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Hi could i please get some help on this ds question?

Is |x−1|<1?

(1) (x−1)^2 ≤ 1
(2) x^2−1 > 0

Many thanks.

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.

Sequence S has the difference between a term and its previous term constant. Thus, it is an arithmetic sequence.

The general term of an arithmetic sequence Tn = a + (n-1)d – where ‘a’ is the first term, n is total terms and d is a common difference.

We have n = 250. Thus, \(T_{250}\) = a + (250 - 1)d = a + 249d.

We have to find n= 200 or a + 199d.

Follow the second and the third step: From the original condition, we have 2 variables (‘a’ and d). 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.

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

Condition (1) tells us that the 150th term of Sequence S is 305: 305 = a +149d – Equation (1)

Condition (2) tells us that the 100th term of Sequence S is −95: -95 = a + 99d – Equation (2)

Equation (1) – Equation (2)

=> 305 – (- 95) = a + 149d – ( a + 99d)

=> 400 = 50d

=> d = 4

Substituting d = 4 into equation (1):

=> 305 = a + 149(4)

=> 305 = a + 596

=>305 – 596 = a

=>a = -291

Therefore, \(200^{th}\) term a + 199d = -291 + 199(4) = -291+796 = 505

The answer is a unique value; both conditions combined are sufficient according to Common Mistake Type 2 which states that the answer should be a unique value.

Both conditions together are sufficient.

Therefore, C is the correct answer.

Answer: C

Que: Is \(3^x > 100\)?

(1) \(3^\sqrt{x} = 9 \)
(2) \(\frac{1}{ 3^x} > 0.01\)

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 \(3^x >100\)’

Follow the second and the third step: From the original condition, we have 1 variable (‘x’). 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 \(3^\sqrt{x} = 9\)

=> \(3^\sqrt{x} = 3^2\)

=> \(\sqrt{x} = 2\)

=> x = 4

Hence, \(3^x = 3^4 = 81 > 100\) - NO

The answer is unique NO 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 \(\frac{1}{3^x} > 0.01\)

For any value of x, \(3^x\) is positive, also \(\frac{1}{ 0.01}\) is positive. Thus, taking reciprocal and reversing the inequality:

=> \(3^x < \frac{1}{0.01 }\)

=> \(3^x < 100\) - NO

The answer is unique NO 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.

Also, according to Tip 1, it is about 95% likely that D is the answer when condition (1) = condition (2).

Each condition alone is sufficient.

Therefore, D is the correct answer.

Answer: D

Que: Gold gym has few members in two batches, A and B. The gym can divide the members in batch A into eight groups of x members each. However, if it divides the members in batch B into four groups of y members each, three members will be leftover. How many members are in the gym?

(1) \(x = \frac{(y – 1)}{ 2}\)
(2) Number of members in batch B is seven more than that in batch A.

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. [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.

Let us assign variable to the number of members: ‘a’ in batch A and ‘b’ in batch B

Thus, we have a = 8x and b = 4y + 3

We have to find the value of a + b

Follow the second and the third step: From the original condition, we have 4 variables (a, b, x, and y) and 2 equations (a = 8x and b = 4y + 3). 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.

Recall 3- Principles and Choose C as the most likely answer.

Let’s look at both conditions combined together.

Condition (1) tells us that \(x = \frac{(y – 1) }{ 2}\)

=> y = 2x + 1

Condition (2) tells us that the number of members in batch B is seven more than that in batch A

=> b = a + 7

=> 4y + 3 = 8x + 7

=> 4y = 8x + 4

=> y = 2x + 1

Thus, a + b = 8x + 4y + 3

=> 8x + 4(2x + 1) + 3

=> 8x + 8x + 4 + 3

=> 16x + 7

The value of ‘x’ is unknown.

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

Both conditions together are not sufficient.

Therefore, E is the correct answer.

Answer: E

Que: How many distinct positive factors do the integer m have?

(1) m = \(p^3 q^2\) , where p and q are distinct positive prime numbers.
(2) The only positive prime factors of m are 2 and 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.

We have to find the number of distinct positive factors of the integer m

Follow the second and the third step: From the original condition, we have 1 variable (m). 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 m = \(p^3 q^2\), where p and q are distinct positive prime numbers.

The number of positive factors of a number N, expressed in its prime form as \(N = p^x q^y\), where p and q are distinct primes, is given by (x + 1) (y + 1).

=> m = \(p^3 q^2\). Thus, the number of positive factors = (3 + 1) (2 + 1) = 12

The answer is unique; 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 the only positive prime factors of m are 2 and 3.

Here, m can take multiple possible values. For example, 2 × 3, \(2^2 × 3^3\), \(2^4 × 3\), etc. all have 2 and 3 as the only two prime factors. However, the number of factors of each of the above numbers is different.

The answer is not a unique value; 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 not sufficient.

Therefore, A is the correct answer.

Answer: A

Que: If 2 < m < 3, is the tenths digit of the decimal representation of m equal to 8?

