Math Revolution Approach (DS)

[GMAT math practice question]

If the average (arithmetic mean) of 100 numbers is 50, what is the standard deviation of the numbers?

1) The smallest number is 50
2) The largest number is 50

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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. We often encounter this type of question in the GMAT quant exam these days. If the mean is equal to either the maximum or the minimum, or the range ( = Max – Min ) is zero, then the standard deviation is zero.

Condition 1)
All data items are greater than or equal to 50, and their average is 50.
It means that all data items are equal to 50.
Since all of the data items are equal, their standard deviation is 0.
This is sufficient.

Condition 2)
All data items are less than or equal to 50, and their average is 50.
It means that all data items are equal to 50.
Since all of the data items are equal, their standard deviation is 0.
This is sufficient too.

Therefore, the answer is D.

Answer: D

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.

[GMAT math practice question]

If the average (arithmetic mean) price of apples, bananas and oranges is $3.00 per pound, what is their median price?

1) The price of apples is $3.00 per pound.
2) The price of bananas is $2.97 per pound.

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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 (a for apples, b for bananas and o for oranges) and 1 equation ( ( a + b + o ) / 3 = 3), 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.

This question is a new type of GMAT question. The average is provided in the original condition and the question asks for the value of the median. The same average applies for each of the conditions.

Conditions 1) & 2)
If a = 3.00 and b = 2.97, then ( a + b + o ) / 3 = 3.
So,
3.00 + 2.97 + o = 9
and
o = 3.03. Therefore, the median is 3.00.
Thus, both conditions together are sufficient.

Condition 1)
We consider three cases.
Case 1: a = b = c = 3.00
Since all prices are the same, the median is 3.00.

Case 2: b < 3.00
If b < 3.00, then we must have c > 3.00.
Therefore, the median is 3 since b < a < c.

Case 3: b > 3.00
If b > 3.00, then we must have c < 3.00.
Therefore, the median is 3 since c < a < b.

Thus, condition 1) is sufficient on its own.

Condition 2)
If a = 3.00, b = 2.97 and c = 3.03, then the median is 3.00.
If a = 2.00, b = 2.97 and c = 3.00, then the median is 2.97.
Since we don’t have a unique solution, condition 2) is not sufficient on its own.

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.

[GMAT math practice question]

Is x>0?

1) (x+y)^2 > (x-y)^2
2) x+y > x-y

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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) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
Condition 1) tells us that
(x+y)^2 > (x-y)^2
=> x^2 + 2xy + y^2 > x^2 - 2xy + y^2
=> 2xy > -2xy
=> 4xy > 0
=> xy > 0

Condition 2) tells us that
x + y > x – y
=> y > -y
=> 2y > 0
=> y > 0.

Since xy > 0 and y > 0, we have x > 0.

Thus, both conditions 1) & 2) together are sufficient.

In general, there are many questions involving integers and statistics to which we need to apply CMT(Common Mistake Type) 4.

Condition 1):
If x = -2 and y = -1, then the answer is “yes”.
If x = 1 and y = 2, then the answer is “no”.

Thus, condition 1) is not sufficient.

Condition 2):
If x = -3 and y = -1, then the answer is “yes”.
If x = -1 and y = -3, then the answer is “no”.
Thus 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.

[GMAT math practice question]



Events A, B, C and D are all possible outcomes of an experiment. In the table above, P(x) denotes the probability that event x occurs. What is P(A)?
1) P(A)+P(B)+P(C)=0.75
2) P(C)+P(D)=0.25

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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. We then recheck the question.

Since P(A) = a = 1 – ( c + d ) – 0.25 = 0.75 – (c + d), then P(A) can be determined from the value of c + d. Thus, the answer is B.

Condition 1)
If P(A) = 0.5, P(B) = 0.25 and P(C) = 0, then we have P(A) = 0.5.
If P(A) = 0.4, P(B) = 0.25 and P(C) = 0.1, then we have P(A) = 0.4.
Thus, condition 1) is not sufficient.

Condition 2)
Since P(A) + P(B) + P(C) + P(D) = 1, P(B) = 0.25 and P(C) + P(D) = 0.25 from condition 2), we have P(A) = 0.5.
Thus, condition 2) is sufficient.

