Math Revolution Approach (DS)

[GMAT math practice question]

If x and y are integers, is x + y an even integer?

1) x is an odd integer.
2) x^2 + y^2 has a remainder of 2 when it is divided by 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.

Since the question includes 2 variables (x and y) and no equation, C is most likely to be the answer. Since this is an integer question (one of the key question areas), we should also consider choices A and B by CMT 4(A).

Conditions 1) & 2)
Since x^2 + y^2 has a remainder of 2 when it is divided by 4, x^2 + y^2 must be even. Since x is odd, x^2 is odd and so y^2] must also be odd. Therefore, y is odd, and \(x + y\) is even. The answer is ‘yes’.

Condition 1)
Since we don’t know whether y is even or odd, this is not sufficient.

Condition 2)
The condition tells us that x^2+y^2=4k+2=2(2k+1) is even. Since x^2+y^2=(x+y)^2-2xy, and 2xy is even, this implies that (x+y)^2 is also even. But this can only happen if x+y is even. So, the answer is ‘yes’.
This condition is sufficient.

Therefore, the answer is B.

Normally, in problems which require 2 or more additional 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).

Answer: B 

[GMAT math practice question]

Is x^2 + xy > 5xy – 4y^2?

1) x > 2y
2) 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, and then recheck the numbers of variables and equations.

Modifying the original condition and question:
x^2 + xy > 5xy – 4y^2
⇔ x^2 - 4xy + 4y^2 > 0
⇔ (x – 2y)^2 > 0
⇔ x ≠ 2y
The question, ‘Is x^2 + xy > 5xy – 4y^2 ?’, is equivalent to asking if x ≠ 2y.

Condition 1)
This is sufficient because x > 2y implies that x ≠ 2y.

Condition 2)
If x=3 and y=1, then (x-2y)^2=1>0, and the answer is ‘yes’.
If x=2 and y=1, then (x-2y)^2=0, and the answer is ‘no’.
This is NOT sufficient.

Therefore, the answer is A.

Answer: A

[GMAT math practice question]

Is x>0?

1) x+1>0
2) |x+4| < |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.

Since we have 1 variable (x) and 0 equations, D is most likely to be the answer.

Condition 1)
x + 1 > 0 x > -1
Since the range of the question does not include the range of the condition, this is not sufficient.

Condition 2)
|x+4| < |x-2|
⟺|x+4|^2 < |x-2|^2
⟺(x+4)^2 < (x-2)^2
⟺x^2 + 8x + 16 < x^2 -4x + 4
⟺8x + 16 < -4x + 4
⟺12x < -12
⟺x < -1
The answer is ‘no’.
By CMT (Common Mistake Type) 1, ‘no’ is also an answer.
This is sufficient.

Therefore, the answer is 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.

Answer: B

[GMAT math practice question]

If 2x+y≠0, is x/2x+y<1 ?

1) |x|=10|y|
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.

Since we have 2 variables (x and y) and 0 equations, C is most likely to be the answer.

Conditions 1) & 2)
|x|=10|y|
⇔ x = ±10y

Case 1: If x=10y, then x/2x+y= 10y/20y+y= 10y/21y=10/21<1,
and the answer is ‘yes’.
Case 2: If x= -10y, then
x/2x+y = -10y/-20y+y = -10y/-19y =10/19<1,
and the answer is ‘yes’.

Both conditions, applied together, are sufficient.

Since this is an inequality question (one of the key question areas), CMT (Common Mistake Type) 4 (A) tells us
that we need to also consider conditions 1) and 2) separately.

Condition 1)

|x|=10|y|
⇔ x = ±10y

Case 1: If x = 10y, then
x/2x+y= 10y/20y+y= 10y/21y=10/21<1,
and the answer is ‘yes’.

Case 2: If x = -10y, then
x/2x+y= -10y/-20y+y= -10y/-19y=10/19<1
and the answer is ‘yes’.

Since the answer is ‘yes’ in both cases, this condition is sufficient.

