Overview of GMAT Math Question Types and Patterns on the GMAT

[GMAT math practice question]

Is x|y|=xz?

1) x, y, and z are positive
2) y^2=z^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.

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|y| = xz
⇔ x(|y|-z) = 0
⇔ x = 0 or |y| = z
⇔ x = 0 or y = z or y = -z

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) & 2):

Condition 2) tells us that y = z or y = -z.
Since condition 1) states that x, y, z > 0, we can only have y = z.
Thus, both conditions are sufficient, when taken together.

Therefore, the answer is C.

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: C

Hiii
That's a very nice problem, Kudo to you.
See my approach to this problem.

[GMAT math practice question]

Is the positive integer x an even number?

1) The smallest prime factor of x is 3.
2) The greatest prime factor of x is 17.

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

Condition 1)
Since all prime factors of x are greater than or equal to 3, 2 is not a prime factor of x, and x is not an even number.
The answer is ‘no’.
This condition is sufficient by CMT (Common Mistake Type) 1, which tells us that the unique answer ‘no’ also means that the condition is sufficient.

Condition 2)

If x = 2*17 = 34, then the answer is ‘yes’.
If x = 17, then the answer is ‘no’.
Since we do not have a unique answer, condition 2) is not sufficient.

Therefore, the answer is A.
Answer: A

St1: since smallest prime factor is 3, 2 is not a factor. Hence odd. Hence sufficient.

St2: since greatest prime factors is 17, 2 can be a factor but not necessarily.

Eg. 17,34. It can be even or odd. Insufficient.

Answer is A

[GMAT math practice question]

If x=343y, where y is a positive integer, and x/196 is a terminating decimal, what is the smallest possible value of y?

A. 1 B. 3 C. 5 D. 7 E. 9

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x/196 = (343*y)/196 = (7^3*y)/(14^2) = (7^3*y)/(2^2)(7^2) = (7y)/4
As the denominator has only 2 as a prime factor, it is a terminating decimal, regardless of the value of y.
Thus, the smallest possible value of y is 1.

Therefore, the answer is A.

Answer: A

[GMAT math practice question]

Is x^y<1?

1) x>1
2) y<1

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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.

Conditions 1) & 2):
If x = 4 and y = -1, then xy = 1/4 < 1, and the answer is ‘yes’.
If x = 4 and y = ½, then xy = 2 > 1, and the answer is ‘no’.

Since we don’t have a unique answer, both conditions are not sufficient when taken together by CMT (Common Mistake Type) 2.

Therefore, the answer is E.

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

[GMAT math practice question]

If n is a positive integer, is 91 a factor of n?

1) 91 is a factor of n^2
2) 91 is a factor of 2n

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

Condition 1)
Since 91 = 7*13, n^2 is a multiple of both 7 and 13. Since 7 and 13 are prime numbers, n must be a multiple of both 7 and 13.
Condition 1) is sufficient.


Condition 2)
If 91 is a factor of 2n, then 91k = 2n for some integer k. Since 91 is odd, k must be an even integer. Write k = 2a, for some integer a. Then
2n = 91*2a. It follows that n = 91*a.
Thus, n is a multiple of 91.
Condition 2) is sufficient.

This is a CMT(Common Mistake Type) 4(B) question. Condition 2) is easy to understand and condition 1) is difficult to figure out. If you are unable to figure out condition 2), you should choose D as the answer.

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]

If x and y are integers, is x^2+11x+13y an even number?

1) x=11
2) y=13

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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, and then recheck the question.

Now, x^2+11x+13y = x(x+11)+13y. As x(x+11) is always an even integer (one of x and x + 11 must be even), x^2+11x+13y is even precisely when 13y is even. This is equivalent to y being even, so the question can be restated as ‘Is y an even number?’.

Condition 1)
If x = 11 and y = 1, then x^2+11x+13y = 255 is an odd integer.
If x = 11 and y = 2, then x^2+11x+13y = 268 is an even integer.
Since the question does not have a unique answer, condition 1) is not sufficient.

Condition 2)
If y = 13, then y is odd, and the answer to the question is ‘no’.
Condition 2) is sufficient.


Therefore, B is the answer.

Answer: B

[GMAT math practice question]

Events E and F are independent. Is the probability that both events E and F will occur less than 0.6?

1) The probability that event E will occur is 0.4.
2) The probability that event F will occur is 0.5.

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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.

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

The definition of independent events E and F is P(E∩F) = P(E)P(F).
The question asks if P(E)P(F) < 0.5.

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

Conditions 1) & 2):
P(E ∩F) = P(E)P(F) = 0.4 * 0.5 = 0.2 < 0.6.
Thus, both conditions together are sufficient.

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

Condition 1):
Since P(E) = 0.4 and P(F) ≤ 1, P(E ∩F) = P(E)P(F) ≤ 0.4 * 1 = 0.4 < 0.6.
Condition 1) is sufficient on its own.

