Overview of GMAT Math Question Types and Patterns on the GMAT

[Math Revolution GMAT math practice question]

x is a positive number. What is the median of x, √x and x^2?

1) x^2=x
2) x^2+x+1=3x

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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.

If x >1, then √x < x < x^2 and x is their median.
If 0 < x <1, then √x > x > x^2 and x is their median.
If x = 1, then √x = x = x^2 and x is their median.
Thus, the question asks for the value of x.

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^2=x
=> x^2-x=0
=> x(x-1)=0
=> x = 0 or x = 1
Since x is positive, x = 1.
Condition 1) is sufficient.

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

Therefore, D is the answer.
Answer: D

FYI, Tip 1) of the VA method states that D is most likely to be the answer if conditions 1) and 2) provide the same information.

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.

[Math Revolution GMAT math practice question]

How many positive two-digit integers have a remainder of 1 when divided by 2, a remainder of 2 when divided by 3, and a remainder of 4 when divided by 5?

A. 1
B. 2
C. 3
D. 4
E. 5

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Let x be a positive integer with these properties.
Since x = 2p + 1 for some integer p, the possible values of x are x = 1, 3, 5, 7, … .
Since x = 3q + 2 some integer q, the possible values of x are x = 2, 5, 8, 11, … .
Since x = 5r + 4 for some integer 4, the possible values of x are x = 4, 9, 14, 19, … .
The first possible 2-digit number is thus 19. To find the others, note that the least common multiple of 2, 3 and 5 is lcm(2,3,5) = 30.
Thus, there are three possible 2-digit numbers with these properties:
19, 49 = 19 + 30 and 79 = 49 + 30.

Therefore, C is the answer.
Answer: C

[Math Revolution GMAT math practice question]

Tom saved $10,000 at a constant compound interest rate of r percent annually. After 10 years, the balance is double the principal. What will be the balance 30 years after the deposit?

A. $30,000
B. $40,000
C. $50,000
D. $60,000
E. $80,000


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After 10 years, we have 10,000(1+r)^{10} = 20,000 or (1+r)^{10} = 2.
So, after 30 years the balance is 10,000(1+r)^{30} = 10,000((1+r)^{10})^3= 10,000*2^3 =10,000*8 = 80,000.

Therefore, the answer is E.
Answer: E


[Math Revolution GMAT math practice question]

What is the solution set of (1+|x|)(1+x) > 0?

A. x > -1
B. x < -1
C. x < 0
D. x > 0
E. x > 1

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Since 1+|x| > 0, we can divide both sides of the inequality by 1 + |x| to obtain 1+x > 0 or x > -1.

Therefore, the answer is A.
Answer: A


[Math Revolution GMAT math practice question]

How many perfect squares lie between 2^4 and 2^8, inclusive?

A. 12
B. 13
C. 15
D. 18
E. 20

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We need to count the number of integers n satisfying 2^4 ≤ n^2 ≤ 2^8 or 2^2 ≤ n ≤ 2^4.
The number of integers satisfying 2^2 ≤ n ≤ 2^4 is 2^4 – 2^2 + 1 = 16 – 4 + 1 = 13.

Therefore, the answer is B.
Answer: B

[Math Revolution GMAT math practice question]

x and y are positive integers. Is y an even integer?

1) x^2+x=y+2
2) x = 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.

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 y = x^2 + x – 2 and x = 2, we have y = 4.
Since this answer is unique, 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)
Since y = x^2 + x – 2 = (x-1)(x+2) and x is an integer, one of x – 1 and x + 2 is an even integer.
Thus, y is always an even integer and condition 1) is sufficient.

Condition 2)
Since it provides no information about y, 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. 

[Math Revolution GMAT math practice question]

n is a positive integer. Is n(n+1)(n+2)/4 an even integer?

1) n is an even integer
2) 1238 ≤ n ≤ 1240

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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.

Asking for n(n+1)(n+2)/4 to be an even integer is equivalent to asking for n(n+1)(n+2) to be a multiple of 8. If n is an even integer, n and n+2 are consecutive even integers and a product of two consecutive even integers is a multiple of 8. Thus, condition 1) is sufficient.

Condition 2)
If n = 1238, n(n+1)(n+2)=1238*1239*1240 is a multiple of 8 since 1240 is a multiple of 8.
If n = 1239, n(n+1)(n+2)=1239*1240*1241 is a multiple of 8 since 1240 is a multiple of 8.
If n = 1240, n(n+1)(n+2)=1240*1241*1242 is a multiple of 8 since 1240 is a multiple of 8.
Thus, condition 2) is sufficient.

Therefore, the answer is D.
Answer: D

Note: This question is a CMT4(B) question. Condition 1) is easy to understand and condition 2) is hard. When one condition is easy to understand, and the other is hard, D is most likely to be the answer.

[Math Revolution GMAT math practice question]

How many 4-digit numbers greater than 3,000 have the digits: 1, 3, 5, and 7?

A. 6
B. 9
C. 12
D. 15
E. 18

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We need to count the 4-digit numbers with thousands digits 3, 5 and 7.
The number of 4-digit numbers beginning with 3 is 6.
The number of 4-digit numbers beginning with 5 is 6.
The number of 4-digit numbers beginning with 7 is 6.
Thus, the total number of such 4-digit numbers is 18 = 6 + 6 + 6.

Therefore, the answer is E.
Answer: E

[Math Revolution GMAT math practice question]

What is the smallest integer n such that 28 is a factor of n!?

