The Ultimate Q51 Guide [Expert Level]

[GMAT math practice question]

If a is an integer, is a-b an integer?

1) 95 is a factor of a
2) b is 20 percent of a

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

Under the condition that a is an integer, if a – b is an integer, b is an integer. Thus, the question asks if b is an integer.

Since we have 2 variables and 0 equation, C is most likely to be the answer.
Condition 1) & 2)
b = 95 * (20/100) = 95 / 5 = 19.
Both conditions are sufficient.
Thus, C is the answer as expected.

Normally for cases where we need 2 more equations, such as original conditions with 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, we have 1 equation each in both con 1) and con 2). Therefore, C has a high chance of being the answer, which is why we attempt to solve the question using 1) and 2) together. Here, there is 70% chance that C is the answer, while E has 25% chance. These two are the key questions. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer according to DS definition, we solve the question assuming C would be our answer hence using con 1) and con 2) together. (It saves us time). Obviously, there may be cases where the answer is A, B, D or E.

Answer: C

Hi,
The question does not say a and b are distinct numbers.
So, a = b = 2 or 4 is also a valid solution for a, b.
And when sq(a) + sq(b) is divided by 3, we get 8/3 or 32/3, respectively, which gives 2 as remainder (as was expected here).
But now, sq(a) - sq(b) becomes 0. So, 0/3 will give remainder 0.
Though, this does not change the answer, yet, we need to be careful in taking this condition into consideration, too.

KS

[GMAT math practice question]

Is x>0?

1) x^3+x-1=0
2) x^3>1

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Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember that equal number of variables and independent equations ensures a solution.
Since we have 1 variable and 0 equation, D is most likely to be the answer.

Condition 1)
x^3+x-1=0
x^3+x = 1
x(x^2+1) = 1
x = 1/(x^2+1) > 0.
This is sufficient.

Condition 2)
x^3 > 1
x^3 – 1 > 0
( x – 1 )( x^2 + x + 1 ) > 0
x – 1 > 0, since x^2 + x + 1 > 0.
x > 1 > 0.
This is sufficient.

Therefore, D is the answer as expected.

For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both con 1) and con 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Thus, D is most likely to be the answer using con 1) and con 2) separately according to DS definition. Obviously, there may be cases where the answer is A, B, C or E.


Answer: D

[GMAT math practice question]

If mn>0, is m^3n^4>0?

1) m>0
2) n>0

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Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember that equal number of variables and independent equations ensures a solution.
Since we have 2 variables and 1 equation, D is most likely to be the answer.
In an inequality question, an inequality in the original condition is counted as an equation.

The question m^3n^4>0? is equivalent to m > 0?, after dividing both sides by m^2n^4. Since mn>0, m>0 is equivalent to n>0 too.

Condition 1)
m > 0 is same as the question.
This is sufficient.

Condition 2)
We have m > 0 from mn > 0 and n > 0. It is same as the question.
This is sufficient too.
By tip 1), D is the answer, due to 1) = 2).
Therefore, D is the answer as expected.

For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both con 1) and con 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Thus, D is most likely to be the answer using con 1) and con 2) separately according to DS definition. Obviously, there may be cases where the answer is A, B, C or E.

Answer: D

[GMAT math practice question]

There are 3 machines with the same work rate. If it took t hours for 3 machines to work and took t+2 hours for 2 machines when working together, what is the value of k?

A. 4
B. 5
C. 6
D. 7
E. 9

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Assume X is the time one machine takes to work.

1/T=3/X and 1/T+2=2/X

X = 3T and X = 2(T+2).
We have 3T = 2T + 4 or T = 4.

Thus, the answer is A.

Answer: A

[GMAT math practice question]

Tom traveled the entire 90 km trip. If he travelled the first 18 km of the trip at a constant rate of 36 km per hour and the remaining trip at a constant rate of 72 km per hour, what is his average speed?

A. 30 km/h
B. 36 km/h
C. 45 km/h
D. 48 km/h
E. 60 km/h

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90 / (18/36 + 72/72) = 90 / 1.5 = 60

Therefore, the answer is E.
Answer: E

[GMAT math practice question]

If the average (arithmetic mean) exam score of Alice, Bob and Cindy is 80, what is their median score of the exam?

1) Alice’s score is 75.
2) Cindy’s score is 80.

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Forget conventional ways of solving math questions. In DS, VA (Variable Approach) method is the easiest and quickest way to find the answer without actually solving the problem. Remember that equal number of variables and independent equations ensures 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.

