The Discreet Charm of the DS

Hi Bunuel,

Can we solve this question using graphical approach.

Since (x-a)^2+(y-b)^2 = r^2 ----(A)

Equation 1 represents equation of circle with radius 1.

Now, if we can maximize the value of both x and y, we can get the maximum value of xy and that could let us know whether statement (1) is sufficient or not.

Here, I require your help. Intuitively, I find that x=y will maximize the equation x^2+y^2=1. But I am not sure about my reasoning as well as the Mathematics behind. Could you please look into this.

Thanks

Useful property: For given sum of two numbers, their product is maximized when they are equal.

(1) says that \(x^2+y^2=1\). So, \(x^2y^2\) will be maximized when \(x^2=y^2\): \(x^2+x^2=1\) --> \(x^2=\frac{1}{2}\) --> the maximum value of \(x^2y^2\) thus is \(\frac{1}{4}\) and the maixmum value of \(xy\) is \(\frac{1}{2}\).

Hope it's clear.

Responding to a pm:

Time taken by Bonnie to complete one work = x hrs
Time taken by Clyde to complete one work = y hrs
x and y are odd integers i.e. they could take values such as 1/3/5/7/9/11...

Question: Is x = y? i.e. is the time taken by Bonnie equal to time taken by Clyde? i.e. is the speed of Bonnie equal to the speed of Clyde?

(1) x^2+y^2<12
This info is not related to work concepts. It's just number properties. x and y are odd integers.
If x = y = 1, this inequality is satisfied.
If x = 1 and y = 3, this inequality is satisfied.

This means x may or may not be equal to y. Not sufficient.

(2) Bonnie and Clyde complete the painting of the car at 10:30 am.r
Together, they take 45 mins to complete the painting of the car. This means, if their rate of work were the same, each one of them would have taken 1.5 hrs working alone. But their time taken is an integer value. We can say that they do not take the same time i.e. x is not equal to y. Hence this statement is sufficient to say \(x \neq y\)

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.

Hope it's clear.

Hi Bunuel,

In the 1st condition if we take ac = 6b and hence multiply the the given expression 6a=3b=7c by 2 we get 12a=6b=14c.

Now substituting 12a=ac=14c

I get 12a=ac --> c = 12 and similarly a = 14 and hence b= 28.

Can you please point where am I going wrong?

Regards

ac=12a (here you cannot reduce by a and write c=12 as you exclude possibility of a=0) --> a(c-12)=0 --> either a=0 OR c=12. So, we get either a=b=c=0 or a=14, b=28 and c=12.

Does this make sense?

Hi Bunnel,

One question here.

in st2 i can consider 8-0 also then why we are considering only least value. in st1 we are considering least value as well as other value. why this is different in st2?

Thanks

Because considering the least value of a (7), while knowing nothing about b is enough to say that the second statement is not sufficient. If b=8, then the answer is yes but if b is 0, then the answer is no.

Hi Bunuel,
Referring to below:
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

The average (arithmetic mean) of these numbers is a prime number --> in any evenly spaced set the arithmetic mean (average) is equal to the median --> mean=median=prime. Since it's not possible that median=2=even, (in this case not all 5 numbers will be positive), then median=odd prime, and we have the same case as above. Sufficient.

I didn't quite get the explanation on the second statement. Do you mind explaining with some numbers?

Thanks,
Mitesh

If mean=median=prime=2, then the 5 consecutive integers would be {0, 1, 2, 3, 4}. But this set is not possible since we are told that the set consists of 5 positive integers and 0 is neither positive nor negative integer.

Hence, mean=median=odd prime --> {Odd, Even, Odd prime, Even, Odd}. So, the set includes 2 consecutive even numbers. Out of 2 consecutive even integers only one is a multiple of 4.

Hope it's clear.

Hi Bunuel, a small doubt instead of \(2x+8\geq{0}\) --> \(x\geq{-4}\) this logic if we take |x+10| to be both positive and negative we get 2 vale by solving equation x=2 and x=-6. however by putting values back in eqtn we can see that only for x=2 equation is satisfying. Is this correct approach?

Yes, that's also a correct way of solving. Good thing you did is that you did not forget to test both values after you got them.

Bunuel
In
2. Is xy<=1/2?
(1) x^2+y^2=1
(2) x^2-y^2=0

If we take (X+Y)^2 instead of (X-Y)^2 for analyzing statement 1, why we get two different answers.
Also, the detailed explanation link is not working perhaps.

The link to the solution is here: https://gmatclub.com/forum/the-discreet ... l#p1039634


The point is that if you consider \((x+y)^2 \ge 0\), you'll get \(xy\geq{-\frac{1}{2}}\), which is not helpful at all. Actually since both \(xy\geq{-\frac{1}{2}}\) (from \((x+y)^2 \ge 0\)) and \(xy \le \frac{1}{2}\) (from \((x-y)^2 \ge 0\)) are true, then we get that \(-\frac{1}{2} \leq{xy} \leq{\frac{1}{2}}\). But only the approach given in the solution above gives you the answer we are looking for, while another one gives you an inequality which is not helpful.

P.S. You posted a question IN the thread, why use report along with it? Pleas do NOT use report button that way. It's for reporting problems, not for questions.,

Bunuel in Statement 2, does this mean that because t = 3/4, which is a fraction, it is not odd and hence 3/4 =x and 3/4= y?

Separate question: I was confused by the wording of St2. For a second it made it sound like Bonnie and Clyde were working together on painting, but question says independent. Which statement to trust more here?

1. From the stem we concluded that IF x = y, then the total time would be (x hours)/2 = (odd hours)/2 (since given that x is odd). (odd hours)/2 means the total time would be 0.5 hours, 1.5 hours, 2.5 hours, ... (2) says that the total time is 3/4 hours. 3/4 hours is NOT in the form of (odd hours)/2, so x does not equal to y.

2. Working simultaneously and independently, means that they work on different parts of the job. For example, one is painting the front and another the back of the car. This ensure that their works does not overlap and they do the job in fastest way possible.