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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 7728
GMAT 1: 760 Q51 V42 GPA: 3.82
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[GMAT math practice question]

If x is a positive integer, is x/30 a terminating decimal?

1) x is divisible by 3
2) x is divisible 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.

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

Note that 30 = 2*3*5. Therefore, for x/30 to be a terminating decimal, 3 must be a factor of x (then the 3 from the denominator will cancel out with a 3 from the numerator). So, the question is asking whether x is a multiple of 3.

As this is precisely condition 1), condition 1) is sufficient.

Note that condition 2) does not tell us whether x is a multiple of 3.

Therefore, A is the answer.
Answer: A

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: 7728
GMAT 1: 760 Q51 V42 GPA: 3.82
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[GMAT math practice question]

If x and y are positive integers and x>y, is x/y an integer?

1) x is a multiple of 10
2) y is a multiple of 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 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):
If x = 10 and y = 2, x / y = 5 is an integer.
If x = 10 and y = 4, x / y = 10 / 4 = 5 / 2 is not an integer.
Since we don’t have a unique solution, 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.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 7728
GMAT 1: 760 Q51 V42 GPA: 3.82
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[GMAT math practice question]

abc0 is a positive, four-digit integer, where a, b and c are 1-digit positive integers. Is abc0 divisible by 4?

1) ab is an odd number
2) bc is an odd 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 (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

We can determine whether an integer is divisible by 4 from its final two digits.
Thus, abc0 is divisible by 4 if and only if c0 is divisible by 4.
Condition 1) tells us nothing about the value of c, so it is not sufficient.
Condition 2) tells us that both b and c are odd numbers.
If c is an odd number, then 10*c + 0 = 10*c is not a multiple of 4, and so abc0 is not a multiple of 4. The answer is ‘no’.
Since ‘no’ is also a unique answer by CMT (Common Mistake Type) 1, condition 2) is sufficient.

Therefore, B is the answer.

Answer: B
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 7728
GMAT 1: 760 Q51 V42 GPA: 3.82
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[GMAT math practice question]

What is the range of the 5 numbers x, y, 10, 15, and 20?

1) x and y lie between 10 and 20, inclusive.
2) The average (arithmetic mean) of the five numbers is 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.
Since we have 2 variables (x and y) and 0 equations, C is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.
Conditions 1) & 2):
Since we have 10 ≤ x ≤ 20 and 10 ≤ y ≤ 20 by condition 1), the maximum of the numbers is 20 and the minimum of the numbers is 10.
The range is the difference between the maximum and the minimum, which is 20 – 10 = 10.
Both conditions together are sufficient.
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)
Since 10 ≤ x ≤ 20 and 10 ≤ y ≤ 20 by condition 1), the maximum of the numbers is 20 and the minimum of the numbers is 10.
The range is the difference between the maximum and the minimum, which is 20 – 10 = 10.
Thus, condition 1) is sufficient.

Condition 2)
If x = 10 and y = 20, the range is 10.
If x = 9 and y = 21, the range is 12.
Since we don’t have a unique solution, 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: 7728
GMAT 1: 760 Q51 V42 GPA: 3.82
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[GMAT math practice question]

A school employs math teachers and physics teachers. If 4 of these teachers are selected randomly, what is the probability that at least one math teacher is selected?

1) The ratio of the number of physics teachers to the number of math teachers is 2 to 1.
2) The sum of the number of physics teachers and the number of math teachers is 24.

=>

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

Let m be the number of math teachers and p be the number of physics teachers. Since we have 2 variables 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) yields p : m = 2 : 1 => p = 2m.
Condition 2) gives m + p = 24.
Combining these equations gives m + p = m + 2m = 3m = 24.
So, m = 8 and p = 16.
The probability that at least one math teacher is selected is 1 – pC4 / m+pC4 = 1 - 16C4 / 24C4.
Both conditions together are sufficient.

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) (p = 2m)
If m = 8 and p = 16, the probability is 1 – pC4 / m+pC4 = 1 - 16C4 / 24C4.
If m = 4 and p = 8, the probability is 1 – pC4 / m+pC4 = 1 - 8C4 / 12C4.
These values are different. Since we don’t have a unique solution, condition 1) is not sufficient.

Condition 2) (m + p = 24)
If m = 8 and p = 16, the probability is 1 – pC4 / m+pC4 = 1 - 16C4 / 24C4.
If m = 12 and p = 12, the probability is 1 – pC4 / m+pC4 = 1 - 12C4 / 24C4.
These values are different. 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: 7728
GMAT 1: 760 Q51 V42 GPA: 3.82
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[GMAT math practice question]

Katrina draws a regular n-sided polygon. What is the value of n?

