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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8257
GMAT 1: 760 Q51 V42 GPA: 3.82
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[GMAT math practice question]

If gcd(m,n) is the greatest common divisor of m and n, then n=?

1) gcd(m,n)=n
2) gcd(m,gcd(m,n))= 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.

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)
In general, gcd(m,n) = gcd(m,gcd(m,n)).
By condition 1), gcd(m,gcd(m,n)) = n.
Therefore, by condition 2), n = gcd(m,gcd(m,n)) = 30.
Both conditions, together, are sufficient.

Since this question 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)
The condition that gcd(m,n) = n tells us only that m is a multiple of n.
Condition 1) is not sufficient.

Condition 2)

gcd(m,gcd(m,n)) = gcd(m,n) since m is a multiple of gcd(m,n).
Thus, gcd(m,n) = 30. But this does not allow us to determine a unique value of n.
Condition 2) 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.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8257
GMAT 1: 760 Q51 V42 GPA: 3.82
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[GMAT math practice question]

If a, b, and c are consecutive integers and a<b<c, is a an even number?

1) ac is a multiple of 8.
2) abc is a multiple of 8.

=>

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 statement that a is an even integer is equivalent to the statement that ac is a multiple of 8: if a and c are two consecutive even integers, then one of them is a multiple of 4.
Thus, condition 1) is sufficient.

Condition 2)
If a = 2, b = 3, c = 4, then ac = 8, and the answer is “yes”.
If a = 7, b = 8, c = 9, then ac = 7*9, and the answer is “no”.
Since we don’t have a unique answer, condition 2) is not sufficient.

_________________
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Joined: 16 Aug 2015
Posts: 8257
GMAT 1: 760 Q51 V42 GPA: 3.82
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[GMAT math practice question]

For a positive integer m, [m] is defined to be the remainder when 7m is divided by 3. If n is a positive integer, which of the following are equal to 1?

I. [3n+1]
II. [3n]
III. [3n] + 2

A. I only
B. II only
C. I & II only
D.I & III only
E. I, II, &III

=>

Statement I
7(3n+1) = 21n + 7 = 3(7n+2) + 1
Thus, [3n+1] = 1

Statement II
7(3n) = 21n = 3*7n + 0
Thus, [3n] = 0

Statement III
Since [3n] = 0, we have [3n] + 2 = 2.

Thus, only [3n+1] equals 1.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8257
GMAT 1: 760 Q51 V42 GPA: 3.82
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[GMAT math practice question]

When the positive integer n is divided by 8, the remainder is 3, and when n is divided by 5, the remainder is 2. What is the remainder when the smallest possible value of n is divided by 6?

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

=>

If n has remainder 3 when it is divided by 8, then n = 8a + 3 for some integer a. The possible positive integer values of n are 3, 11, 19, 27, 35, … .
If n has remainder 2 when it is divided by 5, then n = 5b + 2 for some integer b. The possible positive integer values of n are 2, 7, 12, 17, 22, 27, 32, … .
The smallest value of n that appears in both lists is 27. Therefore, the smallest possible value for n is 27.
27 = 6*4 + 3, 27 has remainder 3 when it is divided by 6.

_________________
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Joined: 16 Aug 2015
Posts: 8257
GMAT 1: 760 Q51 V42 GPA: 3.82
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[GMAT math practice question]

70% of students at a school joined the soccer club and 60% joined the baseball club. What percent of students joined both clubs?

1) 10% of students joined neither of the clubs.
2) 20 students joined both clubs

=>

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.

For questions related to 2x2 matrices and percentages, we need 3 percentages for sufficiency. Since the original condition gives us 2 percentages, we need 1 additional percentage to solve the problem.

As condition 1) gives information about a percentage, it is sufficient.

As condition 2) does not give information about a percentage, it is not sufficient.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8257
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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[GMAT math practice question]

[x] is the least integer greater than or equal to x. If x is not an integer, what is the value of [x]?

1) The greatest integer less than x is 1
2) The nearest integer to x is 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 (x) and 0 equations, D is most likely to be the answer. So, we should consider each of the conditions on their own first.

[x] can be defined in the following way:
If n – 1 < x ≤ n, where n is an integer, then [x] = n.

Condition 1)
We have 1 < x < 2.
Thus, [x] = 2.
Condition 1) is sufficient, since we have a unique solution.

Condition 2)
If x = 0.1, then [x] = 1.
If x = -0.1, then [x] = 0.
Since we do not have a unique answer, condition 2) 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.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8257
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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[GMAT math practice question]

If m and n are positive integers, what is the value of m + n?

1) m^2< 5
2) m – 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.