Assume A, B and C are exam scores of Alice, Bob and Cindy, respectively.
We have ( A + B + C ) / 3 = 80 or A + B + C = 240.
Since we have 3 variable and 1 equation, C is most likely to be the answer.

Conditions 1) and 2)
Since A = 75, C = 80 and A + B + C = 240, we have B = 85.
Then, the median is C = 80.
Since it is a statistics question, one of the key questions, according to CMT4(A), we should consider A or B if C is chosen too easily.

From the condition 1), A = 75, we can’t figure out the median.
From the condition 2), C = 80, we have A + B = 160.
If one of them is less than 80, the other one must be greater than 80 and C = 80 is the median.
Otherwise, A = B = C = 80 and the median is 80.
The condition 2) is sufficient.
Therefore, B is the answer.

Normally for cases where we need 2 more equations, such as original conditions with 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, we have 1 equation each in both 1) and 2). Therefore, C has a high chance of being the answer, which is why we attempt to solve the question using 1) and 2) together. Here, there is 70% chance that C is the answer, while E has 25% chance. These two are the key questions. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer according to DS definition, we solve the question assuming C would be our answer, hence using 1) and 2) together. (It saves us time). Obviously there may be cases where the answer is A, B, D or E.

Ans: B

[GMAT math practice question]

1/x > 1/y?

1) x < y
2) xy > 0

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

When we modify the question, we can multiply both sides of the inequality by x^2y^2, we have x(y^2)>x^2y? or xy(y-x)>0. Condition 1) states y – x > 0 and condition 2) states xy > 0.

Thus, C is the answer as expected.

Normally for cases where we need 2 more equations, such as original conditions with 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, we have 1 equation each in both 1) and 2). Therefore, C has a high chance of being the answer, which is why we attempt to solve the question using 1) and 2) together. Here, there is 70% chance that C is the answer, while E has 25% chance. These two are the key questions. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer according to DS definition, we solve the question assuming C would be our answer, hence using 1) and 2) together. (It saves us time). Obviously, there may be cases where the answer is A, B, D or E.

Answer: C

[GMAT math practice question]

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

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

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Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember that equal number of variables and independent equations ensures a solution

Condition 1)
x – y = ( x + 3y ) - 4y is a difference of even integers, since x + 3y is even and 4y is even. This is sufficient.

Condition 2)
x = 3, y = 1 ➔ x – y = 2 is even.
x = 3, y = 2 ➔ x – y = 1 is odd.
This is not sufficient.
Therefore, the answer is A.

Normally for cases where we need 2 more equations, such as original conditions with 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, we have 1 equation each in both 1) and 2). Therefore, C has a high chance of being the answer, which is why we attempt to solve the question using 1) and 2) together. Here, there is 70% chance that C is the answer, while E has 25% chance. These two are the key questions. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer according to DS definition, we solve the question assuming C would be our answer, hence using 1) and 2) together. (It saves us time). Obviously, there may be cases where the answer is A, B, D or E.
Answer: A

[GMAT math practice question]

If x=-1 and n is the sum of all prime numbers less than 100, what is the value of x^n+x^{n+1}+x^{n+2}+x^{n+3}+x^{n+4}+x^{n+5}?

A. -2
B. -1
C. 0
D. 1
E. 2

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Whatever the value of n is, (-1)^n+(-1)^{n+1}+(-1)^{n+2}+(-1)^{n+3}+(-1)^{n+4}+(-1)^{n+5} = 0, since two consecutive integer n and n+1 always have an odd integer and an even integer, and so (-1)^n+(-1)^{n+1} = 0.

Thus, the answer is C.

Answer: C

[GMAT math practice question]

Is x > 0?

1) x^3+x^2+x<0
2) x^2<1


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

Since we have 1 variable and 0 equation, D is most likely to be the answer.

Conditions 1) x^3+x^2+x<0
x(x^2+x+1)<0 implies x < 0, since x^2+x+1 > 0.
The answer is no.
Since “no” is also an answer according to CMT (Common Mistake Time) 1, this is sufficient.

Condition 2) x^2<1
We have x^2-1 < 0 and (x+1)(x-1) < 0.
Thus, you get -1 < x < 1 according to the “between” rule in inequalities.
This is not sufficient.

Therefore, A is the answer.


