Overview of GMAT Math Question Types and Patterns on the GMAT

[GMAT math practice question]

(algebra) What is the value of (a-b)/(a+b) –ab + b/c ?

1) a=bc
2) a=1/2

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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 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. 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)
Plugging in a = bc = 1/2 yields
(a-b)/(a+b) – ab + b/c = (bc-b)/(bc+b) – b^2c + b/c = b(c-1) / b(c+1) – (1/2)c + (bc)/c^2 = (c-1)/(c+1) – 1/2c + 1/(2c^2).
Since we don’t know the value of c, both conditions together don’t yield a unique solution and they are not sufficient.

Therefore, E is the answer.
Answer: E

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.

[GMAT math practice question]

(algebra) What is the value of (3mr-nt)/(4nt-7mr)?

1) m/n = 4/3
2) r/t= 9/14

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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. We should simplify the conditions if necessary.

We rearrange (3mr-nt)/(4nt-7mr) to see if we can write in terms of the ratios m/n and r/t given in the conditions:
(3mr-nt)/(4nt-7mr)
= ( (3mr)/(nt) – (nt/nt) ) / ( 4(nt/nt) – 7mr/nt )
= ( 3(m/n)*(r/t) – 1 ) / ( 4 – 7(m/n)(r/t) )

Now, both conditions 1) & 2) together are sufficient since the simplified question requires only the values of (m/n) and (r/t).

Therefore, C is the answer.
Answer: C

[GMAT math practice question]

(function) For which value of x will y=ax^2+20x+b have a minimum in the xy-plane?

1) b=10
2) a=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 numbers of variables and equations.

We can modify the original condition and question as follows:

If a > 0, the function will have a minimum at x = (-20)/(2a) = (-10)/a.
If a < 0, the function has no minimum. So, to answer the question, we need to find the value of a.
Thus, condition 2) is sufficient.

Note: condition 1) cannot be sufficient as it provides no information about the value of a.

Therefore, the answer is B.
Answer: B

[GMAT math practice question]

(Number) What is the units digit of 3^n?

1) n is a multiple of 4
2) n is a multiple of 6

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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. We should simplify the conditions if necessary.

The units digits of 3^n for n = 1, 2, 3, 4, … are 3, 9, 7, 1, 3, 9, 7, 1, …
So, the units digits of 3^n have period 4:
They form the cycle 3 -> 9 -> 7 -> 1.
Thus, 3^n has a units digit of 1 if n is a multiple of 4.

Note that 6 is not a multiple of 4.

Therefore, A is the answer.
Answer: A

[GMAT math practice question]

(number properties) What is the remainder when 1+n+n^2 +…+ n^8 is divided by 5?

1) The remainder when n is divided by 5 is 3
2) n is less than 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. We then recheck the question. We should simplify conditions if necessary.

The easiest way to solve remainder questions is to plug in numbers.
The units digits of 3^n for n = 1, 2, 3, 4, … are 3, 9, 7, 1, 3, 9, 7, 1, …
So, the units digits of 3^n have period 4:
They form the cycle 3 -> 9 -> 7 -> 1.
Thus, if n has remainder 3 when it is divided by 5, 1+n+n^2 +…+ n^8 has the same remainder as 1 + 3 + 9 + 7 + 1 + 3 + 9 + 7 + 1 = 21 when it is divided by 5. It has a remainder of 1 when it is divided by 5.
Condition 1) is sufficient.

Condition 2)
If n = 1, then 1+n+n^2 +…+ n^8 = 9, which has remainder 4 when it is divided by 5.
If n = 3, then 1+n+n^2 +…+ n^8 has remainder 1 when it is divided by 5.
Since condition 2) doesn’t yield a unique solution, it is not sufficient.

Therefore, A is the answer.
Answer: A

[GMAT math practice question]

(Statistics) When playing a coin-tossing game, Tom wins $5 when the coin lands on heads and loses $3 when the coin lands on tails.
After tossing the coin 20 times Tom has won a total of $12. How many times did the coin land on heads?

