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The Ultimate Q51 Guide [Expert Level]

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Math Revolution GMAT Instructor
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Joined: 16 Aug 2015
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 28 Jan 2018, 17:48
[GMAT math practice question]

When a≠0, how many solutions does the equation a(x+b)^2+c=0 have?

1) bc=0
2) |b|+|c|=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, and then recheck the question.

There are three cases to consider:
Case 1: a > 0, c > 0 or a < 0, c < 0
The equation has no roots.

Case 2: c = 0.
The equation has only one root.

Case 3: a > 0, c < 0 or a < 0, c > 0
The equation has two roots.

Condition 1):
If bc = 0, then when
a = 1, b = 0, c = -1, the equation has two roots, and when
a = 1, b = 0, c = 0, the equation has one root.
As the question does not have a unique answer, condition 1) is not sufficient.

Condition 2)
|b| + |c| = 0 ⇔ b = c = 0.
Since c = 0, the equation has only one root.
Condition 2) is sufficient.

Therefore, the answer is B.

Answer: B

_________________

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Joined: 16 Aug 2015
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 29 Jan 2018, 17:32
[GMAT math practice question]

Is x|y|=xz?

1) x, y, and z are positive
2) y^2=z^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.

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

Modifying the question:
x|y| = xz
⇔ x(|y|-z) = 0
⇔ x = 0 or |y| = z
⇔ x = 0 or y = z or y = -z

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.

Conditions 1) & 2):

Condition 2) tells us that y = z or y = -z.
Since condition 1) states that x, y, z > 0, we can only have y = z.
Thus, both conditions are sufficient, when taken together.

Therefore, the answer is C.

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.
Answer: C
_________________

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Math Revolution GMAT Instructor
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 30 Jan 2018, 17:45
[GMAT math practice question]

If x, y, and z are positive integers, is x+y divisible by 2?

1) x+z is divisible by 2
2) y+z is divisible by 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 (x, y and z) and 0 equations, E is most likely to be the answer. So, we should consider conditions 1) and 2) together first.

Conditions 1) and 2):
There are two cases to consider.
Case 1: If x, y, and z are all even, then the answer is ‘yes’
Case 2: If x, y, and z are all odd, then the answer is also ‘yes’
Since no other cases are possible, both conditions are sufficient, when taken together.

Since this is an integer 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 it tells us nothing about y, condition 1) is not sufficient.

Condition 2)
Since it tells us nothing about y, condition 2) is not sufficient.


Therefore, C is the answer.

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.

Answer: C
_________________

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Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 31 Jan 2018, 17:12
[GMAT math practice question]

How many solutions has the equation ||x-3|-2|=1?

A. 0
B. 1
C. 2
D. 3
E. 4

=>

||x-3|-2|=1
⇔|x-3|-2=±1
⇔|x-3|=2±1
⇔|x-3|=1 or |x-3|=3
⇔x-3=±1 or x-3=±3
⇔x=3±1 or x=3±3
⇔x=2, x = 4, x = 0 or x = 6

Thus, the equation has four solutions.

Therefore, the answer is E.

Answer: E
_________________

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Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 31 Jan 2018, 17:53
MathRevolution wrote:
Is p/m>0?

1) p>m
2) pm>0.

==> If you modify the original condition and the question, from is p/m>0?, you multiply m^2 on both sides, you get (Squared number is always positive, so even if you multiply, the inequality sign doesn't change) p/m(m^2)>0(m^2)?, which becomes pm>0?.

The answer is B.
Answer: B

Why is the correct answer not D for this question?

Sent from my ONEPLUS A5000 using GMAT Club Forum mobile app
Expert Post
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Joined: 16 Aug 2015
Posts: 4908
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 01 Feb 2018, 17:27
[GMAT math practice question]

Is n<0?

1) n-1<0
2) |3-n| > |n+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.

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

Condition 1) : n - 1 < 0 ⇔ n < 1
Since the range of the question, n < 0 does not include that of the condition 1), n < 1, the condition 1) is not sufficient.

Condition 2) :
|3-n| > |n+5|
⇔ |3-n|^2 > |n+5|^2
⇔ (3-n)^2 > (n+5)^2
⇔ n^2 -6n + 9 > n^2 +10n + 25
⇔ -16 > 16n
⇔ n < -1
Since the range of the question includes that of the condition 2), the condition 2) is sufficient.

Therefore, B is the answer.

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.

Answer: B
_________________

MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
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Expert Post
Math Revolution GMAT Instructor
User avatar
D
Joined: 16 Aug 2015
Posts: 4908
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 04 Feb 2018, 17:20
[GMAT math practice question]

Is x^2-x>0?

1) x>0
2) x^3+x>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, and then recheck the question.

In inequality questions, the law “Question is King” tells us that if the solution set of the question includes the solution set of the condition, then the condition is sufficient

Modifying the question:
x^2-x>0
⇔ x(x-1) > 0
⇔ x < 0 or x > 1 by the “LLGG” rule.