(1) m + 0.01 < 3
(2) m + 0.05 > 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.

We have to find ‘Is the tenths digit of the decimal representation of m equal to 8’

For any decimal number, say [m•abc], where a, b, c are the digits after the decimal point, the tenths digit refers to the digit a, i.e. the digit just after the decimal point.

We have to find ‘Is a = 8’

Follow the second and the third step: From the original condition, we have 1 variable (m). 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.

Condition (1) tells us that m + 0.01 < 3

=> m < 3 – 0.01 = 2.99

Since 2 < m < 3, therefore

=> 2 < m < 2.99

If m = 2.88 then a = 8 – YES

But if m = 2.68 then a = 6 - NO

The answer is not unique YES or NO; condition (1) alone is not 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 m + 0.05 > 3

=> m > 3 – 0.05 = 2.95

Since 2.95 < m < 3, therefore

Thus, the possible values of ‘m’ are 2.96,2.97…

If m = 2.96 then a = 9 - NO

If m = 2.97 then a = 9 - NO

The answer is unique NO; condition (2) alone is sufficient according to Common Mistake Type 1 which states that the answer should be a unique YES or a NO.

Condition (2) alone is sufficient.

Therefore, B is the correct answer.

Answer: B

Que: If p and q are integers, what is the value of (p + q)?

(1) \(pq = 6\)
(2) \((p + q)^2 = 49\)

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 the value of p + q – ‘p’ and ‘q’ are integers

Follow the second and the third step: From the original condition, we have 2 variables (p and q). 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.

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

Condition (1) tells us that pq is 6 and condition (2) tells us that \((p + q)^2 = 49\)

=> \((p + q)^2 = 49\)

=> \(p + q = ± 7\)

If p = 1 and q = 6 then pq = 6 and \((p + q)^2 = 49\). Therefore, p + q = 7

But if p = -1 and q = -6 then pq = 6 and \((p + q)^2 = 49\). Therefore, p + q = -7

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

Both conditions together are not sufficient.

Therefore, E is the correct answer.

Answer: E

Que: If x is an integer that lies between 100 and 200, inclusive, what is the value of x?

(1) x is a multiple of 36.
(2) x is an even multiple of 45.

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 the value of ‘x’- where ‘x’ is an integer that lies between 100 and 200

Follow the second and the third step: From the original condition, we have 1 variable (x). 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 the x is a multiple of 36.

The possible multiples of ‘36’ between 100 and 200 are: 36 * 3 = 108 , 36 * 4 = 144, 36 * 5 = 180.

Thus, the value of ‘x’ is not unique.

The answer is not a unique value; condition (1) alone is not sufficient according to Common Mistake Type 2 which states that the answer should be a unique value.

Condition (2) tells us that ‘x’ is an even multiple of 45

The possible multiples of ‘45’ between 100 and 200 are: 45 * 3 = 135 , 45 * 4 = 180.

In this, even multiple is 180, and thus x = 180.

Thus, the value of ‘x’ is unique.

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

Condition (2) alone is sufficient.

Therefore, B is the correct answer.

Answer: B

Que: Set S is a set of 14 consecutive integers. Is an integer ‘7’ present in Set S?

(1) The integer −5 is present in the set.
(2) The integer 6 is present in the set.

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 an integer ‘7’ present in Set S’- where Set S is a set of 14 consecutive integers

Follow the second and the third step: From the original condition, we have 1 variable (n) since n, n+1, n+2, …., n+13. 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.

Condition (1) tells us that the integer −5 is present in the set.

We can have 14 consecutive integers including −5 in the following ways:

Set S: {−5, −4, −3, −2, . . . 6, 7, 8}; (‘7’ is present in the set) - YES

OR

Set S: {−7, −6, −5, −4, −3 . . . 4, 5, 6}; (‘7’ is not present in the set) - NO

The answer is not unique, YES and NO, so 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 the integer 6 is present in the set

We can have 14 consecutive integers including 6 in the following ways:

Set S: {−5, −4, −3, −2, . . . 6, 7, 8}; (‘7’ is present in the set) - YES

OR

Set S: {−7, −6, −5, −4, −3 . . . 4, 5, 6}; (‘7’ is not present in the set) - NO

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

Both conditions (1) and (2) tells us that −5 and 6 are present in the set.

We can have 14 consecutive integers including −5 and 6 in the following ways:

Set S: {−5, −4, −3, −2, . . . 6, 7, 8}; (‘7’ is present in the set) - YES

OR

Set S: {−7, −6, −5, −4, −3 . . . 4, 5, 6}; (‘7’ is not present in the set) - NO

The answer is not unique, YES and NO, so both conditions (1) and (2) combined are sufficient according to Common Mistake Type 1 which states that the answer should be a unique YES or a NO.

Both conditions together are not sufficient.

Therefore, E is the correct answer.

Answer: E

Que: If x and y are integers, is x - y an even number?

(1) x + 4y = odd.
(2) 3x + 11y = even.