Overview of GMAT Math Question Types and Patterns on the GMAT

[GMAT math practice question]

(Algebra) The table shows the difference in bag production at a factory between the production number of that day and that of the previous day. A positive number means the number of bags produced on that day is greater than the previous day. A negative number means the number of bags produced that day is less than the previous day. The production number on the 12th day is 42,000. How many bags were produced on the 8th day?

A. 41400
B. 41700
C. 41900
D. 42300
E. 42500



=>

The number of bags produced on the 11th day is 42,000 + 300 = 42,300.
The number of bags produced on the 10th day is 42,300 - 400 = 41,900.
The number of bags produced on the 9th day is 41,900 - 500 = 41,400.
The number of bags produced on the 8th day is 41,400 + 300 = 41,700.

Therefore, B is the correct answer.
Answer: B

[GMAT math practice question]

(Set) Set A is given as {1, 2, 3, … , n} and n is a positive integer. What is the value of n?

1) The number of subsets of A containing both 1 and n is 16.
2) n is less than 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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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

Let’s look at condition 1). It tells us that set A has four elements.

Remember that the number of subsets of a set with n elements is 2n.
Since 2n = 24 = 16, we have n = 4.

The answer is unique, and the condition is sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Let’s look at condition 2). It tells us that we don’t have a unique solution since n = 6 and n = 7 are possible values.

The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Condition 1) ALONE is sufficient.

Therefore, A is the correct answer.
Answer: A

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]

(Equation) What is the value of a?

1) The equation x/3 + a = x/2 - x - 18/6 in terms of x has no solution.
2) a is an integer.

=>

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 (a) and 0 equations, D is most likely the answer. So, we should consider each condition on its own first.

Let’s look at the condition 1). It tells us that a could be any value other than 3.

x/3 + a = x/2 – x - 18/6
⇔ x/3 + a = x/3 + 3
⇔ a = 3.

It means if a = 3, the equation has an infinite number of solutions, and if a ≠ 3, the equation has no solution. Thus, a can be any value other than 3.

The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Let’s look at the condition 2). It tells us that a does not have a unique solution.
Any integer can be the value of a.

The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Conditions 1) & 2) together tell us that we don’t have a unique solution, as a can be any integer other than 3.

The answer is not unique, and the conditions 1) and 2) together are not sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions 1) & 2) together are not sufficient.

Therefore, E is the correct answer.
Answer: E

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]

(Absolute Value) a and b satisfy a – b > 0 and ab < 0. Which of following is the expression of |a| + |b|?

A. a + b
B. a - b
C. –a + b
D. –a - b
E. b - a

=>

Since ab < 0, a and b have different signs.
Since a – b > 0, we have a > b.
Since a and b have different signs with a > b, a is positive, and b is negative.

Then |a| + |b| = a + (-b) = a – b.

Therefore, B is the correct answer.
Answer: B

[GMAT math practice question]

(Set) Among 100 employees, how many employees like both apples and bananas?

1) 53 employees like apples.
2) 72 employees like bananas.

=>

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.




A and B are sets of employees who like apples and banana, respectively. When we set this question up as the above Venn Diagram, we have x + y + z + w = 100.

Since we have 4 variables (x, y, z and w) and 1 equation (x + y + z + w = 100), E is most likely 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.
The question asks what the value of y is.

Conditions 1) & 2) together give us that they are not sufficient together.

Since 53 employees like apples, we have x + y = 53 from condition 1).
Since 72 employees like bananas, we have y + z = 72 from condition 2).

If x = 20, z = 39, w = 8, then we have y = 33.
If x = 21, z = 40, w = 7, then we have y = 32.

The answer is not unique, and both conditions 1) and 2) together are not sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions 1) & 2) together are not sufficient.

Therefore, E is the correct 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]

(Function) a and b are non-zero real numbers. In which quadrant is point (a, b) on located?

1) y = ax and y = b/x have an intersection.
2) a + b > 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.
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 (a and b) and 0 equations, C is most likely 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) together give us:

Since a and b have the same parities in order for y = ax and y = b/x to have an intersection, we have ab > 0 from condition 1).

Since we have a + b > 0 from condition 2), we have a > 0 and b > 0 when we consider both conditions together.

Then point (a, b) is in the first quadrant.

The answer is unique, and conditions 1) and 2) together are sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions 1) & 2) together are sufficient.

Therefore, C is the correct 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]

(Geometry) Quadrilaterals □ADEB and □ACGF are squares with the same sides. What is the measure of the angle ∠x?

