Math Revolution Approach (DS)

[GMAT math practice question]

If x=y-z, is x>y?

1) y<0
2) z<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 can modify the question as follows:
x>y
⇔ x-y > 0
⇔ -z > 0 by the original condition
⇔ z < 0
This is the same as condition 2).

Condition 1)
If x = -1, y = -2, and z = -1, x > y, and the answer is ‘yes’.
If x = -2, y = -1, and z = 1, x < y, and the answer is ‘no’.
Condition 1) is not sufficient since the question does not have a unique answer.

Therefore, B is the answer.

Answer: B

[GMAT math practice question]

If x + y > 0, is xy^2 + x^2y > 0?

1) x > y
2) xy > 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.

Question:
xy^2 + x^2y > 0
=> xy(x+y) > 0
=> xy > 0, since x + y > 0
So, the question asks if xy > 0.

Since xy > 1 > 0 can be derived from condition 2), it is sufficient.
Condition 1) gives us no information about the sign of xy.

Therefore, B is the answer.

Answer: B

[GMAT math practice question]

Is (x-1)^3< (x-1)?

1) x>-1
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 can be modified as follows:
(x-1)^3< (x-1)
=> (x-1)^3- (x-1) < 0
=> (x-1)((x-1)^2- 1) < 0
=> (x-1)((x-1)+1)((x-1)-1) < 0
=> x(x-1)(x-2) < 0
=> x < 0 or 1 < x < 2
So, the question asks whether x < 0 or 1 < x < 2.


Condition 1)
Since the set of the question doesn’t include that of condition 1), it is not sufficient.

Condition 2)
Since the set of the question includes that of condition 2), it is sufficient.

Therefore, the answer is B.

Answer: B

[GMAT math practice question]

What is the median value of the data displayed in the following frequency table?



1) x=2
2) y=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.

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

If y = 1, the median is x.
If y = 2, the median is ( x + 3 ) / 2
If 3 ≤ y ≤ 9, the median is 3.
If y = 10, the median is ( 3 + 4 ) / 2 = 3.5
If y ≥ 10, the median is 4.

Thus, the condition 2) is sufficient.

Therefore, the answer is B.

Answer: B

[GMAT math practice question]

What is the reminder when n^2 is divided by 4?

1) When n is divided by 2, the reminder is 1.
2) When n is divided by 3, the reminder is 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 (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)
Plugging-in numbers is suggested for remainder questions.
The integers which have a remainder of 1 when divided by 2 are odd.
So,
n: 1, 3, 5, 7, 9, …
and
n^2: 1, 9, 25, 49, 81, …
Each value of n^2 has a remainder of 1 when it is divided by 4.
So, condition 1) is sufficient.

Condition 2)
When n = 1, n^2= 1 has a remainder of 1 when it is divided by 4.
When n = 4, n^2=16 has a remainder of 0 when it is divided by 4.
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.

[GMAT math practice question]

If r, s are positive integers, is r/s a terminating decimal?

1) 1/r^2 is a terminating decimal
2) 1/s^2 is a terminating decimal

=>

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 can modify the question as follows:
r/s is a terminating decimal if and only if s has no other prime factors than 2 and 5.
This is the same as condition 2). Condition 1) tells us nothing about the prime factors of s.

Therefore, B is the answer.


Answer: B

[GMAT math practice question]

If x<y, is x(1+x)<y(1+y)?

1) x>1/2
2) x+y>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.

Now,

x(1+x)<y(1+y)
=> x+x^2 - y - y^2 < 0
=> (x-y) + (x^2 - y^2) < 0
=> (x-y) + (x-y)(x+y) < 0
=> (x-y)(1+x+y) < 0
=> 1+x+y > 0, since x < y.

Condition 1)
Since y > x > 1/2, we have x + y + 1 > 0.
Thus, condition 1) is sufficient.

Condition 2)
Since x + y > 1, we have x + y > 0.
Thus, condition 2) is sufficient too.

Therefore, D is the answer.

Answer: D

[GMAT math practice question]

If |x+1/2|=|y+1/2|, what is the value of x+y?

1) xy<0
2) x>0 and 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.

|x+1/2|=|y+1/2|
=> |x+1/2|^2=|y+1/2|^2
=> (x+1/2)^2=(y+1/2)^2
=> x^2 + x + 1/4 = y^2 + y + 1/4
=> x^2 - y^2 + x - y = 0
=> (x-y)(x+y)+(x-y) = 0
=> (x-y)(x+y+1) = 0
=> x=y or x+y=-1

Condition 1)
Since xy < 0, we have x≠y and x+y = -1 from the original condition.

Condition 2)
Since x > 0 and y < 0, we have x≠y and x+y = -1 from the original condition.

Therefore, D is the answer.

By Tip 1), D is most likely to be the answer.

Answer: D

[GMAT math practice question]

Is |x+y|>|x+z|?

1) y>z
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.

Since we have 3 variables (x and y) 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) together:
If x =1, y = 2, and z = 3, the answer is ‘yes’.
If x = 1, y = -2, and z = -3, the answer is ‘no’.