A. 7
B. 8
C. 9
D. 10
E. 11

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Let h be the number of times the coin landed on heads.
Then 20 – h is the number of times the coin landed on tails.
Thus, Tom won 5h – 3(20-h) = 8h – 60 dollars.
Since Tom won $12, this gives the equation 8h - 60 = 12. Solving this equation yields 8h = 72 and h = 9.

Therefore, C is the answer.
Answer: C

[GMAT math practice question]

(Distance / Rate Problems) A lake has a circumference of two kilometers. A and B both departed from the same point on the circumference at the same time and walked in opposite directions. They met 30 minutes after their departure. What is the sum of the speeds of A and B?

A. 3 km/hr
B. 4 km/hr
C. 5 km/hr
D. 6 km/hr
E. 7 km/hr

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Let a and b be the velocities of A and B, respectively.
Then (30/60)a + (30/60)b = (1/2)a + (1/2)b = 2.
Thus, a + b = 4.

Therefore, B is the answer.
Answer: B

[GMAT math practice question]

(algebra) What is the value of the integer a?

1) x - (2/3)(x-4a) = 7 has a positive integer solution
2) a is positive

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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 a) 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)
x - (2/3)(x - 4a) = 7 is equivalent to 3x – 2(x-4a) = 21 or x = 21 – 8a.
The possible pairs (x,a) are (13,1) and (5,2).
Since both conditions together don’t yield a unique solution, they are not sufficient.

Therefore, E is the answer.
Answer: 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.

[GMAT math practice question]

(number properties) A and B are positive integers. G is the greatest common divisor of A and B, and L is the least common multiple of A and B. What is the value of A+B?

1) G/A + G/B = 7/10
2) L=70

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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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have 2 variables (A and B) 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)
Suppose A=aG and B=bG for some integers a, b and G, where a and b are relatively prime.

Then
G/A + G/B = 7/10
=> G/(aG) + G/(bG) = 7/10, since A=aG and B = bG
=> bG/(abG) + aG/(abG) = 7/10, taking a common denominator
=> (aG+bG)/(abG) = 7/10
=> (A+B)/L = 7/10
=> (A+B)/70 = 7/10, since L = 70
=> A+B = 49
Since both conditions together yield a unique solution, they 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)
If A = 2 and B = 5, then A + B = 7.
If A = 6 and B = 15, then A + B = 21.
Since condition 1) doesn’t yield a unique solution, it is not sufficient.

Condition 2)
If A = 14 and B = 35, then A + B = 49.
If A = 2 and B = 35, then A + B = 37.
Since condition 2) doesn’t yield a unique solution, it 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]

(set) X is the set of positive integer multiples of 3, and Y is the set of positive integer multiples of 7. Define X+Y as {x+y|x ∈X and y ∈Y}. How many elements of X+Y are less than or equal to 21?

A. 4
B. 6
C. 8
D. 11
E. 13

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X = { 3, 6, 9, 12, 15, 18, 21, … } and Y = { 7, 14, 21, … }
The following pairs ( x, y ) satisfy x + y <= 21:
(3, 7), (3, 14), (6, 7), (6, 14), (9,7) and (12,7).
There are 6 such pairs, (x,y).

Therefore, B is the answer.
Answer: B

[GMAT math practice question]

(Probability) A={2, 4, 6, 8, 10} is given. What is the number of subsets of A containing 3 elements?

A. 5
B. 10
C. 12
D. 24
E. 32

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The number of subsets is equal to the number of ways ways to choose 3 elements out of 5 elements, which is 5C3 = (5*4*3)/(1*2*3) = 10.

Therefore, B is the answer.
Answer: B

[GMAT math practice question]

(number properties) What is the value of x?

1) the remainder, when 170 is divided by x, is 2
2) the remainder, when 140 is divided by x, is 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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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)
Now, 170 = x*a + 2, so 168 = 2^3*3*7 = x*a. Note that x > 2 since the dividend must be greater than the remainder.
So, x is a factor of 168 greater than 2. The possible values of x are 3, 4, 6, …, 168.
Since condition 1) doesn’t yield a unique solution, it is not sufficient.