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)
Condition 1) tells us that m = 1 or m = 2.
If m = 1, then n = 1 – 1 = 0, which is not a positive integer.
If m = 2, then n = 2 - 1 = 1.
Thus, the unique values are m = 2 and n = 1, and m + n = 3.
Both conditions together are sufficient.

Since this question 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 provides no information about the variable n, condition 1) is not sufficient.

Condition 2)
If m = 2 and n = 1, then m + n = 3.
If m = 3 and n = 2, then m + n = 5.
Since we don’t have a unique solution, condition 2) 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.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8257
GMAT 1: 760 Q51 V42 GPA: 3.82
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[GMAT math practice question]

If n is positive integer, is 35 a factor of n?

1) 35 is a factor of n^2
2) 35 is a factor of 5n

=>

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)
Since 35=5*7 is a factor of n^2, 5 and 7 are factors of n^2.
Since 5 and 7 are prime integers and n^2 is divisible by 5 and 7, n is divisible by both 5 and 7.
Thus, 35 is a factor of n.
Condition 1) is sufficient.

Condition 2)
If n = 35, then 35 is a factor of n.
If n = 7, then 35 is not a factor of n.
Since we don’t have 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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Joined: 16 Aug 2015
Posts: 8257
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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[Math Revolution GMAT math practice question]

Is 1/x>1/y?

1) x < y
2) 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. We then recheck the question.

The question may be modified as follows:
1/x > 1/y
=> xy^2 > x^2y by multiplication by x^2y^2
=> xy^2 - x^2y > 0
=> xy(y-x) > 0

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. 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)
Applying both conditions together yields y > x > 0. So, x>0, y>0 and y-x>0.
It follows that the product xy(y – x) is positive. Therefore, both conditions are sufficient when considered together.

Since this question is an inequality 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)
If x = 2 and y = 3, then 1/x = 1/2, 1/y = 1/3, and the answer is ‘yes’.
If x = -2 and y = 3, then 1/x = -1/2, 1/y = 1/3, and the answer is ‘no’.
Condition 1) is not sufficient on its own.

Condition 2)
This condition provides us with no information about the variable y, so 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) provides 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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Joined: 16 Aug 2015
Posts: 8257
GMAT 1: 760 Q51 V42 GPA: 3.82
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[Math Revolution GMAT math practice question]

If p is a prime number and n is a positive integer, what is the number of factors of 3^np^2?

1) n = 4
2) p > 4

=>

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.

If a question asks for a number of factors, it is very important to check that all of the given prime factors are “different”. By condition 2), p is a prime number different from 3. To determine the number of factors, we need to know the exponents in the prime number factorization. Therefore, we also need condition 1).

Since p is a different prime integer from 3, and n = 4, the number of factors of 3^np^2 is (4+1)(2+1) = 15.
Since we have a unique solution, both conditions together are sufficient.

_________________
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Joined: 16 Aug 2015
Posts: 8257
GMAT 1: 760 Q51 V42 GPA: 3.82
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[Math Revolution GMAT math practice question]

There are 3 ways to make the number 12 using products of two positive integers. These are 1*12, 2*6, and 3*4. In how many ways can 2700 be written as the product of two positive integers?

A. 6
B. 12
C. 15
D. 18
E. 36

=>

2700 = 2^2*3^3*5^2
The number of distinct factors of 2700 is (2+1)(3+1)(2+1) = 36.
Since the order of multiplication does not matter (i.e. 30 * 90 = 90*30), the number of pairs of positive integers that multiply to give 2700 is 36/2 = 18.

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

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MathRevolution wrote:
[Math Revolution GMAT math practice question]

There are 3 ways to make the number 12 using products of two positive integers. These are 1*12, 2*6, and 3*4. In how many ways can 2700 be written as the product of two positive integers?

A. 6
B. 12
C. 15
D. 18
E. 36

=>

2700 = 2^2*3^3*5^2
The number of distinct factors of 2700 is (2+1)(3+1)(2+1) = 36.
Since the order of multiplication does not matter (i.e. 30 * 90 = 90*30), the number of pairs of positive integers that multiply to give 2700 is 36/2 = 18.

Why did you divide by 2 i.e. $$36/2 = 18$$?
IIMA, IIMC School Moderator V
Joined: 04 Sep 2016
Posts: 1370
Location: India
WE: Engineering (Other)
The Ultimate Q51 Guide [Expert Level]  [#permalink]

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1
Hi ammuseeru

Quote:
There are 3 ways to make the number 12 using products of two positive integers. These are 1*12, 2*6, and 3*4. In how many ways can 2700 be written as the product of two positive integers?

Quote:
Why did you divide by 2 i.e. $$36/2 = 18$$

Quote:
Since the order of multiplication does not matter (i.e. 30 * 90 = 90*30), the number of pairs of positive integers that multiply to give 2700 is 36/2 = 18.