Therefore, the answer is B.
Answer: B

[GMAT math practice question]

If the elements of set X are a, b, c and d, is the average (arithmetic mean) of a, b, c, and d contained in set X?

1) The average (arithmetic mean) of every pair of elements of set X is 10.
2) The average (arithmetic mean) of every three elements chosen from set X 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.

Since we have 4 variables (a, b, c and d) and 0 equations, E is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) and 2):
Condition 1) gives rise to the following equations:
( a + b ) / 2 = 10 => a + b = 20
( a + c ) / 2 = 10 => a + c = 20
…
( c + d ) / 2 = 10 => c + d = 20

Condition 2) gives rise to the following equations:
( a + b + c ) / 3 = 10 => a + b + c = 30
( a + b + d ) / 3 = 10 => a + b + d = 30
( a + c + d ) / 3 = 10 => a + c + d = 30
( b + c + d ) / 3 = 10 => b + c + d = 30

Combining these equations yields a = b = c = d = 10.
Therefore, the average is 10, which is an element of set X, and conditions 1) and 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)
Condition 1) gives rise to the following equations:
( a + b ) / 2 = 10 => a + b = 20
( a + c ) / 2 = 10 => a + c = 20
…
( c + d ) / 2 = 10 => c + d = 20

Combining these equations yields a = b = c = d = 10.

Condition 2)

Condition 2) gives rise to the following equations:
( a + b + c ) / 3 = 10 => a + b + c = 30
( a + b + d ) / 3 = 10 => a + b + d = 30
( a + c + d ) / 3 = 10 => a + c + d = 30
( b + c + d ) / 3 = 10 => b + c + d = 30

Combining these equations yields a = b = c = d = 10.

Therefore, the answer is D.

Answer: D

Since conditions 1) and 2) are equivalent, D is the answer by Tip 1) of the VA method.

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.

[GMAT math practice question]

When a positive integer n is divided by 5, the remainder is 2. What is the remainder when n is divided by 3?

1) n is divisible by 2
2) When n is divided by 15, the remainder is 2.

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

Plugging-in numbers is the suggested approach to remainder questions.

Condition 1)
The possible values of n are
n = 2, 4, 6, 8, …
When these are divided by 3, the remainders are 0, 1 and 2.
Since the answer is not unique, condition 1) is not sufficient.

Condition 2)
The possible values of n are
n = 17, 32, 47, 62, …
When these are divided by 3, the remainder is always 2.
Since the answer is unique, condition 2) is sufficient.

Therefore, B is the answer.

Answer: B

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E. 

[GMAT math practice question]

Is the total of the sales prices of 3 products greater than $23?

1) The price of the cheapest of the 3 products is at least $8
2) The price of the second cheapest of the 3 products is at least $12.

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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.

If a question includes the expression “greater than”, we should determine the smallest possible value, since all other possibilities are greater than the minimum. Each of the conditions includes the expression “at least”, so this gives us some minimum values.

Condition 1)
Since the cheapest of the three products costs at least $8, the minimum total cost of the three products is
$8 + $8 + $8 = $24 > $23.
Thus, condition 1) is sufficient.

Condition 2)
Since the second cheapest of the three products costs at least $12, the minimum total cost of the three products is
$0 + $12 + $12 = $24 > $23.
Thus, condition 2) is sufficient.

Answer: D

[GMAT math practice question]

Is x>0?

1) x^3+x^2+x = 1
2) x^2 – 4x - 5 > 0

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

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

Condition 1):
x^3+x^2+x = 1
=> x(x^2+x+1) = 1
=> x = 1 / (x^2+x+1)
=> x = 1 / (x^2+x+1) > 0 since x2+x+1 > 0.

Thus, condition 1) is sufficient.


Condition 2):
x^2 – 4x -5 > 0
=> (x+1)(x-5) > 0
=> x < 1 or x > 5

Since the solution set of the inequality, x > 0, from the question does not include the solution set of the inequality from condition 2), x < 1 or x > 5, 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.

[GMAT math practice question]

Each of three consecutive positive integers is less than 100. Their sum of is a multiple of 10. What is the smallest of the three integers?

1) Their median is a multiple of 9
2) Two of the integers 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.