For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both con 1) and con 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. D is most likely to be the answer using con 1) and con 2) separately according to DS definition. Obviously, there may be cases where the answer is A, B, C or E.

Ans: A

[GMAT math practice question]

P(A) and P(B) are the probabilities of independent events A and B, respectively. What is P(A)∙P(B)?

1) The probability that event A occurs but event B does not occur is 0.35.
2) The probability that event A occurs is 0.5.

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

Since we have 2 variables P(A) and P(B), and no equation, C is most likely to be the answer.

Since events A and B are independent, P(A∩B) = P(A) ∙P(B) and P(B) = P(A∩B) / P(A).

Condition 1) & 2)
Since P(only A) = 0.35 and P(A) = 0.5, we have P(A) ∙P(B) = P(A∩B) = 0.15.
Both conditions 1) and 2) together are sufficient.


Therefore, the answer is C as expected.


Normally for cases where we need 2 more equations, such as original conditions with 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, we have 1 equation each in both 1) and 2). Therefore, C has a high chance of being the answer, which is why we attempt to solve the question using 1) and 2) together. Here, there is 70% chance that C is the answer, while E has 25% chance. These two are the key questions. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer according to DS definition, we solve the question assuming C would be our answer hence using 1) and 2) together. (It saves us time). Obviously, there may be cases where the answer is A, B, D or E.
Answer: C

[GMAT math practice question]

If f(x+y) = f(x)f(y), what is the value of f(0)?

1) f(x) > 0
2) f(x) has integer values

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Forget conventional ways of solving math questions. In DS, VA (Variable Approach) method is the easiest and quickest way to find the answer without actually solving the problem. Remember that equal number of variables and independent equations ensures 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.

To find the value of f(x), we can plug in 0 for the variables x and y.
Then we have f(0+0) = f(0) ∙ f(0) or f(0) = {f(0)}^2
{f(0)}^2 - f(0) = 0
f(0) { f(0) – 1 } = 0
Then f(0) = 0 or f(0) = 1.
Then f(0) becomes an unknown variable and we have 1 variable. D is most likely to be the answer.
Condition 1)
Since f(0) > 0, we have f(0) = 1.
This is sufficient.

Condition 2)
Both 0 and 1 are integers, f(0) = 0 or f(0) = 1.
Thus, this is not sufficient.

Therefore, A is the answer.
Answer: A

[GMAT math practice question]

The probability that an event A occurs is 1/2 and the probability that an even B occurs is 1/3. If event A and event B are independent, what is the probability that event A occurs but event B does not occur?

A. 1/6
B. 1/3
C. 1/2
D. 2/3
E. 5/6

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Since events A and B are independent, P(A∩B) = P(A) ∙P(B).
The probability P(A-B) = P(A) - P(A∩B) = P(A) - P(A) ∙P(B) = 1/2 – (1/2)(1/3) = 1/2 – 1/6 = 2/6 = 1/3.

Thus, the answer is B.
Answer: B

[GMAT math practice question]

Is x < 0?

1) x^5+x^3+1<0
2) x^3+1 < 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 the question includes 1 variable (x) and 0 equations, D is most likely to be the answer.

Condition 1)
x^5+x^3+1<0 implies that x^3(x^2+1)< -1.
Since x^2+1>0 always, x^3< -1/x^2+1<0.
Therefore, x<0.
This is sufficient.

Condition 2)
x^3+1 < 0 is equivalent to (x+1)(x^2-x+1)<0. Since
x^2-x+1>0 always, it follows that x+1<0.
This is sufficient, too.

Therefore, the answer is D, as expected.


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

[GMAT math practice question]

If f(x) = x(x-p)(x-q), is f(1) > 0?

1) f(2) = 0
2) f(4) = 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 the question includes 2 variables (p and q) and 0 equations, C is most likely to be the answer.

Conditions 1) & 2):

The equations f(2) = 0 and f(4) = 0 tell us that 2 and 4 are roots of x(x-p)(x-q) = 0, and so f(x) = x(x-2)(x-4)
Therefore, f(1) = 1(1-2)(1-4) = 1*(-1)*(-3) = 3 > 0, and the answer is ‘yes’.
Therefore, conditions 1) and 2), when taken together, are sufficient.

The answer is C, as expected.

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: C

[GMAT math practice question]

Is x^2 + xy > 5xy – 4y^2?

1) x > 2y
2) x > y

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

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

Modifying the original condition and question:

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

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

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

Therefore, the answer is A.

Answer: A