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The Ultimate Q51 Guide [Expert Level]

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Expert Post
Math Revolution GMAT Instructor
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Joined: 16 Aug 2015
Posts: 5563
GMAT 1: 800 Q59 V59
GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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

When a=x+(1/x) and b=x-(1/x), what is a^2 – b^2?

A. x^2 + 1/x^2
B. x^2 - 1/x^2
C. 1
D. 2
E. 4

=>

a^2 – b^2
= (a+b)(a-b)
= ( x + 1/x + x – 1/x ) ( x + 1/x – ( x – 1/x ) )
= (2x)(2/x)
= 4

Therefore, the answer is E.

Answer : E
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Expert Post
Math Revolution GMAT Instructor
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D
Joined: 16 Aug 2015
Posts: 5563
GMAT 1: 800 Q59 V59
GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 20 May 2018, 18:18
[GMAT math practice question]

If x^2>y^2, is x>y?

1) x>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. We then recheck the question.

The original condition x2 > y2 is equivalent to |x| > |y|.

Since we have 2 variables and 1 equation, D is most likely to be the answer. In inequality questions, inequalities are counted as equations. So, we should consider each of the conditions on their own first.

Condition 1)
x^2 > y^2
=> |x| > |y|
=> x > |y|, since x > 0
⇒ x > |y| ≥ y
⇒ x > y
Thus, x > y.
Condition 1) is sufficient.

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

Therefore, A is the answer.

Answer: A

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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Expert Post
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D
Joined: 16 Aug 2015
Posts: 5563
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 21 May 2018, 17:09
[GMAT math practice question]

Is n an integer?

1) 2n is an integer
2) 1/n is an 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.

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)

If 2n = 2, then n = 1, which is an integer.
If 2n = 1, then n = 1/2, which is not an integer.
Since we don’t have a unique solution, condition 1) is not sufficient.

Condition 2)
If 1/n = 1, then n = 1, which is an integer.
If 1/n = 2, then n = 1/2, which is not an integer.
Since we don’t have a unique solution, condition 2) is not sufficient.

Conditions 1) & 2)
n = 1 satisfies both conditions, and n is an integer.
n = 1/2 satisfies both conditions, and n is not an integer.
Both conditions together are not sufficient.

Therefore, E is the answer.

Answer: E

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.
_________________

MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
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Expert Post
Math Revolution GMAT Instructor
User avatar
D
Joined: 16 Aug 2015
Posts: 5563
GMAT 1: 800 Q59 V59
GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 22 May 2018, 18:08
[GMAT math practice question]

How many 4-digit numbers have only even digits?

A. 500
B. 525
C. 600
D. 625
E. 800

=>

Suppoose abcd is 4-digit number with only even digits.
Then a is one of 2, 4, 6 and 8, and b, c and d may be selected from the digits 0, 2, 4, 6 and 8.
So, there are 4 choices for a, and 5 choices for each of b, c and d.

The total number of 4-digit numbers with only even digits is thus 4*5*5*5 = 500.

Therefore, A is the answer.
Answer: A

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Expert Post
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D
Joined: 16 Aug 2015
Posts: 5563
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 23 May 2018, 18:15
[GMAT math practice question]

How many triples (a,b,c) of even positive integers satisfy a^3 + b^2 + c = 50?

A. one
B. two
C. three
D. four
E. five

=>

Consider the variable a first.
Since 4^3 > 50, we can only have a = 2.
If a = 2, then b^2 + c = 42 since a^3 = 2^3 = 8.
Since 8^2 = 64 > 42, we can only have b = 2, 4 or 6.
If b = 2, then b^2 + c = 2^2 + c = 4 + c = 42 and c = 38.
If b = 4, then b^2 + c = 4^2 + c = 16 + c = 42 and c = 26.
If b = 6, then b^2 + c = 6^2 + c = 36 + c = 42 and c = 6.
Thus, there are three possible triples: ( 2, 2, 38 ), ( 2, 4, 26 ) and ( 2, 6, 6 ).