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

Let the integers be x – 1, x and x + 1 where x is a positive integer. Since their sum, x – 1 + x + x + 1 = 3x is a multiple of 10, x must be a multiple of 10.

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)
The median, x, of the three integers is a multiple of 9 and a multiple of 10.
Since x < 100, we must have x = 90.
Therefore, the smallest of the integers is x – 1 = 89.
Thus, condition 1) is sufficient.

Condition 2)
If x = 30, the integers are 29, 30, 31 and the smallest integer is 29.
If x = 90, the integers are 89, 90, 91 and the smallest integer is 89.
Since we do not obtain a unique answer, condition 2) is not sufficient.


Therefore, the answer is A.

Answer: A

[GMAT math practice question]

When a and b are positive integers, what is the greatest common divisor of a + b and a + 100?

1) a = 100
2) b = 99

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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. We then recheck the question.


Condition 1)
If a = 100, and b = 98, the greatest common divisor of a + b and a + 100 is 2.
If a = 100, and b = 99, the greatest common divisor of a + b and a + 100 is 1.
Thus, condition 1) is not sufficient as it does not yield a unique solution.

Condition 2)
a + b = a + 99 and a + 100 are consecutive integers. Therefore, their greatest common divisor is 1.
Thus, condition 2) is sufficient.


Therefore, the answer is B.
Answer: B

[GMAT math practice question]

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

1) x+3y is even
2) (x-1)(y-1) 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.

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 both conditions are satisfied only when x is even and y is even, x + y must be an even number.
Thus, both conditions together are sufficient.

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

Condition 1)
There are two ways in which x + 3y can be even.
If x and y are both even, then x + y is even.
If x and y are both odd, then x + y is even too.
Thus, condition 1) is sufficient.


Condition 2)
Since (x-1)(y-1) is odd, both x -1 and y – 1 must be odd.
So, x and y must both be even.
It follows that x + y is even.
Thus, condition 2) is sufficient too.

Therefore, D is the answer.

Answer: D

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.

[GMAT math practice question]

A bag contains only black and white balls. What is the probability that a ball chosen randomly from the bag is black?

1) The total number of balls in the bag is 12
2) The number of white balls is 3 times the number of black balls

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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.

We can modify the original condition and question as follows.

Assume b and w are the numbers of black and white balls, respectively.
Then b + w = 12.
The question asks for the value of b / ( b + w ).

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) together:
b + w = 12
w = 3b

Combining these equations yields b + 3b = 4b = 12 or b = 3.
So, w = 9.
Thus, b / ( b + w ) = 3 / 12 = 1/4
Both conditions together are sufficient.

Since this question is a probability 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) : b + w = 12
Since we can't determine the value of b, we can't determine b / ( b + w ) = b / 12.
Condition 1) is not sufficient.

Condition 2) : w = 3b
b / ( b + w ) = b / ( b + 3w ) = b / 4b = 1/4.
Condition 2) is sufficient.

Therefore, B is the answer.

Answer: B

When a question asks for a ratio, if one condition includes a ratio and the other condition just gives a number, the condition including the ratio is most likely to be sufficient. This tells us that B is most likely to be the answer to this question.


Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.

[GMAT math practice question]

What is the median of m, n and 5?

1) m+n=10
2) m=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.

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) together:
Since m + n = 10 and m = 5, we have m = n = 5.
Thus, the median of m, n and 5 is 5.

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): m + n = 10
If one of m and n is less than 5, the other one must be greater than 5. For example, if m < 5, then n > 5.
If one of m and n is 5, the other must also equal 5. For example, if m = 5, then n = 5.
In both cases, the median is 5.
Condition 1) is sufficient.

Condition 2) : m = 5.
If n > 5, the numbers are 5, 5, n in ascending order.
If n < 5, the numbers are n, 5, 5 in descending order.
If n = 5, the numbers are 5, 5, 5.
In each of the possible cases, the median is 5.
Condition 2) is sufficient.

Therefore, D is the answer.

Answer: D

Note: The VA approach tells us that the answer is most likely to be D, since this is a CMT(Common Mistake Type) 4B question.
Condition 1) is easy to check, but condition 2) is more difficult to work with. If you can’t figure out condition 2), you should choose D as the answer.

