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655-705 (Hard)|   Algebra|      
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TheSituation
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If x^2y^3 = 200, what is the value of xy ? 09 Apr 2010, 20:14
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60% (01:59) correct 40% (02:03) wrong based on 719 sessions

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If \(x^2 y^3 = 200\), what is the value of \(xy\)?

(1) y is an integer
(2) x/y = 2.5


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Bunuel
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If x^2y^3 = 200, what is the value of xy ? 14 Apr 2010, 02:53
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If \(x^2 y^3 = 200\), what is the value of \(xy\)?

(1) \(y\) is an integer.

\(y\) can be any positive integer, and for each value of \(y\), there will be two corresponding values of \(x\) to satisfy the equation \(x^2y^3=200\). For instance, if \(y=10\), then \(x^2y^3=x^2*1000=200\). This results in \(x=\frac{1}{\sqrt{5} }\) or \(x=-\frac{1}{\sqrt{5} }\). Consequently, there are infinite possible values for \(xy\). Not sufficient.

(2) \(\frac{x}{y}=2.5\).

Upon cross-multiplying, we get \(x=2.5y\). Plugging this into our original equation, we have \((2.5y)^2*y^3=6.25y^5=200\). Simplifying, we get \(y^5=32\), which implies \(y=2\). Then, \(x=2.5y=5\). Therefore, \(xy=10\). Sufficient.


Answer: B
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If x^2y^3 = 200, what is the value of xy ? 21 Aug 2011, 19:58
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x^2 * y^3 = 200

1. not sufficient

as different y will yield different x. hence different xy.

2. sufficient

x/y = 2.5

(2.5y)^2 * y ^3 = 200

25/4 y ^5 = 200 = >y =2

x =5

Answer is B.
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If x^2y^3 = 200, what is the value of xy ? 21 Aug 2011, 22:44
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heyholetsgo
If x^2y^3=200, what is xy?
1.) y is an integer
2.) x/y= 2.5
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why is it not e? Don't we get 2 solutions for x once we get the value for y?
Show more

Even though, the solution has already been provided above, I want to point out something. The silver lining of DS questions is that you don't need to solve. If you know you will get a unique value, you are done.

Question stem: If \(x^2y^3=200\), what is xy?

Statement 1: y is an integer but x needn't be.
If y = 1, \(x = +-\sqrt{200}\)
If y = 2, x = +-5 etc
We do not know what xy is.

Statement 2:\(\frac{x}{y} = \frac{5}{2}\)
This means \(x = (\frac{5}{2})y\)
This value of x you can put above to get a unique value of y. (Since you will get y^5 = ... An odd power will give you only the positive solution here. )
Once you have the value of y, you get a unique value for x and hence you get a value for xy.
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If x^2y^3 = 200, what is the value of xy ? 08 May 2012, 08:24
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I got this problem wrong. Bunuel, can you explain if I approached it the wrong way.

I factored out 200, which = \(2^3 * 5^2\).

After this its obvious \(xy = 10\)

What am i missing here?
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If x^2y^3 = 200, what is the value of xy ? 09 May 2012, 02:06
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ashish8
I got this problem wrong. Bunuel, can you explain if I approached it the wrong way.

I factored out 200, which = \(2^3 * 5^2\).

After this its obvious \(xy = 10\)

What am i missing here?
Show more

Check my post above: if-x-2y-3-200-what-is-xy-1-y-is-an-integer-2-x-y-2-5-b-92486.html#p713396

First of all \(x^2y^3=2^3*5^2\) has infinitely many solutions for \(x\) and \(y\). For ANY (nonzero) value of \(x\) there exist some \(y\) to satisfy \(x^2y^3=200\) and vise-versa. For example \(x=1\) and \(y=\sqrt[3]{200}\), or \(x=10\) and \(y=\sqrt[3]{2}\), ... As you can see it's not necessary for \(x\) and \(y\) to be integers (3 and 2) to satisfy the given equation.

As for (1): \(y\) is an integer. \(y\) can be any positive integer and there will be two values of \(x\) for each \(y\) to satisfy \(x^2y^3=200\). For example \(y=10\) --> \(x^2y^3=x^2*1000=200\) --> \(x=\frac{1}{\sqrt{5}}\) or \(x=-\frac{1}{\sqrt{5}}\). So there are infinite values of \(xy\). Not sufficient.

Hope it's clear.
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If x^2y^3 = 200, what is the value of xy ? 05 Nov 2024, 01:31
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