Therefore, the answer is C.
Answer: C
_________________

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Expert Post
Math Revolution GMAT Instructor
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D
Joined: 16 Aug 2015
Posts: 5563
GMAT 1: 800 Q59 V59
GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 24 May 2018, 18:33
[GMAT math practice question]

If 123,456=123a+87 and 234,567=123b+6, how many multiples of 123 lie between 123,456 and 234,567?

A. a
B. b
C. a+b
D. a-b
E. b-a

=>

Since 123,456 = 123*1003 + 87, we must have a = 1003.
Since 234,567 = 123*1907 + 6, we must have b = 1907.
The multiples of 123 between 123,456 and 234,567 are 123*1004, 123*1005, …, 123*1907.
Thus, 1907 – 1004 + 1 = 1907 – 1003 = b – a multiples of 123 lie between 123,456 and 234,567.

Therefore, E is the answer.
Answer E
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Expert Post
Math Revolution GMAT Instructor
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D
Joined: 16 Aug 2015
Posts: 5563
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 27 May 2018, 18:24
[GMAT math practice question]

There are 100 employees in a company. Is the standard deviation of their monthly salaries less than $1000?

1) The average of the salaries is $3,000
2) The range of the salaries is $500

=>

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.

Suppose d is the standard deviation of the data set and r is the range of the data set. Then d ≤ r. Since we have many variables, 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 d ≤ r, we have d ≤ 500 < 1000.
Both conditions are sufficient, when taken together.

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

Condition 1)
As there is no relationship between standard deviation and average, condition 1) is not sufficient.

Condition 2)
Since d ≤ r, we have d ≤ 500 < 1000.
Condition 2 is sufficient.

Therefore, B is the answer.

Answer: B
_________________

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Expert Post
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D
Joined: 16 Aug 2015
Posts: 5563
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 28 May 2018, 18:08
[GMAT math practice question]

If x and y are positive integers, what is the remainder when 3^{4x+1}+y is divided by 10?

1) x=2
2) y=3

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.
3^1 ~ 3^5 ~ 3^9 ~ … ~ 3 : Integers of the form 3^{4x+1} always have the remainder of 3 when they are divided by 10.
3^2 ~ 3^6 ~ 3^{10} ~ … ~ 9 : Integers of the form 3^{4x+2} always have the remainder of 9 when they are divided by 10.
3^3 ~ 3^7 ~ 3^{11} ~ … ~ 7 : Integers of the form 3^{4x+3} always have the remainder of 7 when they are divided by 10.
3^4 ~ 3^8 ~ 3^{12} ~ … ~ 1 : Integers of the form 3^{4x} always have the remainder of 1 when they are divided by 10.
Therefore, the remainder when 3^{4x + 1} + y is divided by 10 depends only on the value of y.

Only condition 2) gives us a value for y.
Therefore, B is the answer.
Answer: B
_________________

MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
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Expert Post
Math Revolution GMAT Instructor
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D
Joined: 16 Aug 2015
Posts: 5563
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GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 29 May 2018, 18:34
[GMAT math practice question]

What is the remainder when 7^{100} is divided by 50?

A. 0
B. 1
C. 7
D. 21
E. 49

=>

The remainder when 7^{100} is divided by 50 depends only on the units and tens digits.

The units digits of 7^n cycle through the four values 7, 9, 3, and 1.
The tens digits of 7^n cycle through the four values 0, 4, 4, and 0.

We have the following sequence of units and tens digits for 7^n:

7^1 = 07 ~ 07
7^2 = 49 ~ 49
7^3 = 343 ~ 43
7^4 = 2401 ~ 01
7^5 = 16807~ 07


So, 7^{100} = (7^4)^{25} has the same units and tens digits as 7^4, that is, 01.
Thus, the remainder when 7^{100} is divided by 50 is 1.

Therefore, B is the answer.

