Math Revolution Approach (DS)

[GMAT math practice question]

If x and y are integers, is x^3-y^3 an odd integer?

1) x is an odd number
2) x and y are consecutive integers

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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 and question as follows.
There are two different ways in which x^3-y^3 can be odd:
x is even and y is odd.
x is odd and y is even.

Since condition 2) tells us that x and y are consecutive integers, one of them must be odd, and the other must be even. In both cases, the answer is ‘yes’.
Therefore, condition 2) is sufficient.

As condition 1) gives us no information about y, it is not sufficient.

Therefore, B is the answer.

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]

If x is an integer and |x|+(x/3)<5, what is the value of x?

1) x>-12
2) x<-6

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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 original condition:
There are two cases to consider.
Case 1) x ≥ 0
|x|+(x/3)<5
⇔ x + x/3 < 5
⇔ (4/3)x < 5
⇔ x < 15/4
⇔ x = 0, 1, 2, 3

Case 2) x < 0
|x|+(x/3)<5
⇔ -x + x/3 < 5
⇔ -(2/3)x <5
⇔ x > -15/2
⇔ x > -7.5
⇔ x = -7, -6, -5, -4, -3, -2 or -1

The question asks for the value of x if x is one of -7, -6, -5, …, 0, 1, 2, 3. Since we have 1 variable (x), D is most likely to be the answer.

Condition 1)
All of the possible values of x are greater than -12. Therefore, we do not have a unique solution, and this condition is not sufficient.

Condition 2)
The only possible value of x is -7. Since we have a unique solution, condition 2) is sufficient.

Therefore, B is the answer.

Answer: B

[GMAT math practice question]

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

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

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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

Modifying the original condition gives:

|x+1|=|y+1|
⇔ (x+1)^2=(y+1)^2
⇔ (x+1)^2=(y+1)^2
⇔ (x+1)^2-(y+1)2=0
⇔ (x+1+y+1)(x+1-y-1)=0
⇔ (x+y+2)(x-y)=0
⇔ x+y=-2 or x=y

As we have 2 variables (x and y) and 1 equation in the original condition, D is most likely to be the answer.

Condition 1)
Since xy < 0, x≠y.
So, x + y = -2, and condition 1) is sufficient.

Condition 2)
Since x>1 and y<1, x≠y.
So, x + y = -2.
Condition 2) is sufficient too.

Therefore, the answer is D.

Note: Since conditions 1) and 2) are similar, D is most likely to be the answer by Tip 1).

Answer: D

[GMAT math practice question]

If f(x)=x+√x and z=y^2, is f(z)=y^2+y?

1) z=4
2) y>0

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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.

Modifying the question:
f(z)=y^2+y
⇔ z + √z = y^2 + y
⇔ y2 + √y^2 = y^2 + y
⇔ √y^2 = y
⇔ |y| = y
⇔ y ≥0

Since condition 1) tells us nothing about y, and condition 2) tells us that y > 0, only condition 2) is sufficient.

Therefore, the answer is B.

Answer: B

[GMAT math practice question]

If x<-1, is √(-x)|x|/y=1?

1) y=|x|
2) y=-x

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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.

Modifying the question:
Since x<-1, x is negative and we have |x| = -x. So,
√(-x)√|x|/y=1
⇔√(-x)√(-x)/y=1
⇔√x^2/y=1
⇔|x|/y=1
⇔-x/y=1
⇔ y=-x


Condition 1)
y=|x|
⇔ y=-x
Since this is equivalent to the question, condition 1) is sufficient.

Condition 2)
Since this is also equivalent to the question, condition 2) is sufficient too.

The answer is D.

Note: D is most likely to be the answer if conditions 1) and 2) are equivalent.

Answer: D

[GMAT math practice question]

I, …, E, …, M, …, S, …, K, …, W, … Z
The frequencies of the appearance of the letters from ‘A’ to ‘Z’ in a book are listed in decreasing order as shown above. ‘I’ is the most frequently occurring letter. The book contains a total of 100,000 alphabetic characters. Is the probability of randomly selecting an ‘S’ from the alphabetic characters appearing in the book greater than 1/27?

1) The probability of selecting ‘M’ is greater than 1/27.
2) The probability of selecting ‘K’ is greater than 1/26.

