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[GMAT math practice question]

(Inequalities) What is the summation of the maximum and the minimum values of \(6x - 37\)?

1) \(x\) satisfies \(2 < \sqrt{3(x-4)} ≤ 5\).

2) \(x\) is an integer.
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MathRevolution wrote:
[GMAT math practice question]

(Number Properties) What is the solution of (\(x, y\))’s satisfying \(\sqrt{500} = \sqrt{x} +\sqrt{y}\) and \(x < y\)?

1) \(x\) and \(y\) are positive integers.
2) \(x = 5t^2\) with \(0 < t < 5.\)


=>

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.

Condition 1)
\(x = 5 = 5*1^2, y = 5*9^2 = 405\) and \(x = 5*2^2, y = 5*8^2 = 320\) are possible solutions.

Condition 2)
\(x = 5 = 5*1^2, y = 5*9^2 = 405\) and \(x = 5*2^2, y = 5*8^2 = 320\) are possible solutions.

Conditions 1) & 2)

\(x = 5 = 5*1^2, y = 5*9^2 = 405\) and \(x = 5*2^2, y = 5*8^2 = 320\) are possible solutions.

Since both conditions together do not yield a unique solution, they are not sufficient.

Therefore, E is the 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.

Originally posted by MathRevolution on 08 Apr 2020, 18:42.
Last edited by MathRevolution on 20 Apr 2021, 02:51, edited 1 time in total.
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Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink]
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[GMAT math practice question]

(Inequalities) What is the value of (\(x, y\)) satisfying \(\sqrt{x^2+4y}\), with the integer portion equaling \(5\).

1) \(x\) and \(y\) are the numbers of eyes on two dice.

2) \(x\) and \(y\) are positive integers.
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MathRevolution wrote:
[GMAT math practice question]

(Inequalities) What is the summation of the maximum and the minimum values of \(6x - 37\)?

1) \(x\) satisfies \(2 < \sqrt{3(x-4)} ≤ 5\).

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

Condition 1)

\(2 < \sqrt{3(x-4)} ≤ 5\)

\(=> 4 < 3(x - 4) ≤ 25\) (squaring)

\(=> \frac{4}{3} < x - 4 ≤ \frac{25}{3}\) (dividing by \(3\))

\(=> \frac{16}{3} < x ≤ \frac{37}{3}\) (adding \(4\))

Then even though x has a maximum value, x doesn’t have a minimum value.

Since condition 1) does not yield a unique solution, it is not sufficient.

Condition 2)

Since condition 2) does not yield a unique solution; obviously, it is not sufficient.

Conditions 1) & 2)

Since we have \(\frac{16}{3} < x ≤ \frac{37}{3}\) from condition 1), the possible values of \(x\) are \(6, 7, 8, …, 12.\)

Then the maximum and minimum values of \(x\) are \(6\) and \(12\), respectively.

Thus, their sum is \(6 + 12 = 18.\)

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.

Originally posted by MathRevolution on 09 Apr 2020, 19:08.
Last edited by MathRevolution on 06 May 2021, 04:15, edited 1 time in total.
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Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink]
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[GMAT math practice question]

(Inequalities) \(M\) denotes the maximum of \(x\) and m the minimum of \(x\). What is the integer part of \(\sqrt{M-m}? \)

1) The integer part of \(\sqrt{3x-2}\) is \(9. \)

2) \(x\) is a positive integer.
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MathRevolution wrote:
[GMAT math practice question]

(Inequalities) What is the value of (\(x, y\)) satisfying \(\sqrt{x^2+4y}\), with the integer portion equaling \(5\).

1) \(x\) and \(y\) are the numbers of eyes on two dice.

2) \(x\) and \(y\) are 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.

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 the integer part of \(\sqrt{x^2+4y},\) we have

\(5≤\sqrt{x^2+4y} < 6\)

=> \(25≤\sqrt{x^2 + 4y} < 36\)

\(X = 1, y = 6\) and \(x = 2, y = 6\) are possible solutions.

Since both conditions together do not yield a unique solution, they are not sufficient.

