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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
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[GMAT math practice question]

(ratio) There is $$a%$$ of saline solution of $$500g$$. Alice wants to have $$b%$$ of saline solution by boiling saline solution where $$b > a$$. What is the lost weight of the saline solution by boiling?

$$1) a = 20$$

$$2) \frac{a}{b} = \frac{2}{5}$$
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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8261
GMAT 1: 760 Q51 V42 GPA: 3.82

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MathRevolution wrote:
[GMAT math practice question]

(algebra) What is the value of $$(x+y)(2x-y)-(x-y)(2x+y)$$?

$$1) x=\frac{3}{2}$$

$$2) xy=2$$

=>

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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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

The expression of the question $$(x+y)(2x-y)-(x-y)(2x+y)$$ is equivalent to $$2xy$$ as below.
$$(x+y)(2x-y)-(x-y)(2x+y)$$
$$= 2x^2 + xy – y^2 – (2x^2 –xy –y^2)$$
$$= 2xy$$
Thus, condition 2) is sufficient since $$2xy = 2*2 = 4.$$

Condition 1)
Since we don’t have any information about the variable y, condition 2) is not sufficient, obviously.

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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
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[GMAT math practice question]

(function) A function $$f(x)$$ satisfies $$f(x)f(y)=f(x+y)+f(x-y)$$ for every real numbers $$x, y$$. What is the value of $$f(0)f(1)f(2)f(3)$$?

$$1) f(0)=2$$

$$2) f(1)=1$$
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8261
GMAT 1: 760 Q51 V42 GPA: 3.82

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MathRevolution wrote:
[GMAT math practice question]

(ratio) There is $$a%$$ of saline solution of $$500g$$. Alice wants to have $$b%$$ of saline solution by boiling saline solution where $$b > a$$. What is the lost weight of the saline solution by boiling?

$$1) a = 20$$

$$2) \frac{a}{b} = \frac{2}{5}$$

=>

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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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

Let $$x$$ be the lost weight of $$b%$$ saline solution by boiling.

The question asks the value of $$x$$ such that $$(\frac{a}{100})500 = (\frac{b}{100})(500-x).$$

$$(\frac{a}{100})500 = (\frac{b}{100})(500-x)$$

$$=> 500a = b(500-x)$$

$$=> 500a = 500b – bx$$

$$=> bx = 500(b-a)$$

$$=> x = \frac{500(b-a)}{b}$$

$$=> x = (500)(1-\frac{a}{b})$$

Thus, condition 2) is sufficient.

When a question asks for a ratio, if one condition includes a ratio and the other condition just gives a number, the condition including the ratio is most likely to be sufficient. This tells us that B is most likely to be the answer to this question.

Condition 1) is not sufficient obviously, since we don’t have any information about the variable a.

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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8261
GMAT 1: 760 Q51 V42 GPA: 3.82

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MathRevolution wrote:
[GMAT math practice question]

(function) A function $$f(x)$$ satisfies $$f(x)f(y)=f(x+y)+f(x-y)$$ for every real numbers $$x, y$$. What is the value of $$f(0)f(1)f(2)f(3)$$?

$$1) f(0)=2$$

$$2) f(1)=1$$

=>

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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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

If $$x = 1, y = 0$$, then we have $$f(1)f(0) = f(1) + f(1) = 2f(1).$$

If $$x = 1, y = 1,$$ then we have $$f(1)f(1) = f(2) + f(0)$$

If $$x = 2, y = 1$$, then we have $$f(2)f(1) = f(3) + f(1)$$

Since we have $$4$$ variables $$(f(0), f(1), f(2)$$ and $$f(3)$$) and $$3$$ equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
If $$f(1) = 1,$$ then we have$$f(2) = f(1)f(1) – f(0) = 1 – 2 = -1, f(3) = f(2)f(1) – f(1) = (-1)*1 – 1 = -2$$ and $$f(0)f(1)f(2)f(3) = 2*1*(-1)(-2) = 4.$$

If $$f(1) = 2$$, then we have $$f(2) = f(1)f(1) – f(0) = 4 – 2 = 2, f(3) = f(2)f(1) – f(1) = 2*1 – 2 = 0$$ and $$f(0)f(1)f(2)f(3) = 2*2*(2)*0 = 0.$$

Since condition 1) does not yield a unique solution, it is not sufficient.

Condition 2)
Since $$f(1) = 1,$$ we have $$f(0) = f(1)f(0) = f(1) + f(1) = 2, f(2) = f(1)f(1) – f(0) = 1 – 2 = -1, f(3) = f(2)f(1) – f(1) = (-1)*1 – 1 = -2$$ and $$f(0)f(1)f(2)f(3) = 2*1*(-1)*.(-2) = 4.$$

Condition 2) is sufficient since it yields a unique solution.

