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Is xy an integer?

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Is xy an integer?  [#permalink]

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New post 30 Apr 2015, 03:06
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A
B
C
D
E

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Is xy an integer?

(1) x is the ratio of the area of a square to the area of the largest possible circle inscribed within that square.

(2) y is the ratio of the area of a circle to the area of the largest possible square inscribed within that circle.


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Re: Is xy an integer?  [#permalink]

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New post 30 Apr 2015, 12:15
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Bunuel wrote:
Is xy an integer?

(1) x is the ratio of the area of a square to the area of the largest possible circle inscribed within that square.

(2) y is the ratio of the area of a circle to the area of the largest possible square inscribed within that circle.


Kudos for a correct solution.


The stem is a yes or no question so as long as we can give a single definite solution to the question we have sufficiency.

1) we have no information on y so not sufficient. Eliminate A and D.
2) we have no information on x so not sufficient. Eliminate B.

Together we know that each x and y are going to be fixed values regardless of the actual size of the shapes being referenced that will be multiplied together which in turn will give us a single definite answer to the question of if it is an integer or not. This provides is enough information to provide sufficiency, select answer C.
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Re: Is xy an integer?  [#permalink]

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New post 30 Apr 2015, 18:14
1
Algebra way :
The answer choice is between C or E

A. x is the ratio of the area of a square to the area of the largest possible circle inscribed within that square
It implies radius of circle =squareside/2 i.e. r1=s1/2

B. y is the ratio of the area of a circle to the area of the largest possible square inscribed within that circle.
It implies diagonal of the square=diameter of the circle i.e. s2=sqrt(2)*r2

Solving for the ratios ,x=(s1/2)^2/(3.14*(s1/2)^2) which results in 4/3.14
similarly, y =(3.14*(r2^2))/((r2^2)*2) which results in 3.14/2
Multiplying together it leads to 2 an integer
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Re: Is xy an integer?  [#permalink]

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New post 01 May 2015, 12:58
Answer C
Only combining both we'll get a integer value of XY
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Re: Is xy an integer?  [#permalink]

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New post 01 May 2015, 13:54
1
Statement 1:
Nothing about y

Statement 2:
Nothing about x

Combined, when the ratios are multiplied together, the pi from both statements cancel and we are left with an integer.
sufficient

Answer: C
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Re: Is xy an integer?  [#permalink]

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New post 04 May 2015, 03:00
Bunuel wrote:
Is xy an integer?

(1) x is the ratio of the area of a square to the area of the largest possible circle inscribed within that square.

(2) y is the ratio of the area of a circle to the area of the largest possible square inscribed within that circle.


Kudos for a correct solution.


MANHATTAN GMAT OFFICIAL SOLUTION:

The question asks whether a particular product (xy) is an integer. Note that this is a Yes-No question.

Statement 1: NOT SUFFICIENT. This statement only refers to x. Without knowing anything about y, you cannot know whether xy is an integer.

Statement 2: NOT SUFFICIENT. Likewise, this statement only refers to y, so it cannot be sufficient.

Statements 1 and 2 TOGETHER: SUFFICIENT. The super-fancy way to get the answer is to realize that x is a fixed number, completely determined by its definition in the statements. The same is true of y. Why is this the case? All circles are the same shape, so they are all similar to each other. Likewise, all squares are the same shape and are similar to each other. So when you inscribe a circle inside a square (to touch all four sides of the square), there

There should be only one “shape” to the picture in your mind of a square with a circle inscribed inside it, touching all four walls. All that’s different is how large or small that picture is, so the ratio of the square’s area to the circle’s area is fixed:
Image
The same is true for y, the ratio of the circle’s area to the area of an inscribed square:
Image
So x and y are fixed. You don’t know what their values are, but you don’t care: in theory, you could calculate those values. And then you could determine whether the product is an integer or not.

The longer way to get the answer is to actually figure out these ratios.

Take x first. Call the side of the square 1. Then the radius of the inscribed circle is 1/2, and the area of the circle is \(\pi*r^2= \frac{\pi}{4}\). The area of the square is 1^2 = 1, so the ratio of the square’s area to the circle’s area is \(1:\frac{\pi}{4}\), or \(\frac{4}{\pi}\). That’s the value of x.

Now take y. Call the side of the square 1 again. Then the diameter of the circle is the diagonal of the square, which is \(\sqrt{2}\). The radius of the circle is \(\frac{\sqrt{2}}{2}\), and the area of the circle is \(\pi*r^2 = \pi(\frac{\sqrt{2}}{2})^2 = \frac{\pi}{2}\). That’s the value of y, since the area of the square is just 1, and you want the ratio of the circle’s area to the square’s area.

Finally, the product of x and y is \((\frac{4}{\pi})(\frac{\pi}{2}) = 2\), which is indeed an integer.

The correct answer is C.

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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: Is xy an integer?  [#permalink]

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