Therefore, E is the answer.
Answer: E

Originally posted by MathRevolution on 10 Apr 2020, 17:11.
Last edited by MathRevolution on 04 Apr 2021, 03:40, edited 1 time in total.
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[GMAT math practice question]

(Number Properties) \(a, b, \)and \(c\) are \(3\) different unit numbers. What is the \(3\)-digit number \(abc\)?

1) The \(5\)-digit number \(ababc\) is a multiple of \(12\).

2) The \(2\)-digit number \(ab\) is equal to \(c^2\).
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Expert Reply
MathRevolution wrote:
[GMAT math practice question]

(Inequalities) \(M\) denotes the maximum of \(x\) and m the minimum of \(x\). What is the integer part of \(\sqrt{M-m}? \)

1) The integer part of \(\sqrt{3x-2}\) is \(9. \)

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

Condition 1)

\(9 ≤ \sqrt{3x-2} < 10\)

=> \(81 ≤ 3x - 2 < 100\) (squaring)

=> \(83 ≤ 3x < 102\) (adding \(2\))

=> \(\frac{83}{3} ≤ x < \frac{102}{3}\) (dividing by \(3\))

Since \(x\) doesn’t have a maximum value, condition 1) is not sufficient.

Condition 2)

Since \(x\) doesn’t have a maximum value, condition 2) is not sufficient.

Conditions 1) & 2)
When we consider both conditions together, the possible values of \(x\) are \(28, 29, …, 33.\)

Then we have \(M = 33\) and \(m = 29.\)
Thus, we have \(\sqrt{M-m}= \sqrt{4} = 2.\)

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.

Originally posted by MathRevolution on 12 Apr 2020, 04:53.
Last edited by MathRevolution on 28 Apr 2021, 03:52, edited 1 time in total.
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MathRevolution wrote:
[GMAT math practice question]

(Number Properties) \(a, b, \)and \(c\) are \(3\) different unit numbers. What is the \(3\)-digit number \(abc\)?

1) The \(5\)-digit number \(ababc\) is a multiple of \(12\).

2) The \(2\)-digit number \(ab\) is equal to \(c^2\).


=>

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

Since we have \(3\) variables (\(a, b\), and \(c\)) and \(0\) equations, 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.

Conditions 1) & 2)

Since the \(5\)-digit number \(ababc\) is a multiple of \(12\), it is a multiple of both \(3\) and \(4\). It means we have \(a + b + a + b + c = 2a + 2b + c\), which is a multiple of \(3\), and \(10b + c\) is a multiple of \(4\) since we can check the multiplicity of \(3\) with the sum of all digits and the multiplicity of \(4\) with the last two digits.

Since the \(2\)-digit number ab is equal to \(c^2\), we have \(10a + b = c^2.\)

Then, the possible solutions of (\(a, b, c\)) are (\(1, 6, 4\)), (\(4, 9, 7\)), (\(6, 4, 8\)), (\(8, 1, 9\)) from condition 2) since \(a, b\), and \(c\) are different and \(a\) is not equal to \(0\).

When we apply condition 1), we get \(a = 1, b = 6\) and \(c = 4. \) This is because \(a + b + a + b + c\) = multiple of \(3\), and in this case \(1 + 6 + 1 + 6 + 4 = 18\), which is a multiple of \(3\).

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)
\(a = 1, b = 3, c = 2\) and \(a = 7, b = 3, c = 2\) are possible solutions.

Since condition 1) does not yield a unique solution, it is not sufficient.

Condition 2)
\(A = 1, b = 6, c = 4\). and \(a = 4, b = 9, c = 7\) are possible solutions from the above reasoning.

Since condition 2) does not yield a unique solution, it is not 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.

Originally posted by MathRevolution on 12 Apr 2020, 04:56.
Last edited by MathRevolution on 23 Oct 2021, 12:10, edited 1 time in total.
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[GMAT math practice question]

(Number Properties) \(x\) is a \(3\)-digit positive integer. What is the value of \(x\)?