A. 8
B. 10
C. 12
D. 14
E. 16


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Since 2 = 2^1, 4 = 2^2, 6 = 2^1*3, 8 = 2^3, 10=2^1*5, the prime factorization of 10! = 1*2*3*…*10 has the form 28*m for some integer m, where m and 2 are relatively prime. Note that 9! = 2^7*k for some integer k, where k and 2 are relatively prime.

Therefore, the answer is B.
Answer: B

[Math Revolution GMAT math practice question]

If m and n are positive integers, is 3^{4m+2}+n divisible by 5?

1) m=3
2) n=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.

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

The units digits of 3^k have period 4 as they form the cycle 3 -> 9 -> 7 -> 1.
3^{4m+2} has 9 as its units digit if 3^{4m+2} has units digit 9, regardless of the value of m.
Thus, the divisibility of 3^{4m+2}+n by 5 relies on the variable n only.

Therefore, the correct answer is B.
Answer: B

[Math Revolution GMAT math practice question]

If x^2+y^2=28 and xy=11, what is the value of (x+y)^2?

A. 28
B. 39
C. 50
D. 61
E. 72

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(x+y)^2 = x^2 + 2xy + y^2 = x^2 + y^2 + 2xy = 28 + 2*11 = 28 + 22 = 50.

Therefore, the answer is C.
Answer: C

[Math Revolution GMAT math practice question]

Can n be expressed as the difference of 2 prime numbers?

1) (n-17)(n-21) = 0
2) (n-15)(n-17)=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 we have 1 variable (n) and 0 equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
(n-17)(n-21) = 0 is equivalent to the statement n = 17 or n =21
If n = 17, then 17 = 19 – 2 is a difference of two prime numbers and the answer is ‘yes’.
If n = 21, then 21 = 23 – 2 is a difference of two prime numbers and the answer is ‘yes’.
Since it gives a unique answer, condition 1) is sufficient.

Condition 2)
(n-15)(n-17) = 0 is equivalent to the statement n = 15 or n = 17
If n = 15, then 15 = 17 – 2 is a difference of two prime numbers and the answer is ‘yes’.
If n = 17, then 17 = 19 – 2 is a difference of two prime numbers and the answer is ‘yes’.
Since it gives a unique answer, condition 2) is sufficient.

Therefore, D is the answer.
Answer: D

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.

[Math Revolution GMAT math practice question]

If x is integer and 3|x|+x<4, what is the value of x?

1) x<0
2) x>-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.

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

Modifying the original condition:
There are two cases to consider.
Case 1) x ≥ 0:
3|x|+x < 4
=> 3x + x < 4
=> 4x < 4
=> x < 1
=> 0 ≤ x < 1

Case 2) x < 0:
-3x+x < 4
=> -2x < 4
=> x > -2
=> -2 < x <0

Therefore, x is an integer with -2 < x < 1. Thus, the original condition tells us that x = -1 or 0.

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)
Since x < 0, we must have x = -1 as the original condition tells us that x = 0 or x = -1.
Condition 1) is sufficient, because it yields a unique solution.


Condition 2)
Both x = 0 and x = -1 satisfy condition 2).
Since it does not yield a unique solution, condition 2) is not sufficient.

Therefore, A is the answer.

[Math Revolution GMAT math practice question]

In the xy-plane, a circle has center (0,0) and radius 5. Is the point (r,s) inside or on the circle?

1) -3 < r < 3
2) -4 < s < 4

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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 inequality satisfied by points inside or on the circle is r^2+s^2≤5^2=25.

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

Conditions 1) & 2):
Since -3<r<3 and -4<s<4, we have 0≤r^2<3^2=9 and 0≤s^2<4^2=16. Thus, 0≤r^2+s^2<9+16=25 and both conditions together are sufficient.



Therefore, the answer is C.
Answer: C

[Math Revolution GMAT math practice question]

If x, y are integers, is x^2+x+y an odd integer?

1) x is an odd integer
2) y is an odd integer

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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 parity of x^2+x+y = x(x+1) + y is same as the parity of y, since x^2+x = x(x+1) is the product of two consecutive integers and so it is always an even integer.
Thus, asking whether x^2+x+y = x(x+1) + y is odd is equivalent to asking whether y is odd.

Therefore, B is the answer.
Answer: B

[Math Revolution GMAT math practice question]

Three machines have equal constant work rates. It takes h + 3 hours to produce 360 toys when 2 machines work together, and it takes h hours to produce 360 toys when 3 machines work together. How many toys can each machine produce per hour when working on its own?

A. 12
B. 15
C. 18
D. 20
E. 24

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The work rate for each machine is given by 360 / {2(h+3)} = 180 / ( h + 3 ). Another expression for this work rate is 360 / (3h) = 120 / h.
Thus, 180 / ( h + 3 ) = 120 / h or 3 / ( h + 3 ) = 2 / h.
So, 3h = 2(h+3) and h = 6.
The sum of the work rates of the 3 machines is 360 / 6 = 60.
The work rate of each machine is 60/3 = 20 toys / hour.

Therefore, the answer is D.
Answer: D

[Math Revolution GMAT math practice question]

If the median and the range of the data set 1, 1, 2, 2, 3, 3, 4, 4, m are the same, which of the following could be the value of m?

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

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If 3 ≤ m ≤ 4, then the median is 3 and the range of the data is 3. Thus, any number between 3 and 4, inclusive, is possible.
If 2 ≤ m < 3, then the median is m and the range of the data is 3. Thus, there is no possible value of m in the interval 2 ≤ m < 3.
If m < 2, then the median is 2 and the range of the data is greater than 2. Thus, there is no possible value of m less than 2.
If m > 4, then the median is 3 and the range of the data is greater than 3. Thus, there is no possible value of m greater than 4.

Therefore, the answer is E.
Answer: E