Answer : B
_________________

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Expert Post
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D
Joined: 16 Aug 2015
Posts: 5563
GMAT 1: 800 Q59 V59
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 01 Jun 2018, 02:42
[GMAT math practice question]

What is the median of 3 consecutive integers?

1) The product of the integers is 0
2) The sum of the integers is equal to their product

=>

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.

Let the 3 consecutive integers be n – 1, n and n + 1.

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 its own first.

Condition 1)
Since the product of the three integers is 0, one of the integers must be 0. There are three possible lists of consecutive integers:
( -2, -1, 0 ), ( -1, 0, 1) and ( 0, 1, 2 ).
The medians of these lists are -1, 0 and 1.
Since we don’t have a unique solution, condition 1) is not sufficient.

Condition 2)
Since the sum of the integers is equal to their product, there are three possible lists of consecutive integers:
(-3, -2, -1), ( -1, 0, 1) and ( 1, 2, 3).
The medians of these lists are -2, 0 and 2.
Since we don’t have a unique solution, condition 2) is not sufficient.

Conditions 1) & 2)
(-1, 0, 1) is the unique list of three consecutive integers that satisfies both conditions 1) & 2).
Both conditions together are sufficient.

Therefore, C is the answer.

Answer: C

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.
_________________

MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
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Expert Post
Math Revolution GMAT Instructor
User avatar
D
Joined: 16 Aug 2015
Posts: 5563
GMAT 1: 800 Q59 V59
GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 05 Jun 2018, 06:25
[GMAT math practice question]

<x> is defined to be the smallest integer greater than or equal to x. What is the value of <x>?

1) x>0
2) x<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 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)
If x = 0.5, then <x> = 1.
If x = 1.5, then <x> = 2.
Since we don’t have a unique solution, condition 1) is not sufficient.

Condition 2)
If x = 0.5, then <x> = 1.
If x = -0.5, then <x> = 0.
Since we don’t have a unique solution, condition 2) is not sufficient.

Conditions 1) & 2)
Now, if n – 1 < x ≤ n, where n is an integer, then <x> = n
Conditions 1) & 2) tell us that 0 < x < 1. Therefore, <x> = 1.
Since we have a unique solution, both conditions are sufficient, when taken together.

Therefore, C is the answer.
Answer: C

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.
_________________

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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"Unlimited Access to over 120 free video lessons - try it yourself"

Expert Post
Math Revolution GMAT Instructor
User avatar
D
Joined: 16 Aug 2015
Posts: 5563
GMAT 1: 800 Q59 V59
GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 06 Jun 2018, 19:07
[GMAT math practice question]

Is xy<0?

1) |x+y| < |x| + |y|
2) |x| - |y| < | x – y |

=>

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 order to solve this type of question within a limited time, we should memorize the property that | x + y | < |x| + |y| is equivalent to xy < 0.
So, condition 1) is sufficient.

Condition 2)
|x| - |y| < | x – y |
=> |x| < | x – y | + |y|
=> | x – y + y | < | x – y | + |y|
=> ( x – y )y < 0

because of the above property.
If x = 1 and y = -1, then we have xy < 0 and the answer is ‘yes’.
If x = 1 and y = 2, then we have xy > 0 and the answer is ‘no’.
Since we don’t have a unique answer, condition 2) is not sufficient.

Therefore, A is the answer.

Answer: A
_________________

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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"Unlimited Access to over 120 free video lessons - try it yourself"

Expert Post
Math Revolution GMAT Instructor
User avatar
D
Joined: 16 Aug 2015
Posts: 5563
GMAT 1: 800 Q59 V59
GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 07 Jun 2018, 18:11
[GMAT math practice question]

What is the largest of three consecutive even integers?

1) The smallest integer is 6.
2) Their average is 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.

Let 2k – 2, 2k, 2k + 2 be the 3 consecutive even integers. Since we have 1 variable (k) and 0 equations, D is most likely to be the answer. So, we should consider each of the conditions on its own first.