Condition 2):
Since P(E) ≤ 1 and P(F) = 0.5, P(E ∩F) = P(E)P(F) ≤ 1 * 0.5 = 0.5 < 0.6.
Condition 2) is sufficient on its own.

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]

What is the value of the positive integer k?

1) When k is divided by 3, the remainder is 2.
2) When k is divided by 5, the remainder is 1

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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 1 variable (k) 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):
Two possible values of k which satisfy condition 1) are 5 and 8.
Since it does not give us a unique value of k, condition 1) is not sufficient.

Condition 2):
Two possible values of k which satisfy condition 2) are 1 and 6.
Since it does not give us a unique value of k, condition 2) is not sufficient.

Conditions 1) & 2):
Two possible values of k which satisfy both conditions 1) & 2) are 11 and 26.
Since they do not give us a unique value of k, conditions 1) & 2) together are not sufficient.

Therefore, E is the answer.

Answer: E

[GMAT math practice question]

How many arrangements of the letters ‘s’, ‘u’, ‘c’, ‘c’, ‘e’, ‘s’, ‘s’ in a straight line are possible?

A. 60
B. 120
C. 180
D. 240
E. 420

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The seven letters ‘s’, ‘u’, ‘c’, ‘c’, ‘e’, ‘s’, ‘s’ include three s’s and two c’s.
The number of permutations is 7! / (3!*2!) = 420.

Therefore, the answer is E.

Answer: E

[GMAT math practice question]

A data set contains 5 values: 11, 13, 15, 19, and x. What is the value of x?

1) The median of the 5 values is 15
2) The range of the data set is 8

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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 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.

Note: if the question doesn’t include the phrase, “in increasing order”, then x could be any number. In particular, it could be greater than or less than 19.



Condition 1):
Since the median is 15, we only know that x ≥ 15.
Condition 1) is not sufficient.

Condition 2):
Since the range of the values is 8, we have 11 ≤ x ≤ 19.
Condition 2) is not sufficient.

Conditions 1) & 2):
Both conditions 1) & 2) together yield 15 ≤ x ≤ 19. But this does not tell us the exact value of x. We do not have a unique solution.
Thus, both conditions together are not sufficient.

Therefore, E is the answer.

Answer: E

[GMAT math practice question]

The 20% acid liquid solution is produced by adding a gallons of 10% acid liquid solution to b gallons of 50% acid liquid solution. To get 10 gallons of 20% acid liquid solution, how many gallons of 50% acid liquid solution are needed?

A. 1
B. 1.5
C. 2
D. 2.5
E. 3

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We have a + b = 10.

( 0.1a + 0.5b ) / 10 = 0.2
⇔ 0.1a + 0.5b = 2
⇔ 0.1(10-b) + 0.5b = 2
⇔ 1 – 0.1b + 0.5b = 2
⇔ 0.4b = 1
⇔ b = 2.5

Therefore, D is the answer.

Answer: D

[GMAT math practice question]

When n is divided by 5, what is the remainder?

1) When n is divided by 3, the remainder is 2
2) When n+1 is divided by 5, the remainder is 3

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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 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.
It is suggested we plug in numbers for the remainder questions.

Condition 1): n=3p+2=2,5,8……..
n = 2 : The remainder of n is 2 when it is divided by 5.
n = 5 : The remainder of n is 0 when it is divided by 5.

Since we don’t have a unique solution, condition 1) is not sufficient.

Condition 2):
We have n + 1 = 5q + 3 for some integer, since n + 1 has the remainder 3 when it is divided by 5.
It is equivalent to n = 5q + 2=2,7,12,17,….
The remainder is 2 when n is divided by 5.
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]

Is x/y^3>0?

1) x-y>0
2) xy>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.

The question can be modified as follows:
x/y^3>0 ⇔ xy>0.
This can be seen by multiplying both sides by y^4.
It is same as condition 2).

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

Therefore, the answer is B.

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 at least one of 3 consecutive integers a multiple of 5?

1) The sum of the integers is divisible by 5
2) The product of the integers is divisible by 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.

Write the 3 consecutive integers as n-1, n, n+1.

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

Condition 1)
Since ( n – 1 ) + n + ( n + 1 ) = 3n is divisible by 5, and 3 and 5 are prime numbers, n is a multiple of 5, since 3 is not divisible by 5.
Condition 1) is sufficient.

Condition 2)
Since (n-1)n(n+1) is divisible by 5 and 5 is a prime number, one of n-1, n and n+1 is a multiple of 5.
Condition 2) is sufficient.

Therefore, D is the answer.

Answer: 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.

Can someone provide detailed solution to this problem? Any strategies for setting up these kind of word problems? Thanks! Bunuel or chetan2u

Of all of the apples inspected last week at a certain orchard, 4.5 percent failed to pass inspection. Of the apples that failed inspection, 2/3 of them were rotten and the rest was unmatured. If all of the apples that were rotten or unmatured failed inspection, how many of apples inspected last week at the plant were unmatured?
(1) 450 of apples inspected last week at the orchard failed to pass inspection.
(2) 9,550 of apples inspected last week at the orchard passed inspection.