The Ultimate Q51 Guide [Expert Level]

[GMAT math practice question]

How many solutions has the equation ||x-3|-2|=1?

A. 0
B. 1
C. 2
D. 3
E. 4

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||x-3|-2|=1
⇔|x-3|-2=±1
⇔|x-3|=2±1
⇔|x-3|=1 or |x-3|=3
⇔x-3=±1 or x-3=±3
⇔x=3±1 or x=3±3
⇔x=2, x = 4, x = 0 or x = 6

Thus, the equation has four solutions.

Therefore, the answer is E.

Answer: E

Why is the correct answer not D for this question?

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[GMAT math practice question]

Is n<0?

1) n-1<0
2) |3-n| > |n+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.

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 - 1 < 0 ⇔ n < 1
Since the range of the question, n < 0 does not include that of the condition 1), n < 1, the condition 1) is not sufficient.

Condition 2) :
|3-n| > |n+5|
⇔ |3-n|^2 > |n+5|^2
⇔ (3-n)^2 > (n+5)^2
⇔ n^2 -6n + 9 > n^2 +10n + 25
⇔ -16 > 16n
⇔ n < -1
Since the range of the question includes that of the condition 2), the condition 2) is sufficient.

Therefore, B is the answer.

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: B

[GMAT math practice question]

Is x^2-x>0?

1) x>0
2) x^3+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.

In inequality questions, the law “Question is King” tells us that if the solution set of the question includes the solution set of the condition, then the condition is sufficient

Modifying the question:
x^2-x>0
⇔ x(x-1) > 0
⇔ x < 0 or x > 1 by the “LLGG” rule.

Condition 1): x > 0
Since the solution set of the question includes the solution set of condition 1), condition 1) is sufficient.

Condition 2):
x^3+x>0
⇔ x(x^2+1)>0
⇔ x>0, since x2+1 > 0 is always true.
Condition 2) is equivalent to the question, so it is sufficient.
Therefore, the answer is D.

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]

Is xy< -(x/y)?

1) xy<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:
xy < -(x/y)
⇔ xy^3 < -xy (multiplying both sides by y^2)
⇔ xy^3 + xy < 0
⇔ xy(y^2+1) < 0
⇔ xy < 0 since y^2+1 > 0

Condition 1): xy < 0
Condition 1) is same as the question.
This condition is sufficient.

Condition 2):
Since this condition tells us nothing about x, it is not sufficient.

Therefore, 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]

Is the positive integer x divisible by 60?

1) x is divisible by 6
2) x is divisible by 10

=>

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.
The question asks if x = 60t for some integer t.
x = 60t
⇔ x = 2^2*3*5*t


Conditions 1) (x = 2*3m for some integer m) and 2) (x = 2*5n for some integer n) do not tell us whether 22 is a factor of x. So, they are not sufficient, when taken together. For example, if
x = 60, then the answer is ‘yes’, but if
x = 30, then the answer is ‘no’.

Both conditions together are not sufficient, as the question does not have a unique answer.

Therefore, E 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: E

[GMAT math practice question]

What is the value of (x+3y)/(3x-y)?

1) x-2y = 1
2) x-2y = 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.

When the question asks for a ratio, a fraction, a percent, a proportion or a rate, if one of conditions provides a ratio and the other condition provides a number, the condition with a ratio could be sufficient.

This question asks for a ratio.
Condition 1) provides a number and condition 2) provides the ratio, x/y = 2.
Thus, condition 2) is likely to be sufficient.

Condition 1) :
If x = 3 and y = 1, then (x+3y)/(3x-y) = 6/8 = 3/4
If x = 5 and y = 2, then (x+3y)/(3x-y) = 16/13
Since we do not obtain a unique answer, condition 1) is not sufficient.

Condition 2) :
Since x = 2y, (x+3y)/(3x-y) = 5y/5y = 1.
Condition 2) is sufficient.

Therefore, B is the answer.

Answer: B

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If you need 2018 OFFICIAL GMAT REVIEW BOOK, you can purchase it from me for just ₹750/-
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Posted from my mobile device

[GMAT math practice question]

If x and y are integers, is 4x^2-y^2 an odd number?

1) x is an odd number
2) y 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, and then recheck the question.

Modifying the question:
In order for 4x^2 – y^2 to be odd, y^2 must be odd since 4x^2 is always even. This is equivalent to y being odd. So, the question asks if y is odd. Thus, the answer is B.

Condition 1)
If y is an odd number, then both 2x + y and 2x – y are odd numbers and (2x+y)(2x-y) is an odd number.
If y is an even number, both 2x + y and 2x – y are even numbers and (2x+y)(2x-y) is an even number.
Since the question does not have a unique answer, condition 1) is not sufficient.