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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 which are the probability of S, P(S), that of M, P(M) and that of K, P(K) and we have 1 equation which is P(M) > P(S) > P(K), C is most likely to be the answer and so we should consider both conditions 1) & 2) together first.

Conditions 1) & 2)
Since P(M) > P(S) > P(K) > 1/26 > 1/27, both conditions together are sufficient.

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

Condition 1)
If P(M) > P(S) = 1/27, the answer is ‘no’.
If P(M) = 1/25 and P(S) = 1/26 > 1/27, the answer is ‘yes’.
Thus, condition 1) is not sufficient.

Condition 2)
Since P(S) > P(K) > 1/26 > 1/27, condition 2) is sufficient.

Therefore, B is the answer.

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]

What is the median value of a list of 5 data values?

1) The 3 highest values are 15, 18, and 20
2) The smallest value is 10

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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:
The median of 5 data values is the third highest value. So, the question is asking for the third highest value.


Condition 1)
Condition 1) tells us that the third highest value is 15.
Therefore, condition 1) is sufficient.

Condition 2)
Condition 2) tells us nothing about the third highest value.
This condition is NOT sufficient.

Therefore, A is the answer.

Answer: A

[GMAT math practice question]

What is the value of (p-q)(p+q)?

1) p-q=5
2) p and q are prime numbers

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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) and 2) together first.

Conditions 1) and 2)
Since p and q are prime numbers satisfying p – q = 5, one of them is an odd prime and the other is an even prime. The unique even prime is 2.
Therefore, q = 2 and p = 7, and
(p-q)(p+q) = (7-2)(7+2) = 5*9 = 45.

Both conditions 1) and 2) together are sufficient.

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.

Answer: C

[GMAT math practice question]

9^{a+2b}/3^{a+3b}=?

1) a+b=2
2) a-b=3

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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.


Modifying the question:
9^{a+2b}/3^{a+3b}=?
⇔ (3^2)^{a+2b}/3^{a+3b}=?
⇔ 3^{2a+4b}/3^{a+3b}=?
⇔ 3^{a+b}=?

Condition 1)
Since the exponent is a + b, this condition is sufficient.

Condition 2)
The value of a – b does not tell us anything about the value of a + b.
For example, if a = 6 and b = 3, a + b = 9; and if a = 7 and b = 4, a + b = 11.
Since there is no unique answer, condition 2) is NOT sufficient.

Therefore, the answer is A.

Answer: A

[GMAT math practice question]

Is a=0?

1) ab=-3a
2) b>0

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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) and 2) together first.

Conditions 1) and 2)
ab = -3a
⇔ ab + 3a = 0
⇔ a(b+3) = 0
⇔ a = 0 since b > 0 and b ≠ -3.

The answer is 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.

Answer: C

[GMAT math practice question]

If x, y, and z are consecutive integers and x<y<z, is y an even number?

1) xz is an odd number
2) xyz is a multiple of 8

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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.

Conditions 1) and 2)
Since xz is an odd number, both x and z are odd integers.
Since xz is an odd number, it follows from condition 2) that y is a multiple of 8.
Therefore, y is an even number.
Conditions 1) and 2) 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)
Since xz is an odd integer, x and z are odd integers. As the three integers are consecutive, y must be even.
Thus, condition 1) is sufficient.

Condition 2)
If x = 7, y = 8, z = 9, the answer is ‘yes’.
If x = 2, y = 3, z = 4, the answer is ‘no’.
Since we do not have a unique answer, condition 2) is NOT sufficient.

Therefore, A is the answer.

Normally, in problems which require 2 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]

If the average (arithmetic mean) of a, b, c, and d is m, is their standard deviation greater than 1?

1) a=1
2) m=4

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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 average of a,b,c and d is
( a + b + c + d ) / 4 = m.
Since we have 5 variables and 1 equation, E is most likely to be the answer. So, we should consider conditions 1) and 2) together first.

Conditions 1) and 2):
Standard deviation
= √ { (a–m)^2 + (b-m)^2 + (c-m)^2 + (d-m)^2 } / 4
≥ √ ( 3^2 / 4 )
= √ ( 9 / 4 )
= 3/2
> 1
Both conditions 1) and 2) together are sufficient.