I am not an expert, but here are my two cents. Hope you have not overlooked highlighted text.
The reason we divided by 2 is that the pairs can also be interchanged within themselves. It is same
as 36C2.
_________________
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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8257
GMAT 1: 760 Q51 V42 GPA: 3.82
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[Math Revolution GMAT math practice question]

If x and y are integers such that (xy)^2 + x^2 – 2xy – 2x + 2 = 0, what is the value of y?

A. -2
B. -1
C. 0
D. 1
E. 2

=>

(xy)^2 + x^2 – 2xy – 2x + 2 = 0
=> (xy)^2 – 2xy + 1 + x^2 – 2x + 1 = 0
=> (xy – 1)^2 + (x – 1)^2 = 0
=> xy = 1 and x = 1.
Thus , x = 1 and y = 1.

_________________
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Joined: 16 Aug 2015
Posts: 8257
GMAT 1: 760 Q51 V42 GPA: 3.82
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[Math Revolution GMAT math practice question]

1. (inequality) Is x>0?
1) x = |x|
2) x2 - 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.

Condition 1)
x = |x|
=> x ≥ 0
In inequality questions, the law “Question is King” tells us that if the solution set of the question does not include the solution set of a condition, then the condition is not sufficient.
Condition 1 is not sufficient.

Condition 2)
x2-x =0
=> x(x-1) = 0
=> x = 0 or x = 1
If x = 0, then the answer is “no”.
If x = 1, then the answer is “yes”.
Since we don’t have a unique solution, condition 2) is not sufficient.

Conditions 1) & 2)
The two conditions give x = 0 or x = 1.
They are not sufficient, when taken together, by the argument above.

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.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8257
GMAT 1: 760 Q51 V42 GPA: 3.82
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[Math Revolution GMAT math practice question]

If the average (arithmetic mean) height of Bob, John and Tom is 180 cm, what is their median height?

1) Bob’s height is 175cm.
2) John’s height is 180cm.

=>

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.

When we have 3 numbers, if one of them is equal to their average, then it is also equal to their median.

Thus, condition 2) is sufficient.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8257
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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[Math Revolution GMAT math practice question]

When a positive integer n is divided by 3, what is the remainder?

1) When n is divided by 5, the remainder is 2
2) When n is divided by 6, the remainder is 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. We then recheck the question.

Since the divisor (6) of condition 2) is a multiple of the divisor (3) of the question, condition 2) is sufficient.

When we encounter questions related to remainders, plugging in numbers is suggested.

Condition 1)
If n = 2, then the remainder when n is divided by 3 is 2.
If n = 7, then the remainder when n is divided by 3 is 1.
Since we don’t have a unique solution, condition 1) is not sufficient.

Condition 2)
The possible values of n are
n = 2, 8, 14, 20, …
All of these values have a remainder of 2 when they are divided by 3.
Thus, condition 2) is sufficient.

_________________
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Joined: 16 Aug 2015
Posts: 8257
GMAT 1: 760 Q51 V42 GPA: 3.82
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[Math Revolution GMAT math practice question]

If m=-1 and n = 1^2 + 2^2 + … + 10^2, what is the value of m^n+m^{n+1}+m^{n+2}+m^{n+3}?

A. -2
B. -1
C. 0
D. 1
E. 2

=>

m^n+m^{n+1}+m^{n+2}+m^{n+3}
= m^n(1+m^1+m^2+m^3)
= (-1)^n(1+(-1)^1+(-1)^2+(-1)^3)
= (-1)^n* 0 = 0
Whatever the value of n is, the answer is 0.

_________________
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Joined: 16 Aug 2015
Posts: 8257
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[Math Revolution GMAT math practice question]

If a=2, b=-2, and c=3, what is the value of a^4^{b-3}c^0?

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

=>

a^4b^{-3}c^0
= 2^4*(-2)^{-3}*3^0
= -(2^4*2^{-3}*1)
= -2

_________________
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Posts: 8257
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[Math Revolution GMAT math practice question]

a, b, c, d, e and f are integers. Is their median greater than their average (arithmetic mean)?

1) a<b<c<d<e<f
2) b-a=d-c=f-e

=>

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 6 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.

If a = 1, b = 2, c =5, d = 6, e = 7, f = 8, we have a median of 5.5 and an average of 29/6. The median is greater than the average, and the answer is “yes”.

If a = 1, b = 2, c =3, d = 4, e = 5, f = 6, we have a median of 3.5 and an average of 3.5. The median is not greater than the average, and the answer is “no”.

Since we don’t have a unique solution, both conditions 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.
_________________ Re: The Ultimate Q51 Guide [Expert Level]   [#permalink] 14 Aug 2018, 17:50

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# The Ultimate Q51 Guide [Expert Level]

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