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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8229
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]

(0.0004)(0.000005)/200,000 is equal to which of the following?

A. 10^{-5}
B. 10^{-4}
C. 10^{-3}
D. 10^{-2}
E. 10^{-1}

=>

(0.0004)(0.000005)/200,000
= {4*5*(0.0001)(0.000001)}/{2*100,000}
= (2*10^1*10^{-4}*10^{-6})/ (2*10^5)
= (10^{-9})/(10^{-5}) = 10^{-4}

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8229
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] 12.20/q51 12.31

120 tickets are to be evenly distributed to n people. If n is greater than 1 and the number of tickets given to each person is greater than 1, what is the greatest possible value of n?

A. 9
B. 10
C. 12
D. 14
E. 15

=>

120 = 2^3*3^1*5^1 or 120 = 8*15.
Thus 15 is the greatest factor among choices.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8229
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]

f(x) is the greatest prime factor of x. If n is a positive integer less than 10, what is the value of n?

1) f(n) = f(1000)
2) f(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 condition on its own first.

Condition 1)
Since 1000 = 2^3*5^3, f(1000) = 5.
So, f(n) = 5 and n = 5.
Thus, condition 1) is sufficient, since it gives a unique solution.

Condition 2)
Condition 2) is the same as condition 1).
Thus, condition 2) is also sufficient.

FYI, Tip 1) of the VA method states that D is most likely to be the answer if conditions 1) and 2) provide the same information.

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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MathRevolution wrote:
[Math Revolution GMAT math practice question]

(0.0004)(0.000005)/200,000 is equal to which of the following?

A. 10^{-5}
B. 10^{-4}
C. 10^{-3}
D. 10^{-2}
E. 10^{-1}

=>

(0.0004)(0.000005)/200,000
= {4*5*(0.0001)(0.000001)}/{2*100,000}
= (2*10^1*10^{-4}*10^{-6})/ (2*10^5)
= (10^{-9})/(10^{-5}) = 10^{-4}

Isn't it (10^{-9})/(10^{5}) = 10^{-14}??
Kindly check..

Posted from my mobile device
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8229
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]

If m and n are positive integers, is m + n an odd number?

1) m/n is an even number
2) m or n is an even number

=>

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

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

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

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2), when taken together. Obviously, there may be occasions on which the answer is A, B, C or D.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8229
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]

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

1) h=1
2) k=2

=>

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

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

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

Attachment: 1231.png [ 5.04 KiB | Viewed 576 times ]

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

Attachment: 12311.png [ 5.23 KiB | Viewed 575 times ]

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

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2), when taken together. Obviously, there may be occasions on which the answer is A, B, C or D.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8229
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]

If 0<x<2 and 0<y<2 on the xy-plane, what is the probability that x+y<1?

A. 1/2
B. 1/4
C. 1/6
D. 1/8
E. 1/10

=>

Attachment: 12.26.png [ 5.03 KiB | Viewed 535 times ]

The outside square contains all points satisfying 0 < x < 2 and 0 < y < 2.
The shaded region consists of all points satisfying 0 < x < 2 and 0 < y < 2 such that x + y < 1. Its area is ½, and the area of the outside square is 4.
Thus, the required probability is (1/2) / 4 = 1/8.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8229
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]

What is the value of the integer n?

1) n is a prime factor of 21
2) n is a factor of 49

=>

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

Condition 1)
n is a prime factor of 21 = 3*7 and n is 3 or 7.
Since it does not give a unique answer, condition 1) is not sufficient.

Condition 2)
If n is a factor of 49 = 7^2, then n is 1, 7 or 49.
Since it does not give a unique answer, condition 2) is not sufficient.

Conditions 1) & 2)
The unique integer satisfying both conditions is n = 7.
Both conditions are sufficient, when taken together.

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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Intern  Joined: 10 Nov 2018
Posts: 2
Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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MathRevolution wrote:
[Math Revolution GMAT math practice question]

[x] is the greatest integer less than or equal to x. <x> is the least integer greater than or equal to x. What is the value of x?

1) [x] = 2
2) <x> = 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.

[x] is analyzed as follows.
If n ≤ x < n + 1 for some integer n, then [x] = n.
<x> is analyzed as follows.
If n – 1 < x ≤ n for some integer n, then <x> = n.

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

Condition 1)
[x] = 2
=> 2 ≤ x < 3
Thus, condition 1) is not sufficient, since it does not yield a unique solution.

Condition 2)
<x> = 2
=> 1 < x ≤ 2
Thus, condition 2) is not sufficient, since it does not yield a unique solution.

Conditions 1) & 2)
Only x = 2 satisfies both conditions.
Since the answer is unique, both conditions together are 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.

Sent from my iPhone using GMAT Club Forum mobile app
Intern  Joined: 10 Nov 2018
Posts: 2
Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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Could you confirm why the answer is not D?

For 1, doesn’t 2=<x<3 mean x can only be 2?

Likewise for 2), hence either condition is sufficient?

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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8229
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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Crb08 wrote:
Could you confirm why the answer is not D?

For 1, doesn’t 2=<x<3 mean x can only be 2?

Likewise for 2), hence either condition is sufficient?

Sent from my iPhone using GMAT Club Forum

Condition 1)
[x] = 2
=> 2 ≤ x < 3
There is a possibilty that x = 2.5 which is not 2.