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

Condition 1):
If a = b = c = d = 1, the standard deviation is 0<1, and the answer is ‘no’.
If a = 1, b = 4, c = 4, d = 7, the standard deviation is 4.5 > 1, and the answer is ‘yes’.
Since we do not have a unique answer, condition 1) is not sufficient.

Condition 2):
If a = b = c = d = 4, m=4, the standard deviation is 0<1, and the answer is
‘no’.
If a = 1, b = 4, c = 4, d = 7, m=4, the standard deviation is 4.5 > 1, and the answer is ‘yes’.
Since we do not have a unique answer, condition 2) is not 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]

If a and b are integers, is a^2+b^3 an odd number?

1) 3a+4b is an odd number
2) a and b are consecutive integers

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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.

We can modify the original condition and question as follows.
There are two different ways in which a2+b3 can be odd:
a is even and b is odd.
a is odd and b is even.

Since condition 2) tells us that a and b are consecutive integers, one of them must be odd, and the other must be even. In both cases, the answer is ‘yes’.
Therefore, condition 2) is sufficient.

Condition 1) implies that a is odd, but tells us nothing about b. Therefore, it is not sufficient.

Therefore, the answer is 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.

Answer: B

[GMAT math practice question]

If p is a prime number and n is a positive integer, what is the number of positive factors of 3^np^2?

1) n=3
2) p is odd.

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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.
Usually, when we encounter questions about numbers of factors and are given a prime factorization, the primes in the prime factorization are different. If p ≠ 3, the number of factors is (n+2)(2+2), which can be determined from n. However, since the question gives no restrictions on p, we should consider the following two cases:

Conditions 1) and 2) together:
Case 1: p ≠ 3, n = 3:
The number of factors is (3+1)(2+1) = 12.

Case 2: p = 3, n = 3
3^nP^2 = 3^33^2 = 3^5
The number of factors is 5+1 = 6.

Since we do not obtain a unique answer, both conditions together are not sufficient.

Therefore, the answer is 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.

Answer: C

[GMAT math practice question]

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

1) y=x^2+3x+2
2) xy is an even number


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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) and 2) together first.

Conditions 1) and 2)
Case 1: x is odd
Since xy is even, y is even.

Case 2: x is even.
Since x^2, 3x and 2 are even, y =x^2+3x+2 is even.
Since we have a unique answer, both conditions together are sufficient.

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):
There are two cases to consider.
Case 1: x is even
Since x^2, 3x and 2 are even, y =x^2+3x+2 is even.

Case 2: x is odd
Since x^2+3x is even and 2 is even, y =x^2+3x+2 is even.
Since we have a unique answer, condition 1) is sufficient.


Condition 2):
If x = 1 and y = 2, y is even.
If x = 2 and y = 1, y is odd.
Since we do not have a unique answer, condition 2) is not sufficient.

The answer is 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.

Answer: A

[GMAT math practice question]

If x and y are integers, is 3x^2+5x+y an even number?

1) x=5
2) y=4

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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) and 2) together first.

Conditions 1) and 2):
Since x = 5 and y = 4, we have 3x^2+5x+y = 3*5^2 + 5*5 + 4 = 75 + 25 + 4 = 104, which is even.
Thus, both conditions together are sufficient.

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):
If x = 5 and y = 1, then
3x^2+5x+y =3*5^2 + 5*5 + 1 = 75 + 25 + 1 = 101, which is odd.
If x = 5 and y = 4, then
3x^2+5x+y =3*5^2 + 5*5 + 4 = 75 + 25 + 4 = 104, which is even.
Since we do not have a unique answer, condition 1) is not sufficient.

Condition 2)
There are two cases to consider.
Case 1: x is even.
Since 3x^2 and 5x are even and y is even, 3x^2+5x+y is even.
Case 2: x is odd.
Since 3x^2+5x is even and y is even, 3x^2+5x+y is even.
Since we have a unique answer, condition 2) is sufficient.

Therefore, the answer is 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.

Answer: B

[GMAT math practice question]

The function y=px^2-4x+q in the x-y plane attains a minimum value. What is the value of x?

1) p = 2
2) q = 5

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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.
y=px^2-4x+q has a minimum value when x = -(-4)/2p = 2/p.
Thus, the question asks for the value of p.

Since only condition 1) gives us information about p, only condition 1) is sufficient.
Therefore, A is the answer.
Answer: A