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.

[GMAT math practice question]

The students at a school took a math exam. Is the average (arithmetic mean) score for the exam higher than the median score?

1) The average (arithmetic mean) score is 75.
2) 51% of the students attained more than the average (arithmetic mean) score.

=>
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 many variables and 0 equations, E is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.


Conditions 1) & 2)
The median score is attained or exceeded by 50% of students.
Since 51% of students scored more than the average, the median is more than the average.
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 we don't know the median, condition 1) is not sufficient.

Condition 2)
Since 51% of students scored more than the average, the median is more than the average.
Condition 2) is sufficient.

Therefore, B is the answer.

Answer: B

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2), when taken together. Obviously, there may be occasions on which the answer is A, B, C or D.

[GMAT math practice question]

Is median of a, b and c equal to their average (arithmetic mean)?

1) a≤b≤c
2) b = ( a + c ) / 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 3 variables and 0 equations, E is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
b is the median of a, b and c since a≤b≤c.
b is the average of a, b and c since b = ( a + c ) / 2.
Thus, the median and the average of a, b and c are the same.

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)
If a = 1, b = 2, and c = 3, the average and the median are 5.
If a = 1, b = 2, and c = 6, the average is 3 and the median is 2. The average and median are different.
Thus, condition 1) is not sufficient.

Condition 2) :
b is the median of a, b and c since b = ( a + c ) / 2 ⇔ a + c = 2b.
The average of a, b and c is ( a + b + c ) / 3 = ( 2b + b ) / 3 = 3b/3 = b.
Thus, the average and the median are the same.
Condition 2) is sufficient.

Therefore, B is the answer.

Answer: B

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2), when taken together. Obviously, there may be occasions on which the answer is A, B, C or D.

[GMAT math practice question]

Alice works from Monday to Friday only. How many days does she work in April?

1) April 1 is Saturday or Sunday
2) April 28 is Saturday or Sunday

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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. We then recheck the question.
When the day on which a specific date, for example, the first of April 2018 is fixed, we can determine the days on which other dates in April fall.
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 April 1st is Saturday, Alice works on the 3rd - 7th, 10th - 14th, 17th - 21st, 24th - 28th. This means she works on 20 days.
If April 1st is Sunday, Alice works on the 2nd - 6th, 9th - 13th, 16th - 20th, 23rd - 27th and 30th. This means she works on 21 days.
Since we don't have a unique solution, condition 1) is not sufficient.

Condition 2)
If April 28th is Saturday, Alice works on the 2nd - 6th, 9th - 13th, 16th - 20th, 23rd - 27th and 30th. This means she works on 21 days.
If April 28th is Sunday, Alice works on the 1st - 5th, 8th - 12th, 15th - 19th, 22nd - 26th, 29th and 30th. This means she works on 22 days.

Since we don't have a unique solution, condition 2) is not sufficient.
Conditions 1) & 2) together:
If the 1st of April is a Saturday or Sunday and the 28th of April is a Saturday or Sunday, the 1st of April must be a Saturday.
So, Alice works on the 2nd - 6th, 9th - 13th, 16th - 20th, 23rd - 27th and 30th. Therefore, she works on 21 days.
Since we have a unique solution, 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.

[GMAT math practice question]

On the number line, 0 lies between x and y. Is x>y?

1) The distance between x and 0 is 2 times the distance between y and 0
2) x+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.
We then recheck the question. Since 0 lies between x and y, we must have xy < 0.

Since we have 2 variables (x and y) and 0 equations, C is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2):
Condition 1) tells us that |x| = 2|y|.
Thus x = ±2y. However, since xy < 0, x = -2y.

Condition 2) tells us that x + y = (-2y) + y = -y > 0. So, y < 0 and x > 0.
Thus, x > y.
Both conditions together are sufficient.

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

Condition 1): |x| = 2|y|
If x = 2 and y = -1, the answer is “yes”.
If x = -2 and y = 1, the answer is “no”.

Condition 1) is not sufficient.

Condition 2): x + y > 0
If x = 2 and y = -1, the answer is “yes”.
If x = -1 and y = 2, the answer is “no”.

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.