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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 7752
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: Math Revolution Approach (DS)  [#permalink]

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[GMAT math practice question]

If f(x) = 4x^2 + px, what is the minimum value of f(x)?

1) f(1) = -4
2) f(2) = 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 (p) 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 f(1) = -4, we have 4 + p = -4 and p = -8.
f(x) = 4x^2 -8x = 4( x^2 - 2x + 1 - 1 ) = 4(x-1)^2 – 4.
The minimum of f(x) is -4.
Condition 1) is sufficient.

Condition 2)
Since f(2) = 0, we have 16 + 2p = 0 and p = -8.
f(x) = 4x^2 -8x = 4( x^2 - 2x + 1 - 1 ) = 4(x-1)^2 – 4.
The minimum of f(x) is -4.
Condition 2) is sufficient.

Therefore, D is the answer.

Answer: D

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).

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: 7752
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: Math Revolution Approach (DS)  [#permalink]

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[GMAT math practice question]

If the operation % is defined by x % y = px^2 + qy^2 (p and q are constants), what is the value of 2 % 4?

1) 1 % 2 = 5
2) 1 % 1 = 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.
2 % 4 = 4p + 16q = 4 ( p + 4q ) = 4( p* 1^2 + q*2^2 ) = 4(1%2)
Thus, condition 1) is sufficient.

Condition 2)
Now, 1%1 = p + q = 2.
If p = 1 and q = 1, then 2 % 4 = 1*2^2 + 1*4^2 = 20.
If p = 3 and q = -1, then 2 % 4 = 3*2^2 + (-1)*4^2 = 12 – 16 = -4.
Since we don’t have a unique solution, condition 2) is not sufficient.

Therefore, A is the answer.
Answer: A
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 7752
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: Math Revolution Approach (DS)  [#permalink]

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[GMAT math practice question]

In the x-y plane, y = f(x) = ax^2+bx+c passes through (1,0) and (2,0). Is f(3)>0?

1) a > 0
2) f(0) > 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 f(x) passes through (1,0) and (2,0), 1 and 2 are roots of f(x) and we have f(x) = a(x-1)(x-2). If f(x) is concave up, then f(3) > 0. Thus, knowing that f(x) is concave up will allow us to answer the question. In addition, since we have 3 variables (a, b, c) and 2 conditions, D is most likely to be the answer.

Condition 1)
If a > 0, then f(x) is concave up.
Thus, condition 1) is sufficient.

Condition 2)
Since 1 and 2 are roots of f(x) = 0, f(0) > 0 implies that f(x) is concave up.
Thus, condition 2) is sufficient, too.

Therefore, D is the answer.

Answer: D

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).
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 7752
GMAT 1: 760 Q51 V42 GPA: 3.82
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[GMAT math practice question]

When the members of group A are divided into groups of 13 people, m subgroups are formed. When the members of group B are divided into groups of 11 people, n subgroups are formed, and 8 people are left over. What is the number of members of group B?

1) m = n
2) The numbers of members of groups A and B are the same.

=>

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)
The conditions give the equations m = n and 13m = 11n + 8.
Plugging the first equation into the second yields 2n = 8 or n = 4.
Thus, group B has 11*4 + 8 = 52 members.
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 equation m = n is not sufficient for determining the value of n.
Thus, condition 1) alone is not sufficient.

Condition 2)
13m = 11n + 8 does not give enough information to find the value of n.
Thus, condition 2) is not sufficient.

Therefore, the answer is C.

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.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 7752
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: Math Revolution Approach (DS)  [#permalink]

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[GMAT math practice question]

|m-n|=?

1) m and n are integers
2) mn=13

=>

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 and n are integers, the pairs of solutions of the equation mn=13 are
m = 1, n = 13; m = 13, n = 1; m = -1, n = -13; and m = -13, n = -1.
If m = 1, n = 13, then | m – n | = | 1 – 13 | = 12.
If m = 13, n = 1, then | m – n | = | 13 – 1 | = 12.
If m = -1, n = -13, then | m – n | = | -1 – (-13) | = 12.
If m = -13, n = -1, then | m – n | = | -13 – (-1) | = 12.