1) One internal angle of the polygon is 108
2) The sum of all the side lengths is 120

=>

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)
The angle sum of the n-sided polygon is ( n – 2 ) * 180 = n * 108.
So, 72*n = 360 o or n = 5.
Condition 1) is sufficient.

Condition 2)
If the length of each side is 24, then n = 120/24 = 5.
If the length of each side is 20, then n = 120/20 = 6.
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: 7728
GMAT 1: 760 Q51 V42 GPA: 3.82
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[GMAT math practice question]

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

1) x>0
2) y>0

=>

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

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

The original condition x^2 > y^2 is equivalent to |x| > |y|.

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

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

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

Therefore, A is the answer.

Answer: A

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 7728
GMAT 1: 760 Q51 V42 GPA: 3.82
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[GMAT math practice question]

Is n an integer?

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

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

Since we have 1 variable (n) and 0 equations, D is most likely to be the answer. So, we should consider each of the conditions on their own first.

Condition 1)

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

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

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

Therefore, E is the answer.

Answer: E

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 7728
GMAT 1: 760 Q51 V42 GPA: 3.82
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[GMAT math practice question]

What is the remainder when a positive integer n is divided by 36?

1) The remainder when n is divided by 12 is 5.
2) The remainder when n is divided by 18 is 11.

=>

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 the divisor (12) of condition 1) is not a multiple of the divisor (36) of the question, condition 1) is not sufficient.
For example, if n = 5, the remainder when n is divided by 36 is 5.
If n = 17, the remainder when n is divided by 36 is 17.
Since we don’t have a unique solution, condition 1) is not sufficient.
Condition 2)
Since the divisor (18) of condition 2) is not a multiple of the divisor (36) of the question, condition 2) is not sufficient.
For example, if n = 11, the remainder when n is divided by 36 is 11.
If n = 29, the remainder when n is divided by 36 is 29.
Since we don’t have a unique solution, condition 2) is not sufficient.
Conditions 1) & 2)
The first integer satisfying both conditions is 29, and the lcm of 12 and 18 is 36. Therefore, the integers satisfying both conditions are 29, 65, 101, ….
Each of these integers has a remainder of 29 when it is divided by 36.
Both conditions together are sufficient.

Therefore, C is the answer.
Answer: C

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 7728
GMAT 1: 760 Q51 V42 GPA: 3.82
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[GMAT math practice question]

Jayden draws a regular n-gon. What is the value of n?

1) Each interior angle of the n-gon is greater than 115°.
2) Each interior angle of the n-gon is less than 125°.

=>

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.
Each interior angle of a regular n-gon measures (n-2)*180° / n.

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)
(n-2)*180° / n > 115°
=> n*180° - 360° > n*115°
=> n*65° > 360°
=> n > 360° / 65°.
So, we must have
n > 5
=> n ≥ 6, since n is an integer.
Since we don’t have a unique solution, condition 1) is not sufficient.

Condition 2)
(n-2)*180° / n < 125°
=> n*180° - 360° < n*125°
=> n*55° < 360°
=> n < 360° / 55°
So, we must have
n < 7, since n is an integer
Since we don’t have a unique solution, condition 2) is not sufficient.

Conditions 1) & 2)
When we consider both conditions together, we have n ≥ 6 and n ≤ 6.
Thus, n = 6.
Both conditions together are sufficient.

Therefore, C is the answer.

Answer: C

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 7728
GMAT 1: 760 Q51 V42 GPA: 3.82
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[GMAT math practice question] 5.22

x+y=?

1) 3x=y
2) √x-1 =-|3-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.

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)
√x-1 =-|3-y| implies that √x-1 + |3-y| = 0.
Since x – 1 ≥ 0, | 3 – y | ≥ 0 and √x-1 + |3-y| =0, we must have √x-1 =0 and |3-y| = 0.
Thus x = 1, y = 3 and x + y = 4.
Both conditions 1) & 2) are sufficient, when taken together.

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

Condition 1)
If x = 1 and y = 3, then x + y = 4.
If x = 2 and y = 6, then x + y = 8.
Since we don’t have a unique solution, condition 1) is not sufficient.

Condition 2)
√x-1 =-|3-y| implies that √x-1 + |3-y| = 0.
Since x – 1 ≥ 0, | 3 – y | ≥ 0 and √x-1 + |3-y| =0, we must have √x-1 =0 and |3-y| = 0.
Thus x = 1, y = 3, and x + y = 4.
So, condition 2) is sufficient.

