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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8454
GMAT 1: 760 Q51 V42
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[Math Revolution GMAT math practice question]

The numbers x, y and z are in the ratio 3 to 4 to 5. What is the value of x?

1) y = 16
2) The difference between y and z is 4

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method I s 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 original condition gives the proportion x : y : z = 3 : 4 : 5, so we can write x = 3k, y = 4k and z = 5k. We have 4 variables and 3 equations.
Therefore, D is most likely to be the answer.

Condition 1) is sufficient since 16 = y = 4k, and so k = 4. Then x = 3k = 12.

Condition 2) is also sufficient since 4 = z – y = 5k – 4k = k. So, x = 3k = 12.

Tip 1) of VA method states that D is most likely to be the answer if conditions 1) and 2) give similar information.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8454
GMAT 1: 760 Q51 V42
GPA: 3.82
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[Math Revolution GMAT math practice question]

What is the value of the 2-digit positive integer n if its tens digit is four times its units digit?

1) n is an even number
2) The units digit of n is greater than 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 only positive two-digit integers n satisfying the original condition are 82 and 41.

Condition 1) is sufficient since 82 is the only one of these integers that is even.
Condition 2) is also sufficient, since 82 the only one of these integers with a units digit that is greater than 1.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8454
GMAT 1: 760 Q51 V42
GPA: 3.82
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[Math Revolution GMAT math practice question]

Which of the following lines is perpendicular to y=(1/2)x?

A. y=-2x
B. y=x/2
C. y=2x
D. y=-x/2
E. y=x

=>

Two lines are perpendicular if and only if the product of their slopes is -1.
Since (1/2)*(-2) = -1, the slope of the line perpendicular to y = (1/2)x is -2.
The only answer choice with a slope of -2 is A.

_________________
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Joined: 16 Aug 2015
Posts: 8454
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GPA: 3.82
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[Math Revolution GMAT math practice question]

If x and y are integers, what is the least possible positive value of 21x+35y?

A. 1
B. 3
C. 5
D. 7
E. 9

=>

21x + 35y = 7*3x + 7*5y = 7(3x+5y) is a multiple of 7. The smallest positive multiple of 7 is 7. So, we look for x and y such that 21x + 35y = 7. This will occur when 3x + 5y = 1.
When x = 2 and y = -1, 3x + 5y = 3*2 + 5*(-1) = 6 – 5 = 1, and 21x + 35y = 7(3x+5y) = 7(1) = 7.

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

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MathRevolution wrote:
[Math Revolution GMAT math practice question]

If x and y are integers, what is the least possible positive value of 21x+35y?

A. 1
B. 3
C. 5
D. 7
E. 9

=>

21x + 35y = 7*3x + 7*5y = 7(3x+5y) is a multiple of 7. The smallest positive multiple of 7 is 7. So, we look for x and y such that 21x + 35y = 7. This will occur when 3x + 5y = 1.
When x = 2 and y = -1, 3x + 5y = 3*2 + 5*(-1) = 6 – 5 = 1, and 21x + 35y = 7(3x+5y) = 7(1) = 7.

Hi MathRevolution,
Given the answer choices, I understand how D would be the correct answer and chose the same but am trying to understand why aren't we considering the negative values or 0 in the question? The question doesn't ask for the least positive value.
Senior Manager  S
Joined: 07 Oct 2017
Posts: 251
Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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Shruti0805 wrote:
MathRevolution wrote:
[Math Revolution GMAT math practice question]

If x and y are integers, what is the least possible positive value of 21x+35y?

A. 1
B. 3
C. 5
D. 7
E. 9

=>

21x + 35y = 7*3x + 7*5y = 7(3x+5y) is a multiple of 7. The smallest positive multiple of 7 is 7. So, we look for x and y such that 21x + 35y = 7. This will occur when 3x + 5y = 1.
When x = 2 and y = -1, 3x + 5y = 3*2 + 5*(-1) = 6 – 5 = 1, and 21x + 35y = 7(3x+5y) = 7(1) = 7.

Hi MathRevolution,
Given the answer choices, I understand how D would be the correct answer and chose the same but am trying to understand why aren't we considering the negative values or 0 in the question? The question doesn't ask for the least positive value.
The question asks: what is the least possible positive value.

Thank you = Kudos
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8454
GMAT 1: 760 Q51 V42
GPA: 3.82
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[Math Revolution GMAT math practice question]

Is xy>0?

1) x^3y>0
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.

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

Since even powers are always non-negative, we can ignore the even exponents.
Condition 1)
x^3y > 0 is equivalent to xy > 0, which is the question itself. Therefore, condition 1) is sufficient.
Condition 2)
x^2y>0 is equivalent to y > 0.
It is not sufficient

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8454
GMAT 1: 760 Q51 V42
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[Math Revolution GMAT math practice question]

Does y=ax^2+bx+c intersect the x-axis?