A. 45° B.55° C. 75° D. 80° E. 90°



=>

Since AB = AC = AD = AF and ∠DAC = ∠BAF, triangles ADC and ABF are congruent. Then we have ∠ADC = ∠ABF.
∠x = 180° - ∠DHB = 180° – (360° - ( ∠E + ∠EDH + ∠EBH ))
= 180° – ( 36° – ( 90° + ∠EDA - ∠ADC + ∠EBA + ∠ABF ) )
= 180° – ( 360° – ( 90° + 90° - ∠ADC + 90° + ∠ADC ) )
= 180° – ( 360° – 270° ) = 90°

Therefore, E is the correct answer.
Answer: E

[GMAT math practice question]

(Arithmetic) a = 4/39. What is the value of 10^6a - a?

A. 10,2500
B. 102,510
C. 102,555
D. 102,564
E. 102,573


=>

10^6a - a = a(10^6 - 1) = 4/39 · 999,999 = 4/13 · 333,333
= 4 · 25641 = 102,564

Therefore, D is the correct answer.
Answer: D

[GMAT math practice question]

(Number Properties) m and n are positive integers. What is the value of mn?

1) 2.03(n/m) = (0.3)^2
2) m and n are relatively prime 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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have 2 variables (m and n) and 0 equations, C is most likely 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) together tell us that m and n are relatively prime integers with 203n = 9m.

We have 203/100*n/m=(3/10)^2=9/100 or 203n = 9m from condition 1).
Since m and n are relatively prime, we have m = 203 and n = 9.

The answer is unique, and conditions 1) and 2) together are sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

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.

Let’s look at condition 1). It tells us that 203n = 9m.
If m = 203 and n = 9, then we have mn = 1827.
If m = 406 and n = 18, then we have mn = 7308.

The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Let’s look at condition 2). It tells us that m and n are relative primes.
If m = 203 and n = 9, then we have mn = 1827.
If m = 2 and n = 3, then we have mn = 6.

The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions 1) & 2) together are sufficient.

Therefore, C is the correct 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]

(Function) f is a function mapping from {1, 2, 3} to {1, 2, 3, 4, 5}. x + f(x) is odd. How many possible cases of f(x) do we have?

A. 9
B. 10
C. 11
D. 12
E. 13

=>

If x is an odd number, then f(x) is an even number.
If x is an even number, then f(x) is an odd number.
Then the possible values of f(1) and f(3) are 2 and 4. The number of possible cases for f(1) and f(3) is 2.
The possible values of f(2) are 1, 3, and 5. The number of possible cases for f(2) = 3.

Thus. the number of possible cases for the function f(x) is 2·2·3 = 12.

Therefore, D is the correct answer.
Answer: D

[GMAT math practice question]

(Number Properties) x, y, and z are positive integers and z < y < x. What is the value of 1/x + 1/y + 1/z?
1) 1/x + 1/y + 1/z is an integer.
2) x = yz and y and z are consecutive integers and 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.
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. Let’s look at both conditions together. However, since the value of condition (1) is equal to the value of condition (2), by Tip 1, we get D as the most likely answer. Let’s look at each condition separately

Let’s look at the condition 1). It tells us that since we have z ≥ 1, y ≥ 2, and x ≥ 3, we have 1/z ≤ 1, 1/y ≤ 1/2, and 1/x ≤ 1/3.
1/x + 1/y + 1/z ≤ 1 + 1/2 + 1/3 = 1 + 3/6 + 2/6 = 1 + 5/6 = 6/6 + 5/6 = 11/6 < 2 from condition 1).
Since the unique positive integer less than 2 is 1, we have 1/x + 1/y + 1/z = 1.

The actual values of x, y, and z are 6, 3, and 2, respectively.

The answer is unique, so the condition is sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Let’s look at the condition 2). It tells us that

Since 2 and 3 are unique consecutive integers and prime numbers, we have y = 3 and z = 2.
If x = 6, y = 3 and z = 2, then 1/x + 1/y + 1/z = 1/6 + 1/3 + 1/2 = 1/6 + 2/6 + 3/6 = 6/6 = 1.

The answer is unique, so the condition is sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Each condition ALONE is sufficient.

Therefore, D is the correct 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.

[GMAT math practice question]

(Geometry) The figure shows the points A, B, C, … , G and ∠B = 70°, ∠C = 68°, ∠D = 78°, ∠E = 82°, ∠F = 86° and ∠G = 88°. What is ∠A?



A. 58°
B. 60°
C. 68°
D. 70°
E. 88°

=>



∠DBE + ∠CEB = ∠BDC + ∠ECD

The sum of all the interior angles of quadrilateral ACDF and triangle GBE is 360° + 180° = 540°.
The measure of the angle ∠A is
540° – ( 70° + 68° + 78° + 82° + 86° + 88° ) = 68°.

Therefore, the correct answer is C.
Answer: C

[GMAT math practice question]

(Number Properties) m is a three-digit positive integer. What is the value of m?