Condition 1)
Since 2k – 2 = 6, we must have 2k = 8 or k = 4.
Then the largest integer is 2k + 2 = 10.
Thus, condition 1) is sufficient.

Condition 2)
The average of three consecutive even integers is their median.
Since 2k = 8, we must have k = 4.
Then, the largest integer is 2k + 2 = 10.
Thus, condition 2) is sufficient.

Therefore, the answer is D.

Answer: D

Note: Since both conditions give us the same information, the answer is D by tip 1).

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.
_________________

MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
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Expert Post
Math Revolution GMAT Instructor
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D
Joined: 16 Aug 2015
Posts: 5563
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 10 Jun 2018, 18:38
[GMAT math practice question]

In the coordinate plane, is the x-intercept of the line ax + by + c = 0 less than 0?

1) ac > 0
2) bc > 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.

We need to plug in 0 for y to get the x-intercept. This yields ax +b*0 + c = 0, or ax + c = 0.
The x-intercept is -c/a and the question asks if –c/a < 0.
Now,
–c/a < 0
=> c/a > 0
=> ac > 0.

Thus, condition 1) is sufficient.

Condition 2)
If a = 1, b = 1 and c = 1, then the x-intercept –c/a = -1 is less than 0.
If a = -1, b = 1 and c = 1, then the x-intercept –c/a = 1 is greater than 0.
Since we don’t have a unique solution, condition 2) is not sufficient.

Therefore, A is the answer.

Answer: A
_________________

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Expert Post
Math Revolution GMAT Instructor
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Joined: 16 Aug 2015
Posts: 5563
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 11 Jun 2018, 18:02
[GMAT math practice question]

If xyz=x, is x=0?

1) yz=0
2) xy=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 have xyz-x=0 or x(yz-1)=0 from the original condition, either x = 0 or yz = 1. Since yz = 0 by condition 1), we must have x = 0. Thus, condition 1) is sufficient.

Condition 2) tells us that xy = 0. This implies that xyz = 0. Since xyz= x, we must have x = 0. Thus, condition 2) is also sufficient.

Therefore, D is the answer.
Answer: D
_________________

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Expert Post
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Posts: 5563
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink]

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New post 12 Jun 2018, 18:15
[GMAT math practice question]

A college has a soccer club, a tennis club and a basketball club. Students can enroll in only one of these three clubs. The ratio of the number of soccer club members to the number of tennis club members is 2:3. The ratio of the number of tennis club members to the number of basketball club members is 4:5. A total of 350 students have joined one of these three clubs. How many students are enrolled in the soccer club?

A. 80
B. 90
C. 100
D.120
E. 150

=>

Let S, T and B be numbers of members in the soccer club, the tennis club and the basketball club, respectively.
Since S:T = 2:3 = 8:12 and T:B = 4:5 = 12:15, we have S:T:B = 8:12:15.
Let S = 8k, T = 12k and B = 15k.
Then S + T + B = 8k + 12k + 15k = 35k = 350,
and k = 10.
The number of members of the soccer club is
S = 8k = 80.

Therefore, the answer is A.
Answer: A

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New post 14 Jun 2018, 18:28
[GMAT math practice question]

Is x<y?

1) 3^x<2^y
2) x>0 and y>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):
Since 3^x < 2^y and x, y are positive, we have 2^x < 3^x < 2^y or 2^x < 2^y. It follows that x < y.
Both conditions together are sufficient.

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 = 1 and y = 2, then the answer is “yes”.
If x = -1 and y = -1, then the answer is “no”.
Thus, condition 1) is not sufficient on its own since it does not give a unique solution.

Condition 2)
If x = 1 and y = 2, then the answer is “yes”.
If x = 2 and y = 1, then the answer is “no”
Thus, condition 2) is not sufficient on its own since it does not give a unique solution.

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] 14 Jun 2018, 18:28

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