1) a<0
2) 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. We then recheck the question.

Since we are asked about a quadratic polynomial, the question asks if the discriminant b^2-4ac is positive. Neither condition 1) nor condition 2) on its own is sufficient to determine this.

Considering both conditions 1) & 2) together yields b^2-4ac > 0 since b^2 ≥ 0 and ac < 0.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8454
GMAT 1: 760 Q51 V42
GPA: 3.82
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[Math Revolution GMAT math practice question]

If n is an integer, is (n+1)^2 an even integer?

1) n-1 is an even integer
2) (n-1)^2 is an even integer

=>

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.

(n+1)^2 is an even integer
=> n+1 is an even integer
=> n is an odd integer

Condition 1)
Since “n-1 is an even integer” is equivalent to “n is an odd integer”, condition 1) is sufficient.

Condition 2)
“(n-1)^2 is an even integer” is equivalent to “n-1 is an even integer”, which is condition 1).
Condition 2) is also sufficient.

Note: Tip 1) of the VA method states that D is most likely to be the answer if condition 1) gives the same information as condition 2).
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Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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MathRevolution wrote:
[Math Revolution GMAT math practice question]

If n is an integer, is (n+1)^2 an even integer?

1) n-1 is an even integer
2) (n-1)^2 is an even integer

=>

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.

(n+1)^2 is an even integer
=> n+1 is an even integer
=> n is an odd integer

Condition 1)
Since “n-1 is an even integer” is equivalent to “n is an odd integer”, condition 1) is sufficient.

Condition 2)
“(n-1)^2 is an even integer” is equivalent to “n-1 is an even integer”, which is condition 1).
Condition 2) is also sufficient.

Note: Tip 1) of the VA method states that D is most likely to be the answer if condition 1) gives the same information as condition 2).

What's the difficulty level of this question?
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8454
GMAT 1: 760 Q51 V42
GPA: 3.82
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[Math Revolution GMAT math practice question]

Is x<0?

1) |x|=-x
2) |x|>x

=>

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.

Condition 1)
|x| = -x
=> x ≤ 0
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.
Since the solution set “x<0” of the question doesn’t include the solution set “x ≤ 0” of the condition 1), condition 1) is not sufficient.

Condition 2)
|x|>x
=> x < 0
Since condition 2) is equivalent to the question, condition 2) is 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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MathRevolution wrote:
[Math Revolution GMAT math practice question]

The numbers x, y and z are in the ratio 3 to 4 to 5. What is the value of x?

1) y = 16
2) The difference between y and z is 4

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method I s 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 original condition gives the proportion x : y : z = 3 : 4 : 5, so we can write x = 3k, y = 4k and z = 5k. We have 4 variables and 3 equations.
Therefore, D is most likely to be the answer.

Condition 1) is sufficient since 16 = y = 4k, and so k = 4. Then x = 3k = 12.

Condition 2) is also sufficient since 4 = z – y = 5k – 4k = k. So, x = 3k = 12.

Tip 1) of VA method states that D is most likely to be the answer if conditions 1) and 2) give similar information.

Is this really a tough question?? I don't think so.
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8454
GMAT 1: 760 Q51 V42
GPA: 3.82
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[Math Revolution GMAT math practice question]

If 5^{m-2}2^{m+2} is an n-digit integer, where m is an integer greater than 2, what is the value of n, in terms of m?

A. n=m-4
B. n=m-2
C. n=m-1
D. n=m
E. n=2m

=>

5^{m-2}2^{m+2} = 5^{m-2}2^{m-2+4} = 5^{m-2}2^{m-2}*2^4 = (5*2)^{m-2}*2^4= (10)^{m-2}*2^4
= 16*(10)^{m-2} = 16*(100…0)= 1600…0 with m-2 0’s.
10^{m-2} has m-2 digits that are 0 and 16*(10)^{m-2} has digits including 1, 6 and m-2 digits that are 0.
Thus, 16*(10)^{m-2} has m – 2 + m = m digits.

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Posts: 8454
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[Math Revolution GMAT math practice question]

2^p/2^q=?

1) p = q + 2
2) pq=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.

Modifying the question:
What is the value of 2^p/2^q= 2^{p-q}?
So, we need to be able to find the value of p-q.

Thus, condition 1), which is equivalent to p – q = 2, is sufficient.

Condition 2):
If p =4 and q = 2, then 2^p/2^q= 2^{p-q} = 2^2 = 4.
If p =8 and q = 1, then 2^p/2^q= 2^{p-q} = 2^7 = 128.
Since we don’t have a unique solution, condition 2) is not sufficient.

_________________
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Joined: 16 Aug 2015
Posts: 8454
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[Math Revolution GMAT math practice question]

If x, y, z are integers, is xyz a multiple of 6?