Since this question is an absolute value 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 we don’t have enough information, condition 1) is not sufficient.

Condition 2)
If m = 1, n = 13, then | m – n | = | 1 – 13 | = 12.
If m = 2, n = 13/2, then | m – n | = | 2 – 13/2 | = 9/2.
Since we don’t have a unique solution, condition 2) is not sufficient.

Therefore, C is the answer.

Answer: C

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 7752
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: Math Revolution Approach (DS)  [#permalink]

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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.

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.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 7752
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: Math Revolution Approach (DS)  [#permalink]

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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.

Therefore, A is the answer.
Answer: A
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 7752
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: Math Revolution Approach (DS)  [#permalink]

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[GMAT math practice question]

In the xy-plane, is the triangle that connects the 3 different points A(3,4), B(p,q), and C(r,s) a right triangle?

1) (p-3)(r-3)+(q-4)(s-4)=0
2) p=3 and s=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.

Condition 2):
If p = 3, then q ≠ 4 since (p,q) is different from (3,4).
If s = 4, then r ≠ 3 since (r,s) is different from (3,4).
So, (p,q) is on the line x = 3 and (r,s) is on the line y = 4.
As these lines are perpendicular AB and AC are perpendicular, and triangle ABC is a right triangle.
Thus, condition 2) is sufficient.

Condition 1) is complicated. If you can’t figure out how to apply condition 1), CMT4(B) tells you to choose D as the answer.

Condition 1)
(p-3)(r-3)+(q-4)(s-4)=0
=> (p-3)(r-3) = -(q-4)(s-4)
=> (p-3)(r-3) / (q-4)(s-4) = -1 or (q-4)(s-4) = 0
=> {(p-3)/(q-4)} * {{r-3)/(s-4)} = -1 or q = 4 or s = 4

Case 1: (p-3)/(q-4)} * {{r-3)/(s-4)} = -1
(p-3)/(q-4) and (r-3)/(s-4) are the slopes of two sides of the triangle.
Since the product of these slopes is -1, the two sides are perpendicular and the triangle is a right triangle.

Case 2: q = 4
If q = 4, then p is not 3. Also, we must have (p-3)(r- 3) = 0. So, r = 3 since (p,q) is different from (3,4).
Thus, (p,q) lies on the line y = 4 and (r,s) lies on the line x = 3.
As these lines are perpendicular, AB and AC are perpendicular, and triangle ABC is a right triangle.

Case 3: s = 4.
A similar argument to the one used for case 2 shows that triangle ABC is a right triangle.

Thus, condition 1) is also sufficient.

Therefore, D is the answer.

Answer: D

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: 7752
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.

Therefore, the answer is A.

Answer: A
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 7752
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: Math Revolution Approach (DS)  [#permalink]

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[GMAT math practice question]

The product of 4 consecutive odd integers is negative. What is the largest odd integer?

1) The smallest integer is negative.
2) The third smallest integer is negative.

=>

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 of the conditions on their own first.

Let the integers be 2n – 3, 2n – 1, 2n + 1 and 2n +3.
Since (2n-3)(2n-1)(2n+1)(2n+3) < 0, either one of the integers is negative, or three of the integers is negative. There are two possible lists of integers: -1, 1, 3, 5 and -5, -3, -1, 1.

Condition 1)
The largest odd integers in the two possible lists are 5 and 1.
Since we don’t have a unique answer, condition 1) is not sufficient.

Condition 2)
If the third smallest integer is negative, then the integers are -5, -3, -1, 1.
The largest integer is 1.
Since we have a unique answer, condition 2) is sufficient.

Therefore, B is the answer.

Answer: B

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 7752
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: Math Revolution Approach (DS)  [#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.

Therefore, A is the answer.

Answer: A
Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 7752
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: Math Revolution Approach (DS)  [#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.

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.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 7752
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: Math Revolution Approach (DS)  [#permalink]

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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 n2 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.

