnie can paint a stolen car in x hours, and Clyde can paint the same car in y hours. They start working simultaneously and independently at their respective constant rates at 9:45am. If both x and y are odd integers, is x=y?

Bonnie and Clyde when working together complete the painting of the car ins \frac{xy}{x+y} hours (sum of the rates equal to the combined rate or reciprocal of total time: \frac{1}{x}+\frac{1}{y}=\frac{1}{T} --> T=\frac{xy}{x+y}). Now, if x=y then the total time would be: \frac{x^2}{2x}=\frac{x}{2}, since x is odd then this time would be odd/2: 0.5 hours, 1.5 hours, 2.5 hours, ....

(1) x^2+y^2<12 --> it's possible x and y to be odd and equal to each other if x=y=1 but it's also possible that x=1 and y=3 (or vise-versa). Not sufficient.

(2) Bonnie and Clyde complete the painting of the car at 10:30am --> they complete the job in 3/4 of an hour (45 minutes), since it's not odd/2 then x and y are not equal. Sufficient.

Answer: B.

Bunuel, I did not understand the reasoning behind statement 2 being sufficient. Line marked in red is my doubt. what does it mean ?

From the stem we got that if x=y then the total time would be: \frac{x^2}{2x}=\frac{x}{2}, since x is odd then this time would be odd/2: 0.5 hours (1/2 hours = 0.5 hours), 1.5 hours (3/2 hours = 1.5 hours), 2.5 hours (5/2 hours = 2.5 hours), 3.5 hours (7/2 hours = 3.5 hours), 4.5 hours (9/2 hours = 4.5 hours), ....

Now, from the second statement we got that they complete the job in 0.75 hours, since the total time (0.75 hours) is NOT odd/2 (0.5 hours, 1.5 hours, 2.5 hours, 3.5 hours, 4.5 hours, ....), then x and y are not equal.

Bunuel, Here's how I did . I don't know why it is wrong. Please advise :

6a=3b=7c

=> 12a=6b=14c

From 1) ac=6b => ac=12a => c=12.. a can be cancelled because it is not inequality. .

so c =12..

however,

ac-12a=0 --> a(c-12)=0 so a=0 OR c=12 .. Is this the reason why my soln is wrong?

Never reduce an equation by a variable (or expression with a variable), if you are not certain that the variable (or the expression with a variable) doesn't equal to zero. We can not divide by zero.

So, if you divide (reduce) ac=12a by a, you assume, with no ground for it, that a does not equal to zero thus exclude a possible solution (notice that both a=0 AND c-12=0 satisfy the equation).

Hope it's clear.

thanks alot. .you rock man!! r u a phd in maths ? _________________

hope is a good thing, maybe the best of things. And no good thing ever dies.

Re: The Discreet Charm of the DS [#permalink]
23 Feb 2013, 15:17

Bunuel wrote:

3. If a, b and c are integers, is abc an even integer?

In order the product of the integers to be even at leas on of them must be even

(1) b is halfway between a and c --> on the GMAT we often see such statement and it can ALWAYS be expressed algebraically as b=\frac{a+c}{2}. Now, does that mean that at leas on of them is be even? Not necessarily, consider a=1, b=3 and c=5. Of course it's also possible that b=even, for example if a=1 and b=7. Not sufficient.

(2) a = b - c --> a+c=b. Since it's not possible that the sum of two odd integers to be odd then the case of 3 odd numbers is ruled out, hence at least on of them must be even. Sufficient.

Answer: B.

1) b = a+c/2 i.e. a+c = even (as it is divisible by 2) and an even# divided by another even# can be odd or even (e.g. 46/2 = 23 an odd, but 48/2 = 24 an even). now if a + c = odd + odd = even and if a+c/2 = odd then all 3 numbers are odd (e.g. a=21, b=23 and c=25) and abc = odd but if a+c = even+even then a+c/2 = odd and abc = even (a=22, b=24 and c=26). so insufficient 2) a = b - c i.e. a + c = b. from number properties we know that i) odd+odd = even, ii) even+odd = odd iii) even+even = even so in any of the 3 cases you will end up with atleast one number that is even and hence abc = even. sufficient. correct ans. B _________________

___________________________________________ Consider +1 Kudos if my post helped

Re: The Discreet Charm of the DS [#permalink]
23 May 2013, 14:59

Hi Brunnel, I have trouble with question 9 for the second statement, 2x-3<3y-4, when I substitute x=-5; y= -1; the equation holds but when I substitute x=-2; y=-5; the equation collapsed. So, wouldn't the answer be E? Please help explain, Thanks.

Re: The Discreet Charm of the DS [#permalink]
23 May 2013, 15:09

Expert's post

smartyman wrote:

Hi Brunnel, I have trouble with question 9 for the second statement, 2x-3<3y-4, when I substitute x=-5; y= -1; the equation holds but when I substitute x=-2; y=-5; the equation collapsed. So, wouldn't the answer be E? Please help explain, Thanks.