1) x+y+z is a multiple of 6
2) x, y, and z are consecutive integers

=>

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) & 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 and y are two consecutive integers, one of them, x or y is an even integer, and xy is a multiple of 2.
Since x, y and z are three consecutive integers, one of them, x, y or z is a multiple of 3, and xyz is a multiple of 3.
Therefore, xyz is a multiple of 6.
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)
If x = 1, y = 2 and z = 3, then xyz = 6 is not a multiple of 6.
If x = 2, y = 2 and z = 2, then xyz = 8 is a multiple of 8.
Since we do not have a unique answer, condition 1) is not sufficient.

Condition 2)
Using the same argument as the one given for both conditions together, 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.
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Posts: 8454
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[Math Revolution GMAT math practice question]

What is the perimeter of a rectangle?

1) The square of the diagonal is 52.
2) The area of the rectangle is 24.

=>

Forget conventional ways of solving math qAnswer: CAnswer: Cuestions. 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 we apply VA method to geometry, we need to count the number of variables. For a rectangle, we have two variables for the length and the width of the rectangle. Let x and y be the length of the width of the rectangle, respectively.

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)
We have x^2+y^2 = 52 by Pythagoras’ theorem, and Area = xy = 24.
So, (x+y)^2 = x^2+2xy + y^2 = (x^2+y^2) +2xy = 52 + 48 = 100.
Therefore, x+y = 10 and we can calculate the perimeter.
Both conditions (together) are sufficient.

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Joined: 16 Aug 2015
Posts: 8454
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GPA: 3.82
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[Math Revolution GMAT math practice question]

|√3-2|+|3-√2|+|5+√3|+|1-√2|=?

A. 0
B. 2√2
C. 2√3
D. 9
E. 11

=>

|A|=A when A>0, |0|=0, and |A|=-A when A<0

Since √3-2 < 0, we have |√3-2| = -(√3-2).
Since 3-√2 > 0, we have |3-√2| = 3-√2.
Since 5+√3 > 0, we have |5+√3| = 5+√3
Since 1-√2 < 0, we have |1-√2| = -(1-√2)
So, |√3-2|+|3-√2|+|5+√3|+|1-√2|= -(√3-2) +(3-√2) + (5+√3) -(1-√2)
= -√3+2 + 3-√2 + 5+√3 – 1 +√2 = 9.

_________________
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Joined: 16 Aug 2015
Posts: 8454
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[Math Revolution GMAT math practice question]

For an integer n greater than 1, the set Sn={ kn | k is an integer greater than or equal to 1} is defined. What is the value of n?

1) 2 is contained in the set Sn.
2) 4 is contained in the set Sn.

=>

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)
Only S2 = { 2, 4, 6, … } includes 2.
Therefore, n = 2.
Condition 1) is sufficient.

Condition 2)
Two sets,
S2 = { 2, 4, 6, … }
S4 = { 4, 8, 12, … }
contain 4. So, n could be 2 or 4.
Since we don’t have a unique solution, 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: 8454
GMAT 1: 760 Q51 V42
GPA: 3.82
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[Math Revolution GMAT math practice question]

When f(x)=ax^2+bx+c (a≠0), is x+1 a factor of f(x)?

1) f(1)=0
2) f(-1)=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.

x+1 is a factor of f(x) precisely when f(x) = (x+1)(ax+b) for some numbers a and b. This occurs when f(-1) = 0, but not necessarily when f(1) = 0.

Thus, only condition 2) is 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: 8454
GMAT 1: 760 Q51 V42
GPA: 3.82
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[Math Revolution GMAT math practice question]

What is the standard deviation of a, b and c?

1) a^2+b^2+c^2 = 77
2) a+b+c =15

=>

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 mean of a, b and c is m = ( a + b + c ) / 3. So, a + b + c = 3m.
The variance of a, b and c, that is, the square of the standard deviation of a, b and c, is given by
VAR = SD^2 = { ( a – m )^2 + ( b – m )^2 + ( c – m )^2 } / 3
= { a^2 - 2am + m^2 + b^2 - 2bm + b^2 + c^2 - 2cm + m^2 } / 3
= { a^2 + b^2 + c^2 – 2m(a+b+c) + 3m^2 } / 3
= { a^2 + b^2 + c^2 – 2m*3m + 3m^2 } / 3
= { a^2 + b^2 + c^2 – 3m^2 } / 3
= ( a^2 + b^2 + c^2 ) / 3 – m^2
= ( a^2 + b^2 + c^2 ) / 3 – { ( a + b + c ) / 3 }^2

This value can be calculated using the information given in conditions 1) and 2). Thus, both conditions together are sufficient.

When we have a sum of data values and a sum of the squares of data values, we can always calculate the standard deviation.

_________________ Re: The Ultimate Q51 Guide [Expert Level]   [#permalink] 03 Oct 2018, 18:22

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# The Ultimate Q51 Guide [Expert Level]

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