Condition 2)
If x = -2 and y = 1, then x/2x+y = -2/-4+1 = -2/-3 = 2/3 <1, and the answer is ‘yes’.
If x=-2 and y=3, then x/2x+y = -2/-4+3 = -2/-1 =2>1, and the answer is ‘no’.
This is NOT sufficient.

Note: Since this condition is so trivial, it is unlikely to be sufficient by Tip 4) of the VA method.

The answer is A.

Normally, in problems which require 2 or more additional 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).

Answer: A 

[GMAT math practice question]

A group ticket for admission to art gallery A costs $p for up to 10 people. Each additional person is charged an admission fee of $q. A group ticket for admission to museum M costs $r for up to 15 people. Each additional person is charged an admission fee of $s. If a group of 20 people visit both the art gallery A and the museum M, is the total cost of their admission to the art gallery less than the total cost of their admission to the museum?

1) p<r
2) q<s

=>
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 asks if p + (20 – 10)q = p + 10q is smaller than r + ( 20 – 15 )s = r + 5s.

Since we have 4 variables (p, q, r, and s) and 0 equations, E is most likely to be the answer.
Therefore, we should consider both conditions together first.

Conditions 1) & 2)
If p = 10, q = 20, r = 100, and s = 200, then p+10q=210<1,100=r+5s. The answer is ‘yes’.
If p = 10, q = 20, r = 11, and s = 21, then p+10q=210>116=r+5s. The answer is ‘no’.
So, conditions 1) & 2) are NOT sufficient when taken together.

Therefore, the answer is E, as expected.


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.

Answer: E

[GMAT math practice question]

The average (arithmetic mean) of a list of 5 numbers (arithmetic mean) is 15. If 2 numbers are removed from list, is the average (arithmetic mean) of the 3 remaining numbers less than 15?

1) Each of the 2 numbers removed is greater than 15.
2) The average of the 2 numbers removed is greater than 15.

=>

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

Assume that the five numbers are, a, b, c, d, and e. Then a + b + c + d + e/5 = 15, or a + b + c + d + e = 75.

Since we have 5 variables (a,b,c,d, and e) and 1 equation, E is most likely to be the answer. Thus, we should consider
both conditions together first.

Conditions 1) and 2):
Suppose that a and b are the numbers removed, and that a > 15 and b > 15. Then
( a + b )/2 > 15
⟺a + b > 30
So,
c + d + e = 75 – ( a + b ) < 45, since a + b > 30. Thus,
c + d + e/3 <45/3 = 15,
and the answer is ‘yes’.
Therefore, both conditions are sufficient when applied 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)
Assume that a and b are the numbers removed, and that a > 15 and b > 15. Then
a + b > 30, and so c + d + e = 75 – ( a + b ) < 45, since a + b > 30.
Therefore,
( c + d + e )/3 <45/3 = 15,
and the answer is ‘yes’.
This is sufficient.

Condition 2)

Assume that a and b are the numbers removed, and that their average is greater than 15. That is,
a + b/2 > 15. Then a + b > 30, and so
c + d + e = 75 – ( a + b ) < 45.
Therefore,
c + d + e/3 <45/3 = 15,
and the answer is ‘yes’.
This is sufficient.

Therefore, the answer is D.

Note: By Tip 1) of the VA method, if conditions 1) and 2) provide the same information, we can just select answer D.

Answer: D

[GMAT math practice question]

What is the value of x in the list {36, 47, 51, 65, x}?

1) The median of the 5 values is 51.
2) The range of the 5 values is 29.

=>

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.

Condition 1)
Since the median is 51, x could be any value greater than or equal to 51.
This is NOT sufficient.

Condition 2)
Since the range is 29=65-36, x could be any value between 36 and 65, inclusive.
This is NOT sufficient.

Conditions 1) and 2)
Condition 1) tells us that x could be any value greater than or equal to 51, while condition 2) tells us that x could be any value between 36 and 65, inclusive. Combining these ranges tells us that x could be any value between 51 and 65, inclusive.
Therefore, the two conditions are NOT sufficient, when considered together.
The answer is 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.