Therefore, the answer is B.

Answer: B

[GMAT math practice question]

If x, y and z are positive numbers, is xy+z > x+yz?

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

Conditions 1) & 2):
xy + z > x + yz
⇔ xy – x – yz + z > 0
⇔ x(y-1)-z(y-1) > 0
⇔ (x-z)(y-1) > 0
⇔ x – z > 0 since y > 1.

If x = 2, y = 2, and z = 1, then xy + z = 5, x + yz = 4 and xy + z > x + yz. So, the answer is ‘yes’.
If x = 2, y = 2, and z = 3, then xy + z = 7, x + yz = 8 and xy + z < x + yz. So, the answer is ‘no’.
Since the question does not have a unique answer, both conditions together are not sufficient.

Therefore, E is the answer.

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

Answer: E

[GMAT math practice question]

5 couples sit on 10 chairs around a round table. If each couple must be seated together, how many possible arrangements are there?

A. 256
B. 512
C. 768
D. 1,024
E. 1,080

=>

The number of arrangements of the 5 couples in a circle is (5-1)! = 4!. The members of each couple can be arranged in 2! ways.
Thus, the total number of arrangements is 4! * 2! * 2! * 2! * 2! * 2! = 24*2*2*2*2*2 = 768.

Therefore, the answer is C.

Answer: C

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[GMAT math practice question]

Is xy<15?

1) x<-3
2) y<-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.

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.

Conditions 1) & 2):
Since x < -3 and y < -5, we have xy > 15, and the answer is ‘no’.
Since ‘no’ is also a unique answer by CMT (Common Mistake Type) 1, both conditions are sufficient, when used together.

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

Condition 1)
This condition provides us with no information about the variable y, so it is not sufficient:
If x = -4 and y = 1, then xy = -4 < 15, and the answer is ‘yes’.
If x = -4 and y = -5, then xy = 20 > 15, and the answer is ‘no’.
Since we do not obtain a unique answer, this condition is not sufficient.

Condition 2)
This condition provides us with no information about the variable x, so it is not sufficient:

If x = 1 and y = -5, then xy = -5 < 15, and the answer is ‘yes’.
If x = -4 and y = -5, then xy = 20 > 15, and the answer is ‘no’.
Since we do not obtain a unique answer, this condition 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]

If the average (arithmetic mean) of set A is 100 and the average (arithmetic mean) of set B is 100, what is the range of sets A and B combined?

1) The range of set A is 20.
2) The range of set B is 30.

=>

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.

There is no relationship between the average of a set and its range.
In addition, there is no way of determining the range of a set from the ranges of its subsets. Notice that “E” is in “no rElation”. Many questions about “relation” have the answer E.
Thus, E is likely to be the answer.

We can verify this by considering both conditions together:
If A = { 90, 110 } and B = { 85, 115 }, then the range of A∪B={85,90,110,115} is 30.
If A = { 90, 110 } and B = { 92, 93, 95, 98, 122 }, then the range of A∪B={ 90, 92, 93, 95, 98, 110, 122 } is 32.
Since the range of A∪B is not uniquely determined, both conditions together are not sufficient.

Therefore, the answer is E.
Answer: E

[GMAT math practice question]

Is n an even number?

1) n(n+1) is an even number
2) n(n+2) 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 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):
All integers satisfy condition 1) since n(n+1) is the product of two consecutive integers:
If n = 1, n(n+1) = 1*2 = 2 is an even number, but n is odd, so the answer is ‘no’.
If n = 2, n(n+2) = 2*3 = 6 is an even number and n is even, so the answer is ‘yes’.
Since we do not have a unique answer, condition 1) is not sufficient.

Condition 2)
The product of two integers can only be even if at least one of them is even. Since n and n+2 must both be even, or both be odd, n must be even.
Thus, 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]

What is the least common multiple of 39, 91, 93 and 217?

A. 8461
B. 8462
C. 8463
D. 8464
E. 8465

=>

Since 39 = 3 * 13, the least common multiple must be a multiple of 3.
The sum of the digits of a multiple of 3 is a multiple of 3.
8643 is the only answer choice for which the sum of the digits is a multiple of 3: 8 + 4 + 6 + 3 = 21 = 3 * 7.

Therefore, C is the answer.

Alternatively, since 39 = 3*13, 91 = 7*13, 93 = 3*31, and 217=7*31, the least common multiple of the four numbers is 3*7*13*31 = 8463.

Therefore, the answer is C.

Answer: C