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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.

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

The catch is that they are working independently.

Answer B

Just to clarify more, working independently does not mean they are painting different cars. They are still painting the same car. Only that, the events are mutually exclusive, as you say in probabilistic terms, so that rates are not affected when they work simultaneously.

Another way : 1) more than one solution possible. 2) let's say 1)x=y=1. Work will be completed in 1/2 hour i.e. 10.15 am. 2)x=y=3. Work will be completed in 3/2 hour i.e. 11.15 am. Not possible. B.
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My funda - Area of square is largest among all the quadilateral with same perimeter. Stmt 1 - Only possible values of x and y are 1/Sqrt(2). So sufficient as xy = 1/2 Stmt 2 - Only says x and y are equal. Not sufficient Answer A

You are close to correct reasoning for (1), though from it you can not say that xy=1/2 and the only possible value for x and y are 1/Sqrt(2). Consider the following example: 0^1+1^2=1.

As for (2): x^2-y^2=0 doesn't mean that x=y.

1) x2 + y2 = 1 is a circle with radius 1, and origin as center. We are only concerned with 1st and 3rd quad (think why? ), Draw a line y=x which intersects circle at x=y=1/sqrt(2). We can observe, as x becomes greater than 1/sqrt2, y gets lesser than 1/sqrt2 moving on the circle (below y=x line). Hence xy<= 1/2. Similarly, as y becomes greater than 1/sqrt2, x gets lesser than 1/sqrt2 moving on the circle (above y=x line). Hence xy<= 1/2. Hence sufficient.

For more mathematically inclined, draw graph of xy=1/2. It has only one point of contact with the circle, tangential at x=y=1/sqrt(2).

Attachments

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IMO, if you can imagine drawing the circle and y=x line, it takes less than 30 sec. to figure it out. It only takes so much longer to explain in text. And ofcourse, no need to draw the xy graph. Will wait though for the even faster method, if any.

If you answered this question in less than 30 sec using this approach then all I can say is great job!

As for the easier/faster solution, little hint: it involves simplest algebraic manipulation.
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(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.

(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.

Hi bunuel, Did not got how u solved option 2.Could you please explain in detail. thanks.

(2) 2x - 3 < 3y - 4 --> \(x<1.5y-\frac{1}{2}\) --> \(x<y+(0.5y-\frac{1}{2})\). Now, since \(y\) is a negative number then \(0.5y-\frac{1}{2}=negative\) so, we have that: \(x<y+negative\). \(y+negative\) is less then \(y\) and if \(x\) is less than \(y+negative\) then it must also be less than \(y\) itself: \(x<y\).

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 also 2 and 3: \(2+3=5=prime\). \(x\) can be 12 or 100. Not sufficient.

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

Answer: C.

hey..., can sm1 pls explain how primes of 100 can be 2 and 3?...(2nd last line),,.. thanx..

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 also 2 and 3: \(2+3=5=prime\). \(x\) can be 12 or 100. Not sufficient.

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

Answer: C.

hey..., can sm1 pls explain how primes of 100 can be 2 and 3?...(2nd last line),,.. thanx..

It should be: "... distinct primes of 100 are 2 and 5: \(2+5=7=prime\). \(x\) can be 12 or 100".
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Given: \(6a=3b=7c\) --> least common multiple of 6, 3, and 7 is 42 hence we ca write: \(6a=3b=7c=42x\), for some number \(x\) --> \(a=7x\), \(b=14x\) and \(c=6x\).

(1) ac=6b --> \(7x*6x=6*14x\) --> \(x^2=2x\) --> \(x=0\) or \(x=2\). Not sufficient.

(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.

Hi Bunuel, I considered \((x+y)^2\) \(>=\) 0 and arrived at \(xy >=\) \(\frac{-1}{2}\). And hence concluded that the statement is insufficient. Please correct me.

Yes, you need to consider \((x-y)^2\geq{0}\) to get sufficiency.
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(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.

This means that the condition doesn't hold good, and hence x < y doesn't hold good.

Could you please point out where exactly I am making a mistake in my process?

The statements in DS questions are true. So, if you pick numbers you should pick such that they satisfy the statement. x and y cannot be -2 and -1 respectively because they do not satisfy 2x - 3 < 3y - 4.
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11. If x and y are integers, is x a positive integer?

(1) x*|y| is a prime number --> since only positive numbers can be primes, then: x*|y|=positive --> x=positive. Sufficient

(2) x*|y| is non-negative integer. Notice that we are told that x*|y| is non-negative, not that it's positive, so x can be positive as well as zero. Not sufficient.

Answer: A.

I think the option should be E Option 1 -x * -y will result in positive, where x is negative.

Can you explain further..??

If x is negative, then x*|y| = negative*non-negative = non-positive, which cannot be prime, since only positive numbers are primes.
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1. B 2. A 3. B 4. D 5. D 6. E 7. D 8. C 9. B 10. C 11. D 12. E

10 correct answers out of 12.

khaadu wrote:

1-b 2-a 3-b 4-a 5-b 6-e 7-d 8-b 9-e 10-a 11-d 12-a

7 correct answers out of 12.

vinayaerostar wrote:

Ans: 1-B 2-A 3-A 4-E 5-D 6-E 7-C 8-A 9-C 10-A 11-D 12-D

4 correct answers out of 12.

Good job everyone! By the way it's better if you post the solutions along with the answers: others will benefit with your approaches and you'll get 1 Kudos point per correct solution.

Will post explanations in couple of days, so that to give some more time to those who want to participate.
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