Answer: E

[GMAT math practice question]

What is the value of x?

1) x^{2x}=1
2) x^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.

00 lies outside the scope of the GMAT exam.

Condition 1)
There are two possible solutions: x = 1 and x = -1. Since the solution is not unique, this condition is not sufficient.

Condition 2)
We have the unique solution x = 1. Since this solution is unique, this condition is sufficient.

Therefore, the answer is 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.
Answer: B

[GMAT math practice question]

For which value of x will y=ax^2+100x+b have a maximum in the xy-plane?

1) b=100
2) a=-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, and then recheck the numbers of variables and equations.

We can modify the original condition and question as follows:

If a < 0, the function will have a maximum at x = (-100)/(2a) = (-50)/a.
If a > 0, the function has no maximum. So, to answer the question, we need to find the value of a.
Thus, condition 2) is sufficient.

Note: condition 1) cannot be sufficient as it provides no information about the value of a.

Therefore, the answer is B.

Answer: B

[GMAT math practice question]

At a certain university, 75% of students have driver’s licences and 60% of the students with driver’s licences are foreign students. What percentage of the total students are foreign students without a driver’s licence?

1) There are 5,000 students at the university.
2) 55% of students at the university are foreigners.

=>
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, and then recheck the question.



45s + 30s + b + d = 100s.
b + d = 25s.

The question asks for the value of b / 100s.

To solve problems involving 2x2 matrices and percentages, we need 3 percentages. Since we have 2 percentages, 75% and 60%, one more percentage is required.

Condition 1) does not provide a percentage, while condition 2) gives us a percentage. So, the answer is most likely to be B.


Condition 1)
100s = 5,000.
s = 50.
This is not sufficient.

Condition 2)
45s + b = 55s
b = 10s
Thus (b/100s)*100 = (10s/100s)*100 = 10 %.
This is sufficient.

Therefore, the answer is B.

Answer: B

[GMAT math practice question]

If n is an integer and n>1,000^2-999^2, n=？

1) n<1,001^2-1,000^2
2) n<502^2-500^2

=>

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, and then recheck the question

We can modify the original condition as follows:

n>1,000^2-999^2
⇔ n > (1000+999)(1000-999)
⇔ n > 1999

Since we have 1 variable (n) and 0 equations, D is most likely to be the answer.

Condition 1)

n<1,001^2-1,000^2
⇔ n < (1001+1000)(1001-1000)
⇔ n < 2001

Since the original condition and condition 1) combine to give 1999 < n < 2001, we must have n = 2000.
We have a unique solution.
Therefore, condition 1) is sufficient.

Condition 2)

n<502^2-500^2
⇔ n < (502+500)(502-500)
⇔ n < 1002*2
⇔ n < 2004

Since the original condition and condition 2) combine to give 1999 < n < 2004, n = 2000, 2001, 2002, or 2003.
We don’t have a unique solution.
So, condition 2) is not sufficient.

Therefore, the answer is 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.

Answer: A

[GMAT math practice question]

m and n are positive integers. If p and q are prime numbers, how many factors does p^mq^n have?

1) m=2 and n=3
2) p=11 and q=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 4 variables (p,q,m, and n) and 0 equations, the answer is most likely to be E.

As E is the most likely answer, we should consider both conditions 1) and 2) together before considering each of them individually. If they are not sufficient when taken together, E is the answer.

Conditions 1) & 2)
The two conditions yield p^mq^n = 11^2x3^3.
Since 11 and 13 are different prime numbers, the number of factors is (2+1)(3+1) = 12.
Both conditions are sufficient, when taken together.

Since this is an integer question (one of the key question areas), we should also consider choices A and B by CMT 4(A).

Condition 1)
This condition does not tell us whether p=q. Therefore, it is not sufficient.