Condition 1): x > 0
Since the solution set of the question includes the solution set of condition 1), condition 1) is sufficient.

Condition 2):
x^3+x>0
⇔ x(x^2+1)>0
⇔ x>0, since x2+1 > 0 is always true.
Condition 2) is equivalent to the question, so it is sufficient.
Therefore, the answer is D.

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.

Answer: D
_________________

MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
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Posts: 4908
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 05 Feb 2018, 17:34
[GMAT math practice question]

Is xy< -(x/y)?

1) xy<0
2) 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, and then recheck the question.

Modifying the question:
xy < -(x/y)
⇔ xy^3 < -xy (multiplying both sides by y^2)
⇔ xy^3 + xy < 0
⇔ xy(y^2+1) < 0
⇔ xy < 0 since y^2+1 > 0

Condition 1): xy < 0
Condition 1) is same as the question.
This condition is sufficient.

Condition 2):
Since this condition tells us nothing about x, it is not sufficient.

Therefore, the answer is A.

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.
Answer: A
_________________

MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 06 Feb 2018, 17:09
[GMAT math practice question]

Is the positive integer x divisible by 60?

1) x is divisible by 6
2) x is divisible by 10

=>

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, and then recheck the question.
The question asks if x = 60t for some integer t.
x = 60t
⇔ x = 2^2*3*5*t


Conditions 1) (x = 2*3m for some integer m) and 2) (x = 2*5n for some integer n) do not tell us whether 22 is a factor of x. So, they are not sufficient, when taken together. For example, if
x = 60, then the answer is ‘yes’, but if
x = 30, then the answer is ‘no’.

Both conditions together are not sufficient, as the question does not have a unique answer.

Therefore, E is the answer.

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.

Answer: E
_________________

MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 08 Feb 2018, 17:27
[GMAT math practice question]

What is the value of (x+3y)/(3x-y)?

1) x-2y = 1
2) x-2y = 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.

When the question asks for a ratio, a fraction, a percent, a proportion or a rate, if one of conditions provides a ratio and the other condition provides a number, the condition with a ratio could be sufficient.

This question asks for a ratio.
Condition 1) provides a number and condition 2) provides the ratio, x/y = 2.
Thus, condition 2) is likely to be sufficient.

Condition 1) :
If x = 3 and y = 1, then (x+3y)/(3x-y) = 6/8 = 3/4
If x = 5 and y = 2, then (x+3y)/(3x-y) = 16/13
Since we do not obtain a unique answer, condition 1) is not sufficient.

Condition 2) :
Since x = 2y, (x+3y)/(3x-y) = 5y/5y = 1.
Condition 2) is sufficient.

Therefore, B is the answer.

Answer: B
_________________

MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 10 Feb 2018, 11:44
Hey guys!
If you need 2018 OFFICIAL GMAT REVIEW BOOK, you can purchase it from me for just ₹750/-
Message me if interested.

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Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 10 Feb 2018, 11:46
Hey guys!
If you need 2018 OFFICIAL GMAT REVIEW BOOK, you can purchase it from me for just ₹750/-
Message me if interested.

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Posts: 4908
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 11 Feb 2018, 17:03
[GMAT math practice question]

If x and y are integers, is 4x^2-y^2 an odd number?

1) x is an odd number
2) y 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, and then recheck the question.

Modifying the question:
In order for 4x^2 – y^2 to be odd, y^2 must be odd since 4x^2 is always even. This is equivalent to y being odd. So, the question asks if y is odd. Thus, the answer is B.

Condition 1)
If y is an odd number, then both 2x + y and 2x – y are odd numbers and (2x+y)(2x-y) is an odd number.
If y is an even number, both 2x + y and 2x – y are even numbers and (2x+y)(2x-y) is an even number.
Since the question does not have a unique answer, condition 1) is not sufficient.


Therefore, the answer is B.

Answer: B
_________________

MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
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Math Revolution GMAT Instructor
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Joined: 16 Aug 2015
Posts: 4908
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 12 Feb 2018, 17:18
[GMAT math practice question]

If x, y and z are positive numbers, is xy+z > x+yz?

1) x>1
2) y>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.

Since we have 3 variables (x and y) and 0 equations, E is most likely to be the answer. So, we should consider conditions 1) & 2) together first.

Conditions 1) & 2):
xy + z > x + yz
⇔ xy – x – yz + z > 0
⇔ x(y-1)-z(y-1) > 0
⇔ (x-z)(y-1) > 0
⇔ x – z > 0 since y > 1.

If x = 2, y = 2, and z = 1, then xy + z = 5, x + yz = 4 and xy + z > x + yz. So, the answer is ‘yes’.
If x = 2, y = 2, and z = 3, then xy + z = 7, x + yz = 8 and xy + z < x + yz. So, the answer is ‘no’.
Since the question does not have a unique answer, both conditions together are not sufficient.

