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Solution: 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 whether \(x^2 - y^2\) is an integer, whether (x-y)(x+y) is an integer

Thus, let’s look at condition (1), it tells us that x=y, from which we get x - y = 0 or (x-y)(x+y)=0 => (x+y)=0, which is an integer.
And we get yes an answer, the answer is unique, yes, the condition is sufficient according to Common Mistake Type 1 which states that the answer should be unique Yes or a NO.

Condition (2) tells us that y=2, from which we cannot determine whether \(x^2 - y^2\) is an integer.

For example, if x=1 and y=2, then we get \(x^2 - y^2\) = \(1^2 - 2^2\)=1 - 4 = -3 which is an integer and we get yes as an answer.

However, if x=0.1 and y=2, then we get \(x^2 - y^2\)= \(0.1^2 - 2^2\)=0.01 - 4=-3.99 which is not an integer and we get no as an answer.

The answer is not unique, yes and no, and condition (1) alone is not sufficient according to Common Mistake Type 1 which states that the answer should be unique Yes or a NO if we get both yes and no as an answer, it is not sufficient.

Condition (1) alone is sufficient

Therefore, A is the correct answer

Answer: A

Que: If \(x^2\) + \(y^2\) + 2xy = 16; Is x divisible by 10?

(1) \(x^2\) + \(y^2\) = 8
(2) y = 2

Solution: To save time and improve accuracy on DS question in GMAT, learn, and apply Variable Approach.

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 ‘Is x divisible by 10’ when \(x^2 + y^2 + 2xy = 16\).

Follow the second and the third step: From the original condition, we have 2 variables (x and y) and 1 equation (\(x^2 + y^2 + 2xy = 16\)). To match the number of variables with the number of equations, we need 1 more equation. Since conditions (1) and (2) will provide 2 equations, D would most likely be the answer.

Recall 3- Principles and Choose D as the most likely answer. Let’s look at each condition separately.


Condition (1) tells us that \(x^2 + y^2\) = 8, if we substitute this equation into \(x^2 + y^2 + 2xy = 16\), then we get 8 + 2xy = 16 => 2xy = 8, or xy = 4.

Also, from \((x+y)^2\) = \(x^2\) + \(y^2\) + 2xy = 8 + 8 = 16, we get x + y = ±4, and since xy = 4, we get x=y=2 or x = y = -2. In both cases, x = 2 or -2 which are not divisible by 10, we get no as an answer.

The answer is unique, no, the condition is sufficient according to Common Mistake Type 1 which states that the answer should be unique Yes or a NO.

Condition (2) tells us that y=2, from which we get \(x^2 + y^2 + 2xy = 16\) => \(x^2\)+\(2^2\)+ 2x(2)=16 => \(x^2\) + 4 + 4x =16 => \(x^2\) + 4x - 12 = 0.

If we factorize that equation we get (x+6)(x-2)=0, or x=-6 or 2. In both cases, x=-6 or 2 which are not divisible by 10, we get no as an answer.

The answer is unique, no, the condition is sufficient according to Common Mistake Type 1 which states that the answer should be unique Yes or a NO.

According to Tip 1, if the value of the condition (1) is equal to the value of condition (2), we get D as an answer.

Each condition alone is sufficient

D is the correct answer.

Answer: D

Que: If the symbol ‘#’ represents addition, subtraction, multiplication, or division, what is the value of 14 # 7?

(1) 25 # 5 = 5
(2) 2 # 1 = 2

Solution: To save time and improve accuracy on DS question in GMAT, learn, and apply a Variable Approach.

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 a ‘value of 14 # 7’ where ‘#’ represents addition, subtraction, multiplication, or division.

Follow the second and the third step: From the original condition, we have 1 variable (#). To match the number of variables with the number of equations, we need 1 equation. Since conditions (1) and (2) will provide 1 equation each, D would most likely be the answer.