Condition 2)
Now, 140 = x*b + 4, so 136 = x*b. Note that x > 4 since the dividend must be greater than the remainder.
So, x is a factor of 136 = 23*17 greater than 4. The possible values of x are 8, 17 and 140.
Since condition 2) doesn’t yield a unique solution, it is not sufficient.

Conditions 1) & 2).
When we consider both conditions together, x is a common factor greater than 4 of 168 = 2^3*3*7 and 136 = 2^3*17. The only possible value of x is 8.

Since both conditions together yield a unique solution, they are sufficient.

Therefore, C is the answer.
Answer: C

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]

(algebra) What is the value of (x+y)(y+z)(z+x)+5?

1) xyz=-3
2) x+y+z=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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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. 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 x + y + z = 0 from condition 2), x + y = -z, y + z = -x, and z + x = -y. Since condition 1) tells us that xyz = -3, (x+y)(y+z)(z+x)+5 = (-x)(-y)(-z) + 5 = -xyz + 5 = -(-3) + 5 = 8.
Since both conditions together yield a unique solution, they are sufficient.

Therefore, C is the answer.
Answer: 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.

[GMAT math practice question]

(number properties) Let aob denote the greatest common divisor of a and b, and let a□b denote the least common multiple of a and b. What is (xoy)□(x□y)?

1) (63o99)x=540
2) 3y-(18□45)=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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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

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)

The following reasoning shows that condition 1) implies that x = 60:
(63o99)x=540
=> (9*7o9*11)x=540
=> 9x=540
=> x=60

The following reasoning shows that condition 2) implies that y = 60:
3y-(18□45)=0
=> 3y-(9*2□9*5)=0
=> 3y-9*2*5=0
=> 3y-90=0
=> y-30=0
=> y=30

Thus, we may calculate
(xoy)□(x□y) = (60o30)□(60□30) = 30□60=60

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.

However, each of the conditions only gives us information about one of the variables, so neither is sufficient on its own.

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]

(absolute values) What is the value of 2x-y?

1) |3x-2y+4| + |-x+2y-2|=0
2) x and y are rational numbers


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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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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

Condition 1) is equivalent to two equations: 3x-2y+4=0 and –x+2y-2 = 0. When we solve this system of occasions, we obtain x = -1 and y = 1/2. Thus, condition 1) is sufficient on its own.
Note the VA tells us this will be the case. The question has two variables, but condition 1) provides 2 equations.

Condition 2) is clearly not sufficient on its own.

Therefore, the answer is A.
Answer: A

[GMAT math practice question]

(algebra) <x,y> denotes x + y/2. What is the value of x?

1) <x,y> = y + x/2
2) <2x,2y>+1=<y,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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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

When we simply condition 2), we have
<2x, 2y>+1=<y, x>-2
=> 2x + 2y/2 + 1 = y + x/2 - 2
=> 2x + y + 1 = y + x/2 – 2
=> (3/2)x = -3
=> x = -2.
Condition 2) is sufficient.

Condition 1)
Since we have <x,y> = y + x/2 = x + y/2, we have y/2 = x/2 or x = y.
Condition 1) is not sufficient, since it does not yield a unique solution.

Therefore, B is the answer.
Answer: B

[GMAT math practice question]

(ratio) There is a% of saline solution of 500g. Alice wants to have b% of saline solution by boiling saline solution where b > a. What is the lost weight of the saline solution by boiling?

1) a = 20
2) a/b = 2/5

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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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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

Let x be the lost weight of b% saline solution by boiling.
The question asks the value of x such that (a/100)500 = (b/100)(500-x).
(a/100)500 = (b/100)(500-x)
=> 500a = b(500-x)
=> 500a = 500b – bx
=> bx = 500(b-a)
=> x = 500(b-a)/b
=> x = (500)(1-a/b)
Thus, condition 2) is sufficient.

When a question asks for a ratio, if one condition includes a ratio and the other condition just gives a number, the condition including the ratio is most likely to be sufficient. This tells us that B is most likely to be the answer to this question.

Condition 1) is not sufficient obviously, since we don’t have any information about the variable a.

Therefore, B is the answer.
Answer: B