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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GMAT 1: 760 Q51 V42 GPA: 3.82

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[GMAT math practice question]

(exponent) What is the value of $$a$$?

$$1) a= 2^x = 3^y$$

$$2) \frac{1}{x} + \frac{1}{y} = \frac{1}{2}$$
_________________

Originally posted by MathRevolution on 23 Sep 2019, 00:58.
Last edited by MathRevolution on 25 Sep 2019, 01:12, edited 1 time in total.
Manager  B
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Hello Max,

Could you help me with the below DS question?? (Specifically, is (2) sufficient?)

If y and x are positive integers, is y divisible by X?

(1) y = x^2 + x
(2) x has the same prime factors as y.

The official answer from the source says (2) is INSUFFICIENT because they can have the same prime factors, but have different numbers of those same prime factors.

But I thought since the question does not say UNIQUE or DISTINCT... That means X and y are the same exact number... So (2) should be sufficient.

Posted from my mobile device
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8261
GMAT 1: 760 Q51 V42 GPA: 3.82

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[GMAT math practice question]

(number properties) If $$a, b$$, and $$c$$ are positive integers, what is the value of $$a+b+c$$?

1) $$a, b$$ and $$c$$ are three consecutive odd numbers in that order

2) $$20≤bc-ab≤24$$
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8261
GMAT 1: 760 Q51 V42 GPA: 3.82

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MathRevolution wrote:
[GMAT math practice question]

(exponent) What is the value of $$a$$?

$$1) a= 2^x = 3^y$$

$$2) \frac{1}{x} + \frac{1}{y} = \frac{1}{2}$$

=>

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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have $$3$$ variables ($$x, y$$, and $$a$$) 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)
Since we have $$2^x = a$$ and $$3^y=a$$, we have $$2=a^{\frac{1}{x}}$$ and $$3=a^{1/y}.$$

Then $$2*3 = a^{\frac{1}{x}}a^{\frac{1}{y}}=a^{\frac{1}{x}+\frac{1}{y}}=a^{\frac{1}{2}}$$ or $$a^{\frac{1}{2}}=6.$$
So we have $$a = 6^2 = 36.$$

Since we have 2 equations from condition 1) and 1 equation from condition 2), even though we have three variables, both conditions together are 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 when the answer is A, B, C, or D.
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Joined: 16 Aug 2015
Posts: 8261
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[GMAT math practice question]

(number properties) What is the value of $$p*q$$?

1) $$p$$ and $$q$$ are prime numbers

2) $$q-p=3$$
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Joined: 16 Aug 2015
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Vinit1 wrote:
Hello Max,

Could you help me with the below DS question?? (Specifically, is (2) sufficient?)

If y and x are positive integers, is y divisible by X?

(1) y = x^2 + x
(2) x has the same prime factors as y.

The official answer from the source says (2) is INSUFFICIENT because they can have the same prime factors, but have different numbers of those same prime factors.

But I thought since the question does not say UNIQUE or DISTINCT... That means X and y are the same exact number... So (2) should be sufficient.

Posted from my mobile device

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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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

Condition 1)
$$y = x^2 + x$$
⇔ $$y = x(x+1)$$
Then, $$y$$ is divisible by $$x$$.
Condition 1) is sufficient.

Condition 2)
If $$x = 2$$ and $$y = 2$$, then $$y$$ is divisible by $$x$$ and the answer is 'yes'.
If $$x = 4$$ and $$y = 2$$, then $$y$$ is not divisible by $$x$$ and the answer is 'no'.
Since condition 2) does not yield a unique answer, it is not sufficient.

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MathRevolution wrote:

Condition 2)
If $$x = 2$$ and $$y = 2$$, then $$y$$ is divisible by $$x$$ and the answer is 'yes'.
If $$x = 4$$ and $$y = 2$$, then $$y$$ is not divisible by $$x$$ and the answer is 'no'.
Since condition 2) does not yield a unique answer, it is not sufficient.

But, 2 and 4 do not have the same prime factors. 2 has only 1 2, and 4 has 2 2s. 2^2 cannot be same as 2 right?

If the question said same UNIQUE prime factors, then this answer would be correct.

If what you say is correct, then can I assume that whenever a question says PRIME FACTORS, it means UNIQUE PRIME FACTORS?

Posted from my mobile device

PS: I also checked with the original source, and they agreed that it was a mistake in their problem. The problem should have specified unique prime factors.
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8261
GMAT 1: 760 Q51 V42 GPA: 3.82

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MathRevolution wrote:
[GMAT math practice question]

(number properties) If $$a, b$$, and $$c$$ are positive integers, what is the value of $$a+b+c$$?

1) $$a, b$$ and $$c$$ are three consecutive odd numbers in that order

2) $$20≤bc-ab≤24$$

=>

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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have $$3$$ variables ($$a, b$$, and $$c$$) and $$0$$ equations, E is most likely 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 $$a, b$$, and $$c$$ are three consecutive odd numbers in that order, we can put $$a = b – 2$$ and $$c = b + 2.$$

$$bc – ab = b(b+2) – (b-2)b = b^2 + 2b – b^2 +2b = 4b$$ and we have $$20≤4b≤24$$ and dividing everything by $$4$$ we get $$5≤b≤6.$$

Since $$b$$ is an odd integer, we have $$b=5.$$

Then we have $$a = 3, b = 5, c = 7$$ and $$a + b + c = 15.$$

Since both conditions together yield a unique solution, they are sufficient.