Recall 3- Principles and Choose D as the most likely answer. Let’s look at each condition separately.

Condition (1) tells us that 25 # 5 = 5, from which we get # = division, since 25 / 5 = 5.

Then 14 # 7: 14 / 7 = 2 which is a unique answer.

The answer is unique and condition (1) alone is sufficient according to Common Mistake Type 2 which states that the answer should be a unique value.


Condition (2) tells us that 2#1 = 2, from which we get # can be multiplication or division.

If # = multiplication, then 2 * 1 = 2,

=> Then 14 # 7: 14 * 7 = 98

But if # = division, then 2/ 1 = 2,

=> Then 14 # 7: 14 / 7 = 2

The answer is not a unique value and condition (2) alone is not sufficient according to Common Mistake Type 2 which states that the answer should be a unique value.

Condition (1) alone is sufficient.

Therefore, A is the correct answer.

Answer: A

Que: If x and y are positive integers, what is the value of x?

(1) \(3^x\) + \(5^y\) = 134
(2) y = 3

Solution: To save time and improve accuracy on DS question in GMAT, learn, and apply Variable Approach.

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.

Forget the conventional way to solve DS questions.

We will solve this DS question using the variable approach.

Remember the relation between the Variable Approach, and Common Mistake Types 3 and 4 (A and B)[Watch lessons on our website to master these approaches and tips]

Step 1: Apply Variable Approach (VA)

Step II: After applying VA, if C is the answer, check whether the question is key questions.

Step III: If the question is not a key question, choose C as the probable answer, but if the question is a key question, apply CMT 3 and 4 (A or B).

Step IV: If CMT3 or 4 (A or B) is applied, choose either A, B, or D.

Let's apply CMT (2), which says there should be only one answer for the condition to be sufficient. Also, this is an integer question and, therefore, we will have to apply CMT 3 and 4 (A or B).

To master the Variable Approach, visit https://www.mathrevolution.com and check our lessons and proven techniques to score high in DS questions.

Let’s apply the 3 steps suggested previously. [Watch lessons on our website to master these 3 steps]

Step 1 of the Variable Approach: Modifying and rechecking the original condition and the question.

We have to find the value of ‘x’ - where 'x' and 'y' are integers

Second and the third step of Variable Approach: From the original condition, we have 2 variables (x and y). To match the number of variables with the number of equations, we need 2 equations. Since conditions (1) and (2) will provide 1 equation each, C would most likely be the answer.

Let’s look at both conditions together.

They tells us that \(3^x\) + \(5^y\) = 134 and y = 3, from which we get 3x + \(5^3\) = 134.

=> 3x + 125 = 134

=> 3x = 134 - 125 = 9 = \(3^2\)

=> x = 3.

The answer is unique, so the conditions combines are sufficient according to Common Mistake Type 2 which states that the answer should be a unique value.

But we know that this is a key question [Integer question] and if we get an easy C as an answer, we will choose A or B.

Let’s take a look at each condition.

Condition (1) tells us that \(3^x\) + \(5^y\) = 134.

=> If y = 1: 3x + 5 = 134

=> 3x = 129

However, 129 cannot be expressed as an exponent of 3 => y ≠ 1

=> If y = 3: 3x + 125 = 134

=> 3x = 9

=> x = 3

Thus, there is only one solution: x = 3

The answer is unique and condition (1) alone is sufficient according to Common Mistake Type 2 which states that the answer should be a unique value.

Condition(2) tells us that y = 3.

=> But x is unknown.

The answer is not a unique value therefore condition (2) alone is not sufficient according to Common Mistake Type 2 which states that the answer should be a unique value.

If the question has both C and A as its answer, then A is the answer rather than C according to the definition of DS questions.


Condition (1) alone is sufficient.

Therefore, A is the correct answer.

Answer: A

Que: If P, Q, & R are numbers on the number line, not necessarily in that order, is |P − R| ≥ 13?