Therefore, E is the answer.

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.

Answer: E
_________________

MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 13 Feb 2018, 17:26
[GMAT math practice question]

5 couples sit on 10 chairs around a round table. If each couple must be seated together, how many possible arrangements are there?

A. 256
B. 512
C. 768
D. 1,024
E. 1,080

=>

The number of arrangements of the 5 couples in a circle is (5-1)! = 4!. The members of each couple can be arranged in 2! ways.
Thus, the total number of arrangements is 4! * 2! * 2! * 2! * 2! * 2! = 24*2*2*2*2*2 = 768.

Therefore, the answer is C.

Answer: C
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 15 Feb 2018, 19:39
Hi MathRevolution, thank you for your useful post on this topic... you've mentioned that there are 5 topics that are integral to scoring Q51. Can you recommend some practise material for these topics where I can practise from to nail these topics cold?

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Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 16 Feb 2018, 04:10
[GMAT math practice question]

Is xy<15?

1) x<-3
2) y<-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.

Since we have 2 variables (x and y) and 0 equations, C is most likely to be the answer. So, we should consider conditions 1) & 2) together first.

Conditions 1) & 2):
Since x < -3 and y < -5, we have xy > 15, and the answer is ‘no’.
Since ‘no’ is also a unique answer by CMT (Common Mistake Type) 1, both conditions are sufficient, when used together.

Since this is an inequality question (one of the key question areas), we should also consider choices A and B by CMT(Common Mistake Type) 4(A).

Condition 1)
This condition provides us with no information about the variable y, so it is not sufficient:
If x = -4 and y = 1, then xy = -4 < 15, and the answer is ‘yes’.
If x = -4 and y = -5, then xy = 20 > 15, and the answer is ‘no’.
Since we do not obtain a unique answer, this condition is not sufficient.

Condition 2)
This condition provides us with no information about the variable x, so it is not sufficient:

If x = 1 and y = -5, then xy = -5 < 15, and the answer is ‘yes’.
If x = -4 and y = -5, then xy = 20 > 15, and the answer is ‘no’.
Since we do not obtain a unique answer, this condition is not sufficient.

Therefore, C is the answer.

Answer: C

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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Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 18 Feb 2018, 18:26
[GMAT math practice question]

If the average (arithmetic mean) of set A is 100 and the average (arithmetic mean) of set B is 100, what is the range of sets A and B combined?

1) The range of set A is 20.
2) The range of set B is 30.

=>

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.

There is no relationship between the average of a set and its range.
In addition, there is no way of determining the range of a set from the ranges of its subsets. Notice that “E” is in “no rElation”. Many questions about “relation” have the answer E.
Thus, E is likely to be the answer.

We can verify this by considering both conditions together:
If A = { 90, 110 } and B = { 85, 115 }, then the range of A∪B={85,90,110,115} is 30.
If A = { 90, 110 } and B = { 92, 93, 95, 98, 122 }, then the range of A∪B={ 90, 92, 93, 95, 98, 110, 122 } is 32.
Since the range of A∪B is not uniquely determined, both conditions together are not sufficient.

Therefore, the answer is E.
Answer: E
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 19 Feb 2018, 17:47
[GMAT math practice question]

Is n an even number?

1) n(n+1) is an even number
2) n(n+2) 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 1 variable (n) and 0 equations, D is most likely to be the answer. So, we should consider each of the conditions on their own first.

Condition 1):
All integers satisfy condition 1) since n(n+1) is the product of two consecutive integers:
If n = 1, n(n+1) = 1*2 = 2 is an even number, but n is odd, so the answer is ‘no’.
If n = 2, n(n+2) = 2*3 = 6 is an even number and n is even, so the answer is ‘yes’.
Since we do not have a unique answer, condition 1) is not sufficient.

Condition 2)
The product of two integers can only be even if at least one of them is even. Since n and n+2 must both be even, or both be odd, n must be even.
Thus, condition 2) is sufficient.

Therefore, B is the answer.

Answer: B

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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Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 20 Feb 2018, 16:59
[GMAT math practice question]

What is the least common multiple of 39, 91, 93 and 217?

A. 8461
B. 8462
C. 8463
D. 8464
E. 8465

=>

Since 39 = 3 * 13, the least common multiple must be a multiple of 3.
The sum of the digits of a multiple of 3 is a multiple of 3.
8643 is the only answer choice for which the sum of the digits is a multiple of 3: 8 + 4 + 6 + 3 = 21 = 3 * 7.

Therefore, C is the answer.

Alternatively, since 39 = 3*13, 91 = 7*13, 93 = 3*31, and 217=7*31, the least common multiple of the four numbers is 3*7*13*31 = 8463.

Therefore, the answer is C.

Answer: C
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Re: The Ultimate Q51 Guide [Expert Level]   [#permalink] 20 Feb 2018, 16:59

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