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GMATH Teacher P
Status: GMATH founder
Joined: 12 Oct 2010
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MathRevolution wrote:
[Math Revolution GMAT math practice question]

(number properties) If $$n$$ is a positive integer, is $$\sqrt{n+1}$$ an even integer?

1) $$n$$ is the product of $$2$$ consecutive odd numbers
2) $$n$$ is an odd number

Beautiful problem, Max. Congrats (and kudos)!

$$n \geqslant 1\,\,\,\operatorname{int}$$

$$\sqrt {n + 1} \,\,\,\mathop = \limits^? \,\,{\text{even}}\,\,\,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\,\,\,\boxed{\,\,n + 1\,\,\,\mathop = \limits^? \,\,\,{{\left( {{\text{even}}} \right)}^2}\,\,}$$

$$\left( 1 \right)\,\,\,n = \left( {2M - 1} \right)\left( {2M + 1} \right) = {\left( {2M} \right)^2} - {\left( 1 \right)^2}\,\,\,\,\,\left[ {M\,\,\operatorname{int} \,} \right]\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\,n + 1 = {\left( {2M} \right)^2}\,\,\,,\,\,\,\,M\,\,\operatorname{int} \,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \,$$

$$\left( 2 \right)\,\,\,\left\{ \matrix{ \,{\rm{Take}}\,\,n = 1\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr \,{\rm{Take}}\,\,n = 3\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr} \right.\,\,$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
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fskilnik wrote:
MathRevolution wrote:
[Math Revolution GMAT math practice question]

(number properties) If $$m$$ and $$n$$ are positive integers, is $$m + n$$ an odd number?

1) $$\frac{m}{n}$$ is an even number
2) $$m$$ or $$n$$ is an even number

$$m,n\,\,\, \geqslant 1\,\,\,{\text{ints}}\,\,\,\,\left( * \right)$$

$$m + n\,\,\,\,\mathop = \limits^? \,\,{\text{odd}}\,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\,\boxed{\,\,\,?\,\,\,:\,\,\,\left( {m\,\,{\text{odd}}\,,\,\,n\,\,{\text{even}}} \right)\,\,\,{\text{or}}\,\,\,{\text{vice - versa}\,\,}\,\,}$$

$$\left( 1 \right)\,\,\,\frac{m}{n} = {\text{even}}\,\,\,\,\left\{ \begin{gathered} \,{\text{Take}}\,\,\left( {m,n} \right) = \left( {2,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{YES}}} \right\rangle \,\, \hfill \\ \,{\text{Take}}\,\,\left( {m,n} \right) = \left( {4,2} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{NO}}} \right\rangle \,\, \hfill \\ \end{gathered} \right.$$

$$\left( 2 \right)\,\,\,m\,\,{\text{even}}\,\,\,{\text{or}}\,\,\,n\,\,{\text{even}}\,\,\,\,\left\{ \begin{gathered} \,\left( {\operatorname{Re} } \right){\text{Take}}\,\,\left( {m,n} \right) = \left( {2,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{YES}}} \right\rangle \,\, \hfill \\ \,\left( {\operatorname{Re} } \right){\text{Take}}\,\,\left( {m,n} \right) = \left( {4,2} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{NO}}} \right\rangle \,\, \hfill \\ \end{gathered} \right.\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\text{E}} \right)$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

P.S.: "A or B" means "only A", "only B" or BOTH.

fskilnik

#2 says m or n even ; why have you taken both as even while proving statement as insufficient?
GMATH Teacher P
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Joined: 12 Oct 2010
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Archit3110 wrote:
#2 says m or n even ; why have you taken both as even while proving statement as insufficient?

Hi Archit3110 ,

Thank you for your interest in my solution.

As I explained in my post scriptum (PS), the word "OR" has not the everyday common use of "exclusive or".

In other words, when it is given that "m is even or n is even", there are three possibilities available:

(i) m is even and n is not even
(ii) m is not even and n is even
(iii) m is even and n is also even

Regards and success in your studies,
Fabio.
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8583
GMAT 1: 760 Q51 V42
GPA: 3.82

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MathRevolution wrote:
[Math Revolution GMAT math practice question]

(number properties) If $$n$$ is a positive integer, is $$\sqrt{n+1}$$ an even integer?

1) $$n$$ is the product of $$2$$ consecutive odd numbers
2) $$n$$ is an odd number

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

The question is equivalent to asking if $$\sqrt{n+1} = 2k$$ for some positive integer $$k$$.
$$\sqrt{n+1} = 2k$$
$$=> n+1 = 4k^2$$
$$=> n = 4k^2-1$$
$$=> n = (2k-1)(2k+1)$$
$$n$$ is a product of two consecutive odd integers.
Thus, condition 1) is sufficient.