Condition 2)
Since this condition does not tell us the values of the exponents, we can’t determine the number of factors. This condition is not sufficient

Note: It is important to look out for the word, “different” in factor or prime factor questions.
For example, suppose we are told that m and n are positive integers, and that p and q are different prime numbers. If we are then asked how many factors p^mq^n has, and given the conditions
1) m=2 and n=3
2) p=11 and q=13,
the answer will be A. As we know that the number of factors of p^mq^n is (m+1)(n+1), the information provided by Condition 1) is sufficient (Condition 2) gives us no information about the exponents, and so is not sufficient).
Answer: C

[GMAT math practice question]

If x and y are integers, is x^2+x+y an even number?

1) x=5
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, and then recheck the question.

Since x^2+x=x(x+1) is always even, the question is really asking, “is y even?”. Condition 1) tells us nothing about the value of y, so it is not sufficient. Since condition 2) tells us that y = 5, it is sufficient, and the answer is B.
Condition 2)
Setting y = 5 yields
x^2+x+y = x^2+x+5 = x(x+1) + 5,
which is odd since x(x+1) is the product of two consecutive integers, one of which must be even.
The answer is ‘no’.
This is sufficient by CMT(Common Mistake Type) 1.

Therefore, the answer is B.
Normally, in problems which require 2 or more additional 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).

Answer: B

[GMAT math practice question]

Is x^2-y^2 < x-y?

1) x - y < 0
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 method is to modify the original condition and the question, and then recheck the question.

Modifying the question yields
x^2-y^2 < x-y?
⇔ x^2 - y^2 – ( x - y ) < 0?
⇔ ( x + y )( x – y ) – ( x - y ) < 0?
⇔ ( x + y – 1 )( x – y ) < 0?

Since we have 2 variables (x and y) and 0 equations, C is most likely to be the answer, and so we should consider conditions 1) & 2) together first.

Conditions 1) & 2)
Since x – y < 0, asking if ( x + y – 1 )( x – y ) < 0 is equivalent to asking if x + y – 1 > 0.
For x = 1, y = 1, x + y – 1 > 0 and the answer is ‘yes’.
For x = 1/4, y = 1/4, x + y – 1 < 0 and the answer is ‘no’.

Thus, both conditions together are not sufficient.

Therefore, the answer is 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.

Answer: C

[GMAT math practice question]

|x-y|=?

1) x and y are different integers
2) xy=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.

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.

Condition 1) & 2)
Since x and y are different integers, xy = 4 has the following four solutions:
x = 1, y = 4
x = 4, y = 1
x = -1, y = -4
x = -4, y = -1
In all cases, |x-y| = 3.
Thus, both conditions 1) and 2) together are sufficient.

Therefore, C is the answer.

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.

Answer: C

[GMAT math practice question]

Is x= a^b>0?

1) a is positive
2) b is a prime 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.

The first step of the VA method is to modify the original condition and the question, and then recheck the question.

Condition 1)
If a > 0, then ab>0. The answer is ‘yes’.
So, this condition is sufficient.

Condition 2)
For a = 2, b = 2, we have a^b = 2^2 = 4 > 0, and the answer is ‘no’.
For a = -2, b = 3, we have a^b = (-2)^2 = -8 < 0, and the answer is ‘yes’.
So, condition 2) is not sufficient.
Therefore, the answer is A.
Normally, in problems which require 2 or more additional 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).

Answer: A

[GMAT math practice question]

Is 1+x+x^2+x^3+x^4<1/(1-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.

The first step of VA(Variable Approach) method is modifying the original condition and the question, and rechecking the number of variables and the number of equations.

Modifying the question:
1+x+x^2+x^3+x^4<1/(1-x)
⇔ (1+x+x^2+x^3+x^4)(1-x)2<(1-x)
⇔ (1-x^5)(1-x)<(1-x)
⇔ 1-x^5-x+x^6<1-x
⇔ x^6-x^5<0
⇔ x^5(x-1)<0
⇔ x(x-1)<0
⇔ 0<x<1

The question asks if 0<x<1.
This question has the unique answer, ‘yes’, if we require both x>0 and x<1.
Thus, both conditions together are sufficient.

Therefore, C is the answer.

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.
Answer: C