(1) |P − Q| = 65
(2) |Q − R| = 52

Solution: To save time and improve accuracy on DS questions in GMAT, learn, and apply the Variable Approach.

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 is |P − R| ≥ 13.

Follow the second and the third step: From the original condition, we have 3 variables (P, Q , and R). 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.

Condition (1) tells us that |P − Q| = 65 => P – Q = ± 65

Condition (2) tells us that |Q − R| = 52 => Q – R = ± 52

Adding both the equations,

=> P – Q + Q – R = ± 65 ± 52

=> 65 + 52 = 117 OR 65 – 52 = 13 OR -65 + 52 = -13 OR -65-52 = -117

=> P – R = ± 117 OR ± 13

=> |P − R| = 117 OR 13

=> Is |P − R| ≥ 13 – YES

The answer is unique, YES, and both conditions combined together are sufficient according to Common Mistake Type 1 which states that the answer should be a unique Yes or a NO.

Both conditions combined together are sufficient.

Therefore, C is the correct answer.

Answer: C

Que: If m and n are integers and p = 13m + 25n, is p odd?

(1) One of m and n is odd.
(2) n is even

Solution: To save time and improve accuracy on DS questions in GMAT, learn, and apply Variable Approach.

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. [Watch lessons on our website to master these 3 steps]

Step 1 of the Variable Approach: Modifying and rechecking the original condition and the question.

We have to find ‘Is p odd’ ? - where 'm' and 'n' are integers and p = 13m + 25n.

Since 13m + 25n=12m + 24n + m + n and 12m + 24n is always even, we should know whether m+n is odd.

Thus, let’s look at condition (1), it tells us that one of ‘m’ and ‘n’ is odd, from which we can get m + n=odd since (m,n)=(odd, even) or (even, odd) and gives yes as an answer. The answer is unique, YES, and condition (1) alone is sufficient according to Common Mistake Type 1 which states that the answer should be a unique YES or a NO.

Condition (2) tells us that (2) n is even, from which we cannot determine whether 13m + 25n is odd.

For example, if (m,n)=(1,2), then 13m + 25n = 13(1) + 25(2)=63= ODD and we get YES as an answer.

However, if (m,n) = (2,2), then 13m + 25n = 13(2) + 25(2)=76=even and we get no as an answer.

The answer is not a unique YES or a NO therefore condition (2) alone is not sufficient according to Common Mistake Type 1 which states that the answer should be a unique YES or a NO.


Condition (1) alone is sufficient.

Therefore, A is the correct answer.

Answer: A

Que: If ‘P, Q, and R’ are points on the number line, what is the distance between Q and R?

(1) The points P and Q are 20 units apart.
(2) The points P and R’ are 25 units apart.

Solution: To save time and improve accuracy on DS questions in GMAT, learn, and apply Variable Approach.

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 ‘distance between Q and R’ when ‘P, Q and R’ are on a number line’.

Follow the second and the third step: From the original condition, we have 3 variables (P, Q, and R). 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 combined together.

Condition (1) tells us that the points P and Q are 20 units apart.

Condition (2) tells us that the points P and R are 25 units apart.

=> If the points are as: P_____Q____R, then the distance between Q and R is 25 – 20 = 5
=> But if the points are as: Q_____P____R, then the distance between Q and R is 25 + 20 = 45

The answer is not unique and both conditions combined together are not sufficient according to Common Mistake Type 2 which states that the answer should be a unique value.

Both conditions combined together are not sufficient

E is the correct answer.

Answer: E

Que: Is 25% of 'n' greater than 20% of the sum of n and \(\frac{1}{2}\) ?

(1) 0 < n < 1
(2) n > 0.5

Solution: Forget the conventional way to solve DS questions. We will solve this DS question using the variable approach.

The first step of the Variable Approach: The first step and the priority is to modify and recheck the original condition and the question to suit the type of information given in the condition.

