The Ultimate Q51 Guide [Expert Level]

[GMAT math practice question]

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

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

=>

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

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

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

Both conditions 1) and 2) together are sufficient.

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

Answer: C

[GMAT math practice question]

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

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

=>

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

Since we have 3 variables (x,y and z) and 0 equations, E is most likely to be the answer. So, we should consider conditions 1) & 2) together first.

Conditions 1) and 2)
Since xz is an odd number, both x and z are odd integers.
Since xz is an odd number, it follows from condition 2) that y is a multiple of 8.
Therefore, y is an even number.
Conditions 1) and 2) are sufficient when taken together.

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

Condition 1)
Since xz is an odd integer, x and z are odd integers. As the three integers are consecutive, y must be even.
Thus, condition 1) is sufficient.

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

Therefore, A is the answer.

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

[GMAT math practice question]

4 points a, b, c and d are placed on the number line in that order. If abcd>0, which of the following must be positive?

I. ab
II. bc
III cd

A. Ⅰonly
B. Ⅱonly
C. Ⅲ only
D.Ⅰ&Ⅲ only
E. Ⅱ&Ⅲ only

=>

There are three cases we need to consider. Note that if a < 0 < b < c < d or a < b < c < 0 < d, then abcd < 0.

Case 1) 0 < a < b < c < d:
ab > 0, bc > 0 and cd > 0

Case 2) a < b < 0 < c < d:
ab > 0, bc < 0, cd > 0

Case 3) a < b < c < d < 0:
ab > 0, bc > 0, cd > 0

Only statements I and III are always true.

Therefore, the answer is D.

Answer : D

[GMAT math practice question]

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

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

=>

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

We can modify the original condition and question as follows.
There are two different ways in which a^2+b^3 can be odd:

i) a is even and b is odd.
ii) a is odd and b is even.

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

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

Therefore, the answer is B.

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

Answer: B

[GMAT math practice question]

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

1) y=x^2+3x+2
2) xy is an even 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.
Since we have 2 variables (x and y) and 0 equations, C is most likely to be the answer. So, we should consider conditions 1) and 2) together first.

Conditions 1) and 2)
Case 1: x is odd
Since xy is even, y is even.
Case 2: x is even.
Since x^2, 3x and 2 are even, y =x^2+3x+2 is even.
Since we have a unique answer, both conditions together are sufficient.
Since this is an integer question (one of the key question areas), we should also consider choices A and B by CMT 4(A).

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

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

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

[GMAT math practice question]

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

1) x=5
2) y=4

=>
Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Since we have 2 variables (x and y) and 0 equations, C is most likely to be the answer. So, we should consider conditions 1) and 2) together first.

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

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

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

Condition 2)
There are two cases to consider.
Case 1: x is even.
Since 3x^2 and 5x are even and y is even, 3x2+5x+y is even.
Case 2: x is odd.
Since 3x^2+5x is even and y is even, 3x2+5x+y is even.
Since we have a unique answer, condition 2) is sufficient.

Therefore, the answer is B.

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

[GMAT math practice question]

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

1) p = 2
2) q = 5

=>

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

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

Since only condition 1) gives us information about p, only condition 1) is sufficient.

Therefore, A is the answer.
Answer: A

[GMAT math practice question]

Working alone at its constant rate, machine A can produce 240 pins in x hours. Working alone at its constant rate, machine B can produce 240 pins in y hours. How many hours will it take machines A and B to produce 240 pins, when working together at their respective constant rates?

1) xy/(x+y)=4
2) x=20

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

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

We have 1/t = 1/x + 1/y = (x+y)/xy or t = xy/(x+y), where t is the time that machines A and B take to produce 240 pins, when working together. The question asks for xy / (x+y).

Thus, only condition 1) is sufficient. Condition 2) is not sufficient as it only gives us information about x.

Therefore, the answer is A.
Answer: A

[GMAT math practice question]

If x and y are integers greater than 1 and x>y, what is the value of x?

1) x+y=10
2) xy=21

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

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

Conditions 1) and 2):
y = 10 – x and xy= x(10-x) = 21
⇔ -x^2 + 10x = 21
⇔ x^2 - 10x + 21 = 0
⇔ (x-3)(x-7) = 0
⇔ x = 3 and y = 7, or x = 7 and y = 3.
Since x > y, we must have x = 7 and y = 3.
Thus x = 7, and conditions 1) and 2) are sufficient, when taken together.

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

Condition 1)
There are two possible solutions: x = 6 and y = 4, and x = 7 and y = 3.
Since the solution is not unique, condition 1) is not sufficient.