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.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 7752
GMAT 1: 760 Q51 V42 GPA: 3.82
Math Revolution Approach (DS)  [#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.

Therefore, the answer is C.
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) 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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[Math Revolution GMAT math practice question]

If x is not 0, is x>1?

1) x/|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)
If x > 0, x/|x| < x is equivalent to x/x < x or x > 1.
If x < 0, x/|x| < x is equivalent to x/(-x) < x or x > -1, and we have -1 < 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.
Since the solution of the question does not include that of condition 1), condition 1) is not sufficient.

Condition 2)
x=|x| is equivalent to x ≥ 0.
Since x is not 0, we must have x > 0.
Since the solution set of the question does not include the solution set of condition 2), condition 2) is not sufficient.

Conditions 1) & 2)
Condition 1) yields x > 1 or -1 < x < 0, while condition 2) yields x > 0.
Thus, applying both conditions together yields x > 0.
Since the solution set of the question does not include the solution set of both conditions, taken together, they are not 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.
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Joined: 16 Aug 2015
Posts: 7752
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Re: Math Revolution Approach (DS)  [#permalink]

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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.

Therefore, C is the answer.
Answer: C
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Joined: 16 Aug 2015
Posts: 7752
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: Math Revolution Approach (DS)  [#permalink]

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[Math Revolution GMAT math practice question]

If m and n are positive integers and √m√n=6, then n=?

1) Both m and n are the squares of integers.
2) m > n

=>
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 1 equation, D is most likely to be the answer. So, we should consider each of the conditions on its own first.

Condition 1)
There are at least two possible pairs of solutions. Two of these pairs are: m = 4, n = 9 and m = 9, n = 4.
Since we don’t have a unique solution, condition 1) is not sufficient.

Condition 2)
There are more than two pairs of solutions. Two of these are m = 9, n = 4 and m = 12, n = 3.
Since we don’t have a unique solution, condition 2) is not sufficient.

Conditions 1) and 2)
There are two possible pairs of solutions: m = 36, n = 1 and m = 9, n = 4.
Since we don’t have a unique solution, both conditions together are not sufficient.
Note: This question includes an example of a “hidden 1”.

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.
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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 7752
GMAT 1: 760 Q51 V42 GPA: 3.82
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[Math Revolution GMAT math practice question]

After t seconds, the height of a ball from the ground is given by the equation h =-9.8t^2+ft+g (f and g are constants). If the ball is at its maximum height, then t=?

1) f = 10
2) g = 10

=>

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

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

Y = aX^2 + bX + c has a maximum or a minimum when X = -b/2a.
Thus h =-9.8t^2+ft+g has a maximum when t = f/(2*9.8).
Condition 1) is sufficient on its own.

Condition 2) is not sufficient since it gives us no information about the value of f.

Therefore, A is the answer.
Answer: A
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Joined: 16 Aug 2015
Posts: 7752
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: Math Revolution Approach (DS)  [#permalink]

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[Math Revolution GMAT math practice question]

Is x>0?

1) x = |x|
2) x^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.

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)
x^2-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.

Therefore, the answer is E.

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.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 7752
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: Math Revolution Approach (DS)  [#permalink]

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[Math Revolution GMAT math practice question]

What is the median of 10 numbers?

1) 6 of the ten numbers are less than or equal to 10.
2) 6 of the ten numbers are greater than or equal to 10.

=>

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

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

Let x1, x2, … , x10 be the 10 numbers, written in increasing order: x1 ≤ x2 ≤ … ≤ x10.
The median of 10 numbers is the average of the 5th number, x5 and the 6th number, x6.

By condition 1), x5 ≤ x6 ≤ 10, and by condition 2), 10 ≤ x5 ≤ x6.
Thus, x5 = x6 = 10.
The median is ( x5 + x6 ) / 2 = 10.
Both conditions together are sufficient.

Therefore, the answer is C.

Answer: C
_________________ Re: Math Revolution Approach (DS)   [#permalink] 02 Aug 2018, 00:37

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