Condition 2)
If $$n = 3$$, then $$\sqrt{3+1} = \sqrt{4}=2$$ and the answer is ‘yes’.
If $$n = 1$$, then $$\sqrt{1+1} = 2$$ is not an integer and the answer is ‘no’.
Condition 2) is not sufficient.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8583
GMAT 1: 760 Q51 V42
GPA: 3.82

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[Math Revolution GMAT math practice question]

(number properties) If $$m$$ and $$n$$ are positive integers, is $$3^{4m+2}+n$$ divisible by $$5$$?

$$1) m=3$$
$$2) n=1$$
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8583
GMAT 1: 760 Q51 V42
GPA: 3.82

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MathRevolution wrote:
[Math Revolution GMAT math practice question]

(number properties) If $$m$$ and $$n$$ are positive integers, is $$m + n$$ an odd number?

1) $$\frac{m}{n}$$ is an even number
2) $$m$$ or $$n$$ is an even number

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have $$2$$ variables ($$m$$ and $$n$$) and $$0$$ equations, C is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
Since $$\frac{m}{n}$$ is even, $$\frac{m}{n} = 2k$$ and $$m = (2k)n$$ for some positive integer $$k$$.
If $$m = 2$$ and $$n = 1$$, then $$m + n = 3$$, is an odd integer and the answer is ‘yes’.
If $$m = 4$$ and $$n = 2$$, then $$m + n = 6$$ is an even integer and the answer is ‘no’.
Since we do not obtain a unique answer, conditions 1) & 2) are not sufficient when considered together.

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2), when taken together. Obviously, there may be occasions on which the answer is A, B, C or D.
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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
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GMAT 1: 760 Q51 V42
GPA: 3.82

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[Math Revolution GMAT math practice question]

(function) In the $$xy$$-coordinate plane, does $$y=a(x-h)^2+k$$ intersect the $$x$$-axis?

$$1) h=1$$
$$2) k=2$$
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
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GPA: 3.82

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MathRevolution wrote:
[Math Revolution GMAT math practice question]

(number properties) If $$m$$ and $$n$$ are positive integers, is $$3^{4m+2}+n$$ divisible by $$5$$?

$$1) m=3$$
$$2) n=1$$

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

The units digits of $$3^k$$ have period $$4$$ as they form the cycle $$3 -> 9 -> 7 -> 1.$$
$$3^{4m+2}$$ has $$9$$ as its units digit if $$3^{4m+2}$$ has units digit $$9$$, regardless of the value of $$m$$.
Thus, the divisibility of $$3^{4m+2}+n$$ by $$5$$ relies on the variable n only.

Therefore, the correct answer is B.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8583
GMAT 1: 760 Q51 V42
GPA: 3.82

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MathRevolution wrote:
[Math Revolution GMAT math practice question]

(function) In the $$xy$$-coordinate plane, does $$y=a(x-h)^2+k$$ intersect the $$x$$-axis?

$$1) h=1$$
$$2) k=2$$

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have $$3$$ variables ($$a, h$$ and $$k$$) and $$0$$ equations, E is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
If $$a = 1$$, then the graph doesn’t intersect the $$x$$-axis shown as below.

Attachment: 1231.png [ 5.04 KiB | Viewed 276 times ]

If $$a = -1$$, then the graph intersects the $$x$$-axis shown as below.

Attachment: 12311.png [ 5.23 KiB | Viewed 276 times ]

Since neither condition gives us information about the value of $$a$$, conditions 1) & 2) are not sufficient, when considered together.

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2), when taken together. Obviously, there may be occasions on which the answer is A, B, C or D.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8583
GMAT 1: 760 Q51 V42
GPA: 3.82

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[Math Revolution GMAT math practice question]

(absolute value) Is $$\sqrt{(x+1)^2}=x+1$$ ?

$$1) x(x-2) = 0$$
$$2) x(x+2) = 0$$
_________________
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Joined: 18 Aug 2017
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Location: India
Concentration: Sustainability, Marketing
GPA: 4
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MathRevolution wrote:
[Math Revolution GMAT math practice question]

(absolute value) Is $$\sqrt{(x+1)^2}=x+1$$ ?

$$1) x(x-2) = 0$$
$$2) x(x+2) = 0$$

Given
$$\sqrt{(x+1)^2}=x+1$$
or say
$$\sqrt{(x+1)^2} = lx+1l lx+1l= x+1 only for values when x=0 or +/-1 so #1: [m]1) x(x-2) = 0$$
x=0 & x=+2

in sufficient
#2
$$2) x(x+2) = 0$$

x=0 & x=-2

in sufficient

from 1 & 2
x=0 sufficient

IMO C
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8583
GMAT 1: 760 Q51 V42
GPA: 3.82

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[Math Revolution GMAT math practice question]

(number properties) What is the value of the integer $$n$$?

1) $$n$$ is a prime factor of $$21$$
2) $$n$$ is a factor of $$49$$
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8583
GMAT 1: 760 Q51 V42
GPA: 3.82

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MathRevolution wrote:
[Math Revolution GMAT math practice question]

(absolute value) Is $$\sqrt{(x+1)^2}=x+1$$ ?

$$1) x(x-2) = 0$$
$$2) x(x+2) = 0$$

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

The question is equivalent to asking if $$x ≥ -1$$ as shown below:
$$\sqrt{(x+1)^2}=x+1$$
$$=> |x+1| = x+1$$
$$=> x ≥ -1$$

Since we have $$1$$ variable ($$x$$) and $$0$$ equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
$$x(x-2) = 0$$
$$=> x = 0$$ or $$x = 2$$
If $$x = 0$$, then $$x ≥ -1$$ and the answer is ‘yes’.
If $$x = 2$$, then $$x ≥ -1$$ and the answer is ‘yes’.
Since it gives a unique answer, condition 1) is sufficient.