Since we don’t have a unique solution, both conditions together are not sufficient.

Therefore, the answer is E.

Answer: E

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.

[GMAT math practice question]

If b≠0, and |ab|=ab, is b>0?

1) a<0
2) a/b>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.

|ab| = ab
=> ab ≥ 0

Condition 1)
Since a < 0, b≠0 and ab ≥ 0, we have b < 0.
Thus, the answer is ‘no’.
Since ‘no’ is also a unique answer by CMT (Common Mistake Type) 1, condition 1) is sufficient.

Condition 2)
If a = 1 and b = 1, the answer is ‘yes’.
If a = -1 and b = -1, the answer is ‘no’.
Since we don’t have a unique solution, condition 2) is not sufficient.

Therefore, A is the answer.

Answer: A

[GMAT math practice question]

Alice studies for a hours on each of Monday, Tuesday, and Thursday. She also studies for b hours on Wednesday, and for c hours on Friday. What is her median study time for the school week?

1) a=10
2) b+c=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.

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

Depending on their values, it is possible to rearrange a, a, a, b, c into ascending order in the following ways:
a, a, a, b, c
a, a, a, c, b
b, a, a, a, c
c, a, a, a, b
b, c, a, a, a
c, b, a, a, a
In every order, the median (middle value) is a. So, the question asks for the value of a.

Thus, only condition 1) is sufficient.

Therefore, A is the answer.

Answer: A

[GMAT math practice question]

If x+2y≠0, is x/x+2y<1?

1) 10|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.

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) together:

10|x|=|y|
=> y = ±10x

Case 1: y = 10x
x/(x+2y) = x/(x+20x) = x/21x = 1/21 < 1

Case 2: y = -10x
x/(x+2y) = x/(x-20x) = x/(-19)x = -(1/19) < 1

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):

10|x|=|y|
=> y = ±10x

Case 1: y = 10x
x/(x+2y) = x/(x+20x) = x/21x = 1/21 < 1

Case 2: y = -10x
x/(x+2y) = x/(x-20x) = x/(-19)x = -(1/19) < 1

Thus, condition 1) only is sufficient.

Condition 2):

Since it gives us no information about y, condition 2) is not sufficient.

Therefore, A is the answer.

Answer: A


[GMAT math practice question]

Is x/yz>0？

1) yz>x2
2) x<y+z

=>

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) together:
If x = 1, y = 2 and z = 3, then the answer is ‘yes’.
If x = -1, y = 2 and z = 3, then the answer is ‘no’.

Since we don’t have a unique answer, both conditions together are not sufficient.

Therefore, E is the answer.

Answer: E

[GMAT math practice question]3.28

Is 2x+y>0?

1) x+y>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.

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) together:
If x = 2, and y = -1, then the answer is ‘yes’.
If x = -1, and y = 2, then the answer is ‘no’.

Since we don’t have a unique solution, both conditions together are not sufficient.

Therefore, the answer is E.

Answer: E

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.

[GMAT math practice question]3.29

4. x=?
1) √x^2-1+√y^2-1=0
2)(x^2-1)^2+(y^2-1)^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 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) together:
√x^2-1+√y^2-1=0
=> x^2 = 1 and y^2 = 1
=> x = ±1 and y = ±1

(x^2-1)^2+(y^2-1)^2=0
=> x^2 = 1 and y^2 = 1
=> x = ±1 and y = ±1

The two conditions yield x = ±1 and y = ±1.
As this does not give us a unique value of x, both conditions together are not sufficient.

Therefore, E is the answer.

Answer: E

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.

[GMAT math practice question]

If 50-√7<x<50+√7, then x=?

1) x is an odd integer
2) √x 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.

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

Since 2< √7< 3, we have 47< x < 53 from the original condition ‘50-√7<x<50+√7’.

Condition 1)
As x is an odd integer, it could be 49 or 51.
Since we don’t have a unique solution, condition 1) is not sufficient.

Condition 2)
If √x is an integer, then x is the square of an integer.
49 is the only perfect square of an integer between 47 and 53.
Thus, x=49 and condition 2) is sufficient.

Therefore, B is the answer.

Answer: B

[GMAT math practice question]

Are there more girls than boys at a school?

1) 3/7 of the number of girls is more than 1/3 of the number of boys
2) 1/3 of the number of girls is more than 2/5 of the number of boys

=>

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 (b for boys and g for girls) 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) together:
From condition 1:
(3/7)g > (1/3)b
=> 9g > 7b

From condition 2:
(1/3)g > (2/5)b
=> 5g > 6b
=> 6g > 5g > 6b
=> g > b

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):

(3/7)g > (1/3)b
=> 9g > 7b

If g = 10 and b = 8, then the answer is ‘yes’.
If g = 8 and b = 8, then the answer is ‘no’.

Thus, condition 1) is not sufficient on its own.


Condition 2):

(1/3)g > (2/5)b
=> 5g > 6b
=> 6g > 5g > 6b
=> g > b

Thus, condition 2) is sufficient on its own.

Therefore, B is the answer.

Answer: B

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.