Since this question is a statistics question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.

Condition 1)
Since there are many possibilities for (a,b,c), the condition is obviously not sufficient.

Condition 2)
If $$a = 3, b = 5$$ and $$c = 7$$, then we have $$a + b + c = 15.$$

If $$a = 1, b = 5$$ and $$c = 5$$, then we have $$a + b + c = 11.$$

Since condition 2) does not yield a unique solution, it is not 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 when the answer is A, B, C or D.
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Joined: 16 Aug 2015
Posts: 8261
GMAT 1: 760 Q51 V42 GPA: 3.82

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[GMAT math practice question]

(algebra) What is the value of $$xyz$$?

$$1) x+ \frac{1}{y} = 2$$

$$2) y – \frac{1}{z} = \frac{1}{2}$$
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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8261
GMAT 1: 760 Q51 V42 GPA: 3.82

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MathRevolution wrote:
[GMAT math practice question]

(number properties) What is the value of $$p*q$$?

1) $$p$$ and $$q$$ are prime numbers

2) $$q-p=3$$

=>

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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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

Since we have $$2$$ variables ($$p$$ and $$q$$) and $$0$$ equations, C is most likely 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 $$p$$ and $$q$$ are prime numbers and $$q – p = 3$$ which is an odd number, $$p$$ and $$q$$ have different parities.
It means that either $$p$$ or $$q$$ is an even prime number. Then we have $$p = 2$$, since $$2$$ is the smallest and unique even prime number.
So $$q = 5.$$

Since this question is a statistics question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.
Condition 1)
Since there many possibilities for $$(p, q),$$ the condition is obviously not sufficient.

Condition 2)
If $$p = 2$$ and $$q = 5$$, then we have $$p*q = 10.$$
If $$p = 1$$ and $$q = 4$$, then we have $$p*q = 4$$.
Since condition 2) does not yield a unique solution, it is not sufficient.

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.
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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8261
GMAT 1: 760 Q51 V42 GPA: 3.82

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[GMAT math practice question]

There are three semi-circles with the same center $$O$$ in the following figure. What is the area of the shaded region?

$$1) BB’ = \frac{AA’}{2}$$

$$2) CC’ = \frac{BB’}{2}$$

Attachment: 9.27.png [ 26.2 KiB | Viewed 138 times ]

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8261
GMAT 1: 760 Q51 V42 GPA: 3.82

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MathRevolution wrote:
[GMAT math practice question]

(algebra) What is the value of $$xyz$$?

$$1) x+ \frac{1}{y} = 2$$

$$2) y – \frac{1}{z} = \frac{1}{2}$$

=>

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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have $$3$$ variables ($$x, y$$ and $$z$$) 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)

Since $$x + \frac{1}{y} = 2$$, we have $$x = 2 – \frac{1}{y} = \frac{(2y-1)}{y}.$$

Since $$y – \frac{1}{z} = \frac{1}{2}$$, we have $$\frac{1}{z} = y – \frac{1}{2} = \frac{(2y-1)}{2}$$ or $$z = \frac{2}{(2y-1).}$$

Then we have $$xyz = [\frac{(2y-1)}{y}]*y*[\frac{2}{(2y-1)}] = 2.$$

Since both conditions 1) & 2) together yield a unique solution, they are 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: 8261
GMAT 1: 760 Q51 V42 GPA: 3.82

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MathRevolution wrote:
[GMAT math practice question]

There are three semi-circles with the same center $$O$$ in the following figure. What is the area of the shaded region?

$$1) BB’ = \frac{AA’}{2}$$

$$2) CC’ = \frac{BB’}{2}$$

Attachment:
9.27.png

=>

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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have three semi-circles, we have 3 variables 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)
Assume r1, r2 and r3 are radii of the circles with diameters $$CC’, BB’$$ and $$AA’,$$ respectively.
We have r2 = 2r1 and r3 = 2r2 = 4r1 from conditions 1) & 2).
Then the shared area is πr2^2/2 - πr1^2/2 = 4πr1^2/2 – πr1^2/2 = (3/2)πr1^2.
However, since we don’t know the value of r1, we can’t determine r1.
So, both conditions together do not yield a unique solution and they are not 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 when the answer is A, B, C or D.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8261
GMAT 1: 760 Q51 V42 GPA: 3.82

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[GMAT math practice question]

(algebra) What is the value of $$x$$?

$$1) [x, y, z]=xy+yz+zx$$

$$2) [1, x, 2]=[4, -3, 2].$$
_________________
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[GMAT math practice question]

(algebra) $$abc ≠ 0$$. What is the value of $$a(\frac{1}{b} + \frac{1}{c}) +b(\frac{1}{c} + \frac{1}{a}) + c(\frac{1}{a} + \frac{1}{b})$$?

$$1) |a+b+c| ≤ 0$$

$$2) a+b+c = 0$$
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# Math Revolution DS Expert - Ask Me Anything about GMAT DS  