Condition 2)
$$x(x+2) = 0$$
$$=> x = 0$$ or $$x = -2$$
If $$x = 0$$, then $$x ≥ -1$$ and the answer is ‘yes’.
If $$x = -2$$, then $$x < -1$$ and the answer is ‘no’.
Since it does not give a unique answer, condition 2) is not sufficient.

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.
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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
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GPA: 3.82

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[Math Revolution GMAT math practice question]

(number properties) Can $$n$$ be expressed as the difference of $$2$$ prime numbers?

$$1) (n-17)(n-21) = 0$$
$$2) (n-15)(n-17)=0$$
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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8583
GMAT 1: 760 Q51 V42
GPA: 3.82

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[Math Revolution GMAT math practice question]

(absolute value) Is $$\frac{x}{y}<0$$?

$$1) |x+y|<|x|+|y|$$
$$2) x+y<0$$
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8583
GMAT 1: 760 Q51 V42
GPA: 3.82

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MathRevolution wrote:
[Math Revolution GMAT math practice question]

(number properties) Can $$n$$ be expressed as the difference of $$2$$ prime numbers?

$$1) (n-17)(n-21) = 0$$
$$2) (n-15)(n-17)=0$$

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have $$1$$ variable ($$n$$) and $$0$$ equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
$$(n-17)(n-21) = 0$$ is equivalent to the statement $$n = 17$$ or $$n =21$$
If $$n = 17$$, then $$17 = 19 – 2$$ is a difference of two prime numbers and the answer is ‘yes’.
If $$n = 21$$, then $$21 = 23 – 2$$ is a difference of two prime numbers and the answer is ‘yes’.
Since it gives a unique answer, condition 1) is sufficient.

Condition 2)
$$(n-15)(n-17) = 0$$ is equivalent to the statement $$n = 15$$ or $$n = 17$$
If $$n = 15$$, then $$15 = 17 – 2$$ is a difference of two prime numbers and the answer is ‘yes’.
If $$n = 17$$, then $$17 = 19 – 2$$ is a difference of two prime numbers and the answer is ‘yes’.
Since it gives a unique answer, condition 2) is sufficient.

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2), when taken together. Obviously, there may be occasions on which the answer is A, B, C or D.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8583
GMAT 1: 760 Q51 V42
GPA: 3.82

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[Math Revolution GMAT math practice question]

Is $$\frac{x}{y}<0$$?

$$1) x^4y^5<0$$
$$2) x^5y^3<0$$
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8583
GMAT 1: 760 Q51 V42
GPA: 3.82

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MathRevolution wrote:
[Math Revolution GMAT math practice question]

(absolute value) Is $$\frac{x}{y}<0$$?

$$1) |x+y|<|x|+|y|$$
$$2) x+y<0$$

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

The question is equivalent to asking if $$xy < 0$$. This can be seen by multiplying both sides of the inequality by $$y^2.$$

Condition 1) is equivalent to $$xy < 0$$ as shown below:
$$|x+y|<|x|+|y|$$
$$=> |x+y|^2<(|x|+|y|)^2$$
$$=> (x+y)^2<|x|^2+2|x||y|+|y|^2$$
$$=> x^2+2xy+y^2<x^2+2|xy|+y^2$$
$$=> 2xy<2|xy|$$
$$=> xy<|xy|$$
$$=> xy<0$$
Thus, condition 1) is sufficient.

Condition 2)
If $$x = -2$$ and $$y = 1$$, then the answer is ‘yes’.
If $$x = -1$$ and $$y = -1$$, then the answer is ‘no’.
Since it does not give a unique answer, condition 2) is not sufficient.

Therefore, the correct answer is A.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8583
GMAT 1: 760 Q51 V42
GPA: 3.82

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MathRevolution wrote:
[Math Revolution GMAT math practice question]

Is $$\frac{x}{y}<0$$?

$$1) x^4y^5<0$$
$$2) x^5y^3<0$$

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The question is equivalent to asking if $$xy < 0$$. This can be seen by multiplying both sides of the inequality by $$y^2$$.

Since we can ignore even exponents in inequalities like $$x^4y^5<0$$, condition 1) is equivalent to the statement $$y < 0$$ and condition 2) is equivalent to the statement $$xy < 0$$.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8583
GMAT 1: 760 Q51 V42
GPA: 3.82

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[Math Revolution GMAT math practice question]

(algebra) If $$x≠y$$, what is the value of $$\frac{( x^2y – xy^2 )}{( x^3 – y^3 )}$$?

$$1) \frac{xy}{( x^2 + xy + y^2)} = \frac{1}{3}$$
$$2) x^2y^2=9$$
_________________ Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS   [#permalink] 06 Jan 2019, 23:54

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# Math Revolution DS Expert - Ask Me Anything about GMAT DS   