To master the Variable Approach, visit https://www.mathrevolution.com and check our lessons and proven techniques to score high in DS questions.

Learn the 3 steps. [Watch lessons on our website to master these 3 steps]

Step 1 of the Variable Approach: Modifying and rechecking the original condition and the question.

We have to find ‘Is 25% of n greater than 20% of the sum of n and \(\frac{1}{ 2}\)’?

=> 25% of n > 20% of (n + \(\frac{1}{ 2}\))

=> \(\frac{25}{100}\) * n > \(\frac{20}{100}\) * (n + \(\frac{1}{ 2}\))

=> \(\frac{(25n)}{100}\) > \(\frac{(20n)}{100}\) + \(\frac{20}{100}\) * 2

=> \(\frac{n}{4}\) > \(\frac{n}{5}\) + \(\frac{1}{10}\)

=> \(\frac{n}{4}\) - \(\frac{n}{5}\) > \(\frac{1}{10}\)

=> \(\frac{(5n – 4n) }{ 20}\) > \(\frac{1}{10}\)

=> \(\frac{n }{ 20}\) > \(\frac{1}{10}\)

=> n > \(\frac{20}{10}\)

=> n > 2

So, we have to know ‘Is n > 2’?


Condition (1) tells us that 0 < n < 1.

=> n ≯ 2

Since the answer is a unique NO, condition (1) alone is sufficient by CMT 1 which states that there should be a unique Yes or a NO.

Condition (2) tells us that n > 0.5.

=> If n = 3 then n > 0.5 and n > 2 - YES

=> But if n = 1 then n > 0.5 but n ≯ 2 - NO

Since the answer is not unique, YES and NO, condition (2) alone is not sufficient by CMT 1 which states that there should be a unique Yes or a NO.

Condition (1) alone is sufficient.

So, A is the correct answer.

Answer: A

SAVE TIME: By Variable Approach[MODIFICATION], check the condition quickly and separately and mark answer as A or B.

Que: If xy = 6, is x < y?

(1) y ≥ 3
(2) y ≤ 3

Solution: To save time and improve accuracy on DS questions in GMAT, learn and apply the Variable Approach.

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.

Let’s apply the 3 steps suggested previously. [Watch lessons on our website to master these 3 steps]

Step 1 of the Variable Approach: Modifying and rechecking the original condition and the question.

We have to find ‘Is x < y’? - where 'xy = 6'

Second and the third step of Variable Approach: From the original condition, we have 2 variables (x, and y) and 1 Equation (xy = 6). To match the number of variables with the number of equations, we need 1 equation. Since conditions (1) and (2) will provide 1 equation each, D would most likely be the answer.

Recall 3- Principles and Choose D as the most likely answer. Let’s look at both conditions together.

Let’s take a look at each condition.

Condition (1) tells us that y ≥ 3.

If we substitute y = \(\frac{6}{x}\) into y≥3, then we get \(\frac{6}{x}\) ≥ 3.

Also y ≥ 3 > 0 and xy = 6>0, so we get x>0 and if we multiply both sides of \(\frac{6}{x}\) ≥ 3 by x, we get 6 ≥ 3x and divide by '3', we get 2 ≥ x. And since y ≥ 3>2≥x, we get y>x and yes becomes an answer.


The answer is unique, YES, and condition (1) alone is sufficient according to Common Mistake Type 1 which states that the answer should be a unique YES or a NO.

Condition (2) tells us that y ≤ 3.

=> If y = 3, then x * 3 = 6 => x = 2 => x < y - YES

=> If y = 1, then x * 1 = 6 => x = 6 => x < y - NO

The answer is not a unique YES or a NO therefore condition (2) alone is not sufficient according to Common Mistake Type 1 which states that the answer should be a unique YES or a NO.

Condition (1) alone is sufficient.

Therefore, A is the correct answer.

Answer: A