Condition 2)
<x> = 2
=> 1 < x ≤ 2
There is a possibility that x = 1.5 which is not 2.

Since we don't have condition that x is an integer, we should consider decimals too.

Happy Studying!!!
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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8229
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 x/y<0?

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

=>

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

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

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

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

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

Therefore, the correct answer is A.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8229
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 x/y<0?

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

=>

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

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

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

_________________
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Joined: 16 Aug 2015
Posts: 8229
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Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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[Math Revolution GMAT math practice question]

If 2x+y=3 and 4x^2-4xy+y^2=1, what is the value of xy?

A. 1/8
B. 1/2
C. 1
D. 2
E. 4

=>

Since 2x+y=3, (2x+y)^2 = 4x^2+4xy+y^2 =9.
When we subtract 4x^2-4xy+y^2=1 from 4x^2+4xy+y^2 =9, we obtain (4x^2+4xy+y^2) - (4x^2-4xy+y^2) = 8xy = 9 - 1 = 8.
So, xy = 1.

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Joined: 16 Aug 2015
Posts: 8229
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Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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[Math Revolution GMAT math practice question]

If x≠y, what is the value of ( x^2y – xy^2 ) / ( x^3 – y^3 )?

1) xy / ( x^2 + xy + y^2) = 1/3
2) x^2y^2=9

=>

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 expression from the question is equivalent to xy / ( x^2 + xy + y^2) as shown below, which is the same as condition 1):
( x^2y – xy^2 ) / ( x^3 – y^3 )
= xy(x-y) / (x-y)(x^2+xy+y^2)
= xy / ( x^2 + xy + y^2 )
Thus, condition 1) is sufficient.

Condition 2)
If x = 3, y = 1, then ( x^2y – xy^2 ) / ( x^3 – y^3 ) = ( 9 – 3 ) / ( 27 – 1) = 6/26 = 3/13.
If x = -3, y = 1, then ( x^2y – xy^2 ) / ( x^3 – y^3 ) = ( 9 + 3 ) / ( - 27 – 1) = -12/28 = -3/7.
Since it does not yield a unique solution, condition 2) is not sufficient.

_________________
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Joined: 16 Aug 2015
Posts: 8229
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]

If x and y are positive, is x>y?

1) 2x>3y
2) -5x<-7y

=>

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 x and y are positive, condition 1) tells us that 3x > 2x > 3y or x > y.
Thus, condition 1) is sufficient.

Condition 2)
-5x < -7y
=> 5x > 7y
This implies that 7x > 5x > 7y or x > y, since x and y are positive.
Condition 2) is sufficient.

FYI, Tip 1) of the VA method states that D is most likely to be the answer if conditions 1) and 2) provide the same information.
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Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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MathRevolution Please check this question again. I am getting 10^{-14}. Some also alluded to same above.

MathRevolution wrote:
[Math Revolution GMAT math practice question]

(0.0004)(0.000005)/200,000 is equal to which of the following?

A. 10^{-5}
B. 10^{-4}
C. 10^{-3}
D. 10^{-2}
E. 10^{-1}

=>

(0.0004)(0.000005)/200,000
= {4*5*(0.0001)(0.000001)}/{2*100,000}
= (2*10^1*10^{-4}*10^{-6})/ (2*10^5)
= (10^{-9})/(10^{-5}) = 10^{-4}

Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8229
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]

If f(a*b) = f(a) + f(b), what is the value of f(1)?

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

=>

Since 1 = 1 * 1, setting a = 1 and b = 1 yields f(1) = f(1*1) = f(1) + f(1) and f(1) = 2f(1), which is equivalent to f(1) = 0.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8229
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]

x, y and z are the sides of the triangle shown and h is its height. Is the perimeter, x + y + z of the triangle greater than 1?

1) h = 1/2
2) x = y = 1/3

Attachment: 1.10.png [ 8.42 KiB | Viewed 257 times ]

=>

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.

In a triangle, the sum of the lengths of two sides is always greater than the length of the other side.
Thus, from triangle ABD, we must have AB + BD > AD = h, and from triangle ABD, we must have AC + CD > AD = h. If h = 1/2, then AB + BC + CA = AB + BD + DC + CA > 1/2 + 1/2 = 1.
Condition 1) is sufficient.

Condition 2):
If AC = 1/2, then AB + BC + CA = 1/3 + 1/3 + 1/2 = 7/6 > 1 and the answer is ‘yes’.
If AC = 1/4, then AB + BC + CA = 1/3 + 1/3 + 1/4 = 11/12 < 1 and the answer is ‘no’.
Since it does not yield a unique solution, condition 2) is not sufficient.

Therefore, the correct answer is A.
_________________
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Joined: 16 Aug 2015
Posts: 8229
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]

x is a positive integer, and the units digits of both (x+2)^2 and (x-2)^2 are 9. What is the units digit of x？

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

=>

Since x is a positive integer and the units digits of both (x+2)^2 and (x-2)^2 are 9, the units digits of x+2 and x-2 must be 7 and 3, respectively.
Thus, the units digit of x is 5.

_________________ Re: The Ultimate Q51 Guide [Expert Level]   [#permalink] 17 Jan 2019, 17:31

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# The Ultimate Q51 Guide [Expert Level]

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