Condition 2):
Either x = 3 and y = 7, or x = 7 and y = 3.
Since x > y, x = 7 and y = 3.
Thus, we have the unique solution, x = 7.

Therefore, condition 2) is sufficient.

Therefore, B is 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.

Answer: B

why did we take square of mod ⇔ (x+1)^2=(y+1)^2 ?

[GMAT math practice question]

The symbol ⊙ represents one of the operations: addition, subtraction, multiplication, and division. Is 2⊙2=4？

1) 1⊙1=1
2) (-1) ⊙ (-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.

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

Since only 2 + 2 = 4 and 2 * 2 = 4, the question asks if ⊙ is addition or multiplication.

Condition 1)
Since 1 * 1 = 1 and 1 / 1 = 1, ⊙ is multiplication or division.
Condition 1) is not sufficient.

Condition 2)
Since (-1) – (-1) = 0, ⊙ is subtraction. The answer is “no”.
CMT1 states that “no” is also an answer.
Condition 2) is sufficient.

Therefore, the answer is B.
Answer: B

[GMAT math practice question]

If m, n and p are positive integers such that (p^m)^n=256, which of the following could be the value of n?

I. 1
II. 2
III. 4

A. I and II
B. I and III
C. II and III
D. II only
E. I, II and III

=>
Since 256 = 2^8, (p^m)^n = p^{mn} = 2^8.
So, p = 2 and mn = 8.

Since n is a factor of 8, n could be any of the values 1, 2, 4, and 8.

Therefore, the answer is E.
Answer: E

[GMAT math practice question]

Is x^3-x^2+x<0?

1) x < 0
2) x^5+x<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, and then recheck the question.

Modifying the question:
x^3-x^2+x<0
⇔ x(x^2-x+1)<0
⇔ x<0 since x^2-x+1>0 always.
So, the question becomes, ‘is x<0?’.

Condition 1) is certainly sufficient.

Condition 2), x^5+x<0, is equivalent to x(x^4+1)<0 or x < 0, since x^4+1 > 0 is always true. So, condition 2) is also sufficient.

Therefore, D is the answer.
Answer: D

[GMAT math practice question]

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

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, and then recheck the question.

Modifying the question:
x^2>xy
⇔ x^2-xy > 0
⇔ x(x-y) > 0
⇔ x > 0 since x > y (which implies that x – y > 0).
So, the question is equivalent to asking ‘is x > 0?’.

Since condition 1) is same as the modified question, it is sufficient.

Condition 2):
Since x > y, condition 2) (y > 0) implies that x > 0.
Condition 2) is also sufficient.

Therefore, the answer is 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.
Answer: D

[GMAT math practice question]

In the 6-digit integer 543,2xy, x and y are chosen from the digits 0, 2, 4, 6 and 8. What is the probability that 543,2xy is divisible by 8?

A. 1/5
B. 6/25
C. 7/25
D. 8/25
E. 9/25

=>

An integer with three or more digits is divisible by 8 if and only if its last three digits from a number that is divisible by 8.

The values of 2xy that are divisible by 8 are 200, 208, 224, 240, 248, 264, 280, and 288.
The total number of 6-digit integers of the form 543,2xy is equal to the number of ways of choosing two digits from 0, 2, 4, 6 and 8, which is 5*5 = 25.
Thus, the probability that 543,2xy is divisible by 8 is 8/25

Therefore, the answer is D.

Answer: D

[GMAT math practice question]

What is the product of all roots of the equation (x+1)^2=|x+1|?

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

=>
Now,
(x+1)^2=|x+1|
⇔ |x+1|^2=|x+1|
⇔ |x+1|^2-|x+1|= 0
⇔ |x+1|(|x+1|-1) = 0
⇔ |x+1| = 0 or |x+1| = 1
⇔ x = -1 or x+1 = ±1
⇔ x = -1 or x = -1 ±1
⇔ x = -1, x= -2 or x = 0

The product of these solutions is (-1)*(-2)*0 = 0.

Therefore, the answer is C.

Answer: D

[GMAT math practice question]

If xy>0, is x^3y^4>0?

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, and then recheck the question.

Modifying the question:
x^3y^4>0
⇔ x>0
since y^4>0 is always true. We can ignore even exponents in inequalities that have 0 on one side.
So, the question becomes, ‘is x>0?’.

Condition 1) is certainly sufficient.

Condition 2):
x > 0 since y > 0 and xy > 0. Thus, condition 2) is also sufficient.

Therefore, D is the answer.

Note: Tip 1) of the VA method states that D is most likely to be the answer if condition 1) gives the same information as condition 2).

Answer: D