Therefore, the answer is B.
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.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 7728
GMAT 1: 760 Q51 V42 GPA: 3.82
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[GMAT math practice question]

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

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

=>

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

Suppose d is the standard deviation of the data set and r is the range of the data set. Then d ≤ r. Since we have many variables, E is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
Since d ≤ r, we have d ≤ 500 < 1000.
Both conditions are sufficient, when taken together.

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

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

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

Therefore, B is the answer.

Answer: B
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 7728
GMAT 1: 760 Q51 V42 GPA: 3.82
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[GMAT math practice question]

If x and y are different positive integers, what is the value of x+y?

1) x^2+y^2=25
2) xy = 12

=>
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.

Condition 1) & 2)
( x + y )^2 = x^2 + y^2 + 2xy = 25 + 2*12 = 25 + 24 = 49
x + y = ±7
Since x and y are positive, x + y = 7.
Both conditions, taken 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 x is a positive integer, there are 5 possible values for x.
Case 1: x = 1
We have x^2 + y^2 = 1 + y^2 = 25 or y^2 = 24. There is no integer solution.

Case 2: x = 2
We have x^2 + y^2 = 4 + y^2 = 25 or y^2 = 21. There is no integer solution.

Case 3: x = 3
We have x^2 + y^2 = 9 + y^2 = 25 or y^2 = 16. Since y is positive, y= 4, and x + y = 3 + 4 = 7.

Case 4: x = 4
We have x^2 + y^2 = 16 + y^2 = 25 or y^2 = 9. Since y is positive, y= 3, and x + y = 4 + 3 = 7.

Case 5: x = 5
We have x^2 + y^2 = 25 + y^2 = 25 or y^2 = 0. There is no positive integer solution.

Thus, we have a unique solution: x + y = 7.

Condition 1) is sufficient.

Condition 2)
If x = 1 and y = 12, then x + y = 13.
If x = 2 and y = 6, then x + y = 8.
Since we don’t have a unique solution, 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: 7728
GMAT 1: 760 Q51 V42 GPA: 3.82
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[GMAT math practice question]

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

1) x=2
2) y=3

=>

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

Only condition 2) gives us a value for y.
Therefore, B is the answer.
Answer: B
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[GMAT math practice question]

What is the median of 3 consecutive integers?

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

=>

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

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

Since we have 1 variable (n) and 0 equations, D is most likely to be the answer. So, we should consider each of the conditions on its own first.

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

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

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

Therefore, C is the answer.

Answer: C

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.
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Joined: 16 Aug 2015
Posts: 7728
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: Math Revolution Approach (DS)  [#permalink]

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[GMAT math practice question]

p, q, x, and y are positive integers. If p^mq^n=72, what is the value of p+q?

1) m>n
2) mn=6

=>

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 1 equation, 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):
We can write 72 = 2^33^2=1^672^1. (find the “hidden 1”)
If p = 2, q = 3, m = 3 and n = 2, then p + q = 5.
If p = 1, q = 72, m = 6 and n = 1, then p + q = 73.
Since we don’t have a unique solution, both conditions, taken 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.
_________________
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Joined: 16 Aug 2015
Posts: 7728
GMAT 1: 760 Q51 V42 GPA: 3.82
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[GMAT math practice question]

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

1) x>0
2) x<1

=>

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

Since we have 1 variable (x) and 0 equations, D is most likely to be the answer. So, we should consider each of the conditions on their own first.

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

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

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

Therefore, C is the answer.
Answer: C

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 7728
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: Math Revolution Approach (DS)  [#permalink]

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[GMAT math practice question]

Is xy<0?

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

=>

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

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.
In order to solve this type of question within a limited time, we should memorize the property that | x + y | < |x| + |y| is equivalent to xy < 0.
So, condition 1) is sufficient.

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

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

Therefore, A is the answer.

Answer: A
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 7728
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: Math Revolution Approach (DS)  [#permalink]

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[GMAT math practice question]

What is the largest of three consecutive even integers?

1) The smallest integer is 6.
2) Their average is 8.

=>

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

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

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

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

Therefore, the answer is D.

Answer: D

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

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 7728
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: Math Revolution Approach (DS)  [#permalink]

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[GMAT math practice question]

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

1) ac > 0
2) bc > 0

=>

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

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

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

Thus, condition 1) is sufficient.

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

Therefore, A is the answer.

Answer: A
_________________ Re: Math Revolution Approach (DS)   [#permalink] 08 Jun 2018, 02:41

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