The question does not ask whether 2x - 3 < 3y - 4, it asks whether x<y. While 2x - 3 < 3y - 4 is given to be true by the second statement. _________________

Re: The Discreet Charm of the DS [#permalink]
05 Jun 2013, 13:19

Bunuel wrote:

10. The function f is defined for all positive integers a and b by the following rule: f(a,b)=(a+b)/GCF(a,b), where GCF(a,b) is the greatest common factor of a and b. If f(10,x)=11, what is the value of x?

Notice that the greatest common factor of 10 and x, GCF(10,x), naturally must be a factor of 10: 1, 2, 5, and 10. Thus from f(10,x)=11 we can get four different values of x:

(1) x is a square of an integer --> x can be 1 or 100. Not sufficient.

(2) The sum of the distinct prime factors of x is a prime number ---> distinct primes of 12 are 2 and 3: 2+3=5=prime, distinct primes of 45 are 3 and 5: 3+5=8\neq{prime} and distinct primes of 100 are 2 and 5: 2+5=7=prime. x can be 12 or 100. Not sufficient.

(1)+(2) x can only be 100. Sufficient.

Answer: C.

I think the answer to the above questions should be "A" not "C". Since the GCD for 10 and x can only be 1,2,5 and 10, the corresponding value for x can be 1, 12, 45 and 100. Given this information, the first clause leaves only 100 as the correct answer, which is a square of 10. Please let me know if there is any flaw in my reasoning.

Re: The Discreet Charm of the DS [#permalink]
05 Jun 2013, 22:17

Bunuel wrote:

9. If x and y are negative numbers, is x<y?

(1) 3x + 4 < 2y + 3 --> 3x<2y-1. x can be some very small number for instance -100 and y some large enough number for instance -3 and the answer would be YES, x<y BUT if x=-2 and y=-2.1 then the answer would be NO, x>y. Not sufficient.

However, the questions is whether x<y. If x< y-(any negative term) doesn't mean that X< Y. For example: if x= -5 y= -3 (Here x<Y) and according to the above equation -5<-3 but -5<-3 + (a negative term, say -3) will make the questions incorrect.

Please tell me where I am going wrong with this. Thanks!

Re: The Discreet Charm of the DS [#permalink]
05 Jun 2013, 23:20

Expert's post

narulajasneet wrote:

Bunuel wrote:

10. The function f is defined for all positive integers a and b by the following rule: f(a,b)=(a+b)/GCF(a,b), where GCF(a,b) is the greatest common factor of a and b. If f(10,x)=11, what is the value of x?

Notice that the greatest common factor of 10 and x, GCF(10,x), naturally must be a factor of 10: 1, 2, 5, and 10. Thus from f(10,x)=11 we can get four different values of x:

(1) x is a square of an integer --> x can be 1 or 100. Not sufficient.

(2) The sum of the distinct prime factors of x is a prime number ---> distinct primes of 12 are 2 and 3: 2+3=5=prime, distinct primes of 45 are 3 and 5: 3+5=8\neq{prime} and distinct primes of 100 are 2 and 5: 2+5=7=prime. x can be 12 or 100. Not sufficient.

(1)+(2) x can only be 100. Sufficient.

Answer: C.

I think the answer to the above questions should be "A" not "C". Since the GCD for 10 and x can only be 1,2,5 and 10, the corresponding value for x can be 1, 12, 45 and 100. Given this information, the first clause leaves only 100 as the correct answer, which is a square of 10. Please let me know if there is any flaw in my reasoning.

x could be 1, 12, 45 or 100.

(1) says that x is a square of an integer --> x could be 1^2=1 or 10^2=100. Two answers, thus the statement is insufficient.

Re: The Discreet Charm of the DS [#permalink]
05 Jun 2013, 23:27

Expert's post

narulajasneet wrote:

Bunuel wrote:

9. If x and y are negative numbers, is x<y?

(1) 3x + 4 < 2y + 3 --> 3x<2y-1. x can be some very small number for instance -100 and y some large enough number for instance -3 and the answer would be YES, x<y BUT if x=-2 and y=-2.1 then the answer would be NO, x>y. Not sufficient.

I have a little doubt around the explanation given for Q9 Clause 2. You mentioned the following as your explanation: (2) 2x - 3 < 3y - 4 --> x<1.5y-\frac{1}{2} --> x<y+(0.5y-\frac{1}{2})=y+negative --> x<y (as y+negative is "more negative" than y). Sufficient.

However, the questions is whether x<y. If x< y-(any negative term) doesn't mean that X< Y. For example: if x= -5 y= -3 (Here x<Y) and according to the above equation -5<-3 but -5<-3 + (a negative term, say -3) will make the questions incorrect.

Please tell me where I am going wrong with this. Thanks!

It's the other way around: if x and y are negative numbers and IF x<y+negative, then x<y. _________________