1) The digits of m are 5, 6, and 7 without repetition.
2) m is a product of two consecutive positive 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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.
We have to find the value of m, if m is a three-digit positive integer.
Follow the second and the third step: From the original condition, we have 3 variables (3-digit integer). To match the number of variables with the number of equations, we need 3 equations. Since conditions (1) and (2) will provide 1 equation each, E would most likely be the answer.

Recall 3 Principles and choose E as the most likely answer.
Let’s look at both conditions together.

Conditions (1) and (2) tell us that the digits of m are 5, 6, and 7 without repetition, and m is a product of two consecutive positive integers, from which we get m = 756.

The possible values of m are 567, 576, 657, 675, 756, and 765 from condition (1).
We must find the prime factors of the possible values, which are 567 = 3^4·7, 576 = 2^6·3^2, 657 = 3^2·73, 675 = 3^3·5^2, 756 = 2^2·3^3·7, and 765 = 3^2·5·17.

Then we see that 756 = 2^2·3^3·7 = 27·28, which is a unique number as a product of two consecutive integers. The answer is unique, so both conditions (1) and (2) combined are sufficient according to Common Mistake Type 2, which states that the number of answers must be only one. So, C seems to be the answer.

However, since this question is a hidden integer question, which is also one of the key questions, we should apply CMT 4(A), which states that if an answer C is found too easily, either A or B should be considered as the answer. Let’s look at each condition separately,

Condition (1) tells us that the digits of m are 5, 6, and 7 without repetition, from which we get that the possible values of m are 567, 576, 657, 675, 756, and 765.

The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Condition (2) tells us that m is a product of two consecutive positive integers, from which we get that m = 132 since m = 11·12 and m = 156 since m = 12·13.

The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Therefore, both conditions (1) and (2) combined are sufficient.
Both conditions (1) and (2) together are sufficient.
Therefore, C is the correct 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]

(Number Properties) If a is the remainder when 1993^{2021} is divided by 10, what is the value of |a - 1| + |a - 2| + |a - 3| + |a - 4| + |a - 5|?

A. 2
B. 3
C. 4
D. 5
E. 6

=>

3^1 = 3~3, 3^2 = 9~9, 3^3 = 27~7, 3^4 = 81~1, 3^5~3, …
So, the units digits of 3^n have a period of 4.
They form the cycle 3 -> 9 -> 7 -> 1.

Thus, 3^n has the units digit of 3 if n has a remainder of 1 when it is divided by 4.
The remainder when 2021 is divided by 4 is 1, so the units digit of 1993^{2021} is 3 and a = 3.

|a - 1| + |a - 2| + |a - 3| + |a - 4| + |a - 5|
= |3 - 1| + |3 - 2| + |3 - 3| + |3 - 4| + |3 - 5|
= 2 + 1 + 0 + 1 + 2
= 6

Therefore, E is the correct answer.
Answer: E

[GMAT math practice question]

(Function) f(x) is a function, mapping positive integers to positive integers. What is the value of f(2) + f(3) + f(5)?

1) f(1) = 1.
2) f(a+b) = f(a) + f(b) + ab.

=>

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.

Now we will solve this DS question using the Variable Approach.
Let’s apply the 3 steps suggested previously.

Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.
We have to find the value of f(2) + f(3) + f(5).

Follow the second and the third step: From the original condition, we have many variables to determine a function f(x). To match the number of variables with the number of equations, we need many equations. Since conditions (1) and (2) will provide 1 equation each, E would most likely be the answer.

Recall 3 Principles and choose E as the most likely answer. Let’s look at both conditions 1) & 2) together.

Since f(1) = 1, we have f(2) = f(1+1) = f(1) + f(1) + 1·1 = 1 + 1 + 1 = 3 using condition 2).

Then we have f(3) = f(2+1) = f(2) + f(1) + 2·1 = 3 + 1 + 2 = 6.
f(5) = f(3+2) = f(3) + f(2) + 3·2 = 6 + 3 + 6 = 15.

Thus, we have f(2) + f(3) + f(5) = 3 + 6 + 15 = 24.

The answer is unique, so both conditions together are sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions 1) & 2) together are sufficient.

Therefore, C is the correct 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) <x> denotes the unit digit of x. For example, <123> = 3. What is <7^19+7^89>?

A. 0
B. 1
C. 3
D. 8
E. 9

=>

The unit digit of the powers of 7 repeats 7->9->3->1 and has a period of 4.
Since 19 = 4·3 + 3 and 19 has a remainder 3 when it is divided by 4, the units digit of 7^19 is 3, 7^19 ~ 7^3 ~ 3.
Since 89 = 4·22 + 1 and 89 has a remainder 1 when it is divided by 4, the units digit of 7^89 is 7, 7^89 ~ 7^1 ~ 7.
< 7^19 + 7^19 > ~ < 3 + 7 > ~ <10> ~ 0.

Therefore, A is the correct answer.
Answer: A

Thank u so much for taking time to help.