1) \(x\) is a multiple of \(9\) and \(\sqrt{3x}\) is an integer.

2) \(x\) is less than \(200\).
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Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink]
What are the best short-term (2 months or less) cram options for the GMAT?
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Expert Reply
=>

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 a three-digit integer, we have \(3\) variables, and 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.

Conditions 1) & 2)

Since \(\sqrt{3x}\) is an integer, we have \(3x = k^2\) for some integer \(k\).
Then \(k\) must be a multiple of \(3\), and we have \(k = 3n\) for some integer as well since \(3\) is a prime number.
Thus, we have \(3x = k^2 = (3n)^2 = 9n^2\) or \(x = 3n^2. \)
Since \(x\) is a \(3\)-digit integer less than \(200\), we have \(100 ≤ x = 3n^2 < 200\) or \(33.3 ≤ n^2 < 66.7\) (dividing all terms by \(3\)).
Then we have \(34 ≤ n^2 ≤ 66\) (because the answer is an integer according to the original condition) and squares between \(34\) and \(66\), inclusive are \(36, 49,\) and \(64.\)
The possible values of \(x\) are \(3*36 = 108, 3*49 = 147\) and \(3*64 = 192.\)

Since both conditions together do not yield a unique solution, 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.
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[GMAT math practice question]

(Function) What is the integer closest to \(r=\frac{1}{f(1)}+\frac{1}{f(2)}+…+\frac{1}{f(50)}\)?

1) \(f(a)=\sqrt{a}+\sqrt{a+1}\)

2) \(r\) is an irrational number.
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[GMAT math practice question]

(Statistics) The standard deviation of the salaries of the workers in company \(A\) is \(\sqrt{5}\) and that of the workers in company \(B\) is \(\sqrt{30}.\) What is the standard deviation of the salary of company \(A\) and B’s workers together?

1) The number of workers in company \(A\) is \(20\) and that of company \(B\) is \(30\).
2) The averages of the salaries of the workers in companies \(A\) and \(B\) are the same.
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MathRevolution wrote:
[GMAT math practice question]

(Function) What is the integer closest to \(r=\frac{1}{f(1)}+\frac{1}{f(2)}+…+\frac{1}{f(50)}\)?

1) \(f(a)=\sqrt{a}+\sqrt{a+1}\)

2) \(r\) is an irrational number.


=>

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. If we determine the value of \(f(x)\), then we can get a solution.

Condition 1)
Since \(f(a)= \sqrt{a}+\sqrt{a+1}, \)

we have
\(\frac{ 1}{f(a)}=\frac{1}{\sqrt{a}+\sqrt{a+1}}\)

\(= \frac{(\sqrt{a} - \sqrt{a+1})}{(\sqrt{a} + \sqrt{a+1})(\sqrt{a} - \sqrt{a+1})} \)(multiplying both the denominator and numerator by the conjugate)

\(=\frac{(\sqrt{a} - \sqrt{a+1}}{a - (a+1)}\) (multiplying the denominator)

\(=\frac{(\sqrt{a} - \sqrt{a+1})}{a - a - 1}\) (multiplying -1 through the bracket)

\(= \frac{(\sqrt{a} - \sqrt{a+1})}{-1}\) (adding like terms)

\(= -\sqrt{a} + \sqrt{a+1}\) (dividing by -1)

Then \(\frac{1}{f(1)}+\frac{1}{f(2)}+⋯+\frac{1}{f(50)}=(-\sqrt{1}+\sqrt{2})+(-\sqrt{2}+\sqrt{3})+⋯+(-\sqrt{50}+\sqrt{51})\\
= -1+\sqrt{51}.\)

Since \(7 < \sqrt{51} < 7.5\), we have \(6 < \sqrt{51}-1 < 6.5\) and the integer closest to \(r\) is \(6\).

Since condition 1) yields a unique solution, it is sufficient.

Condition 2)

Since we don’t have any specific definition of f, condition 2) does not yield a unique solution, and it is not sufficient.

Therefore, A is the answer.
Answer: A

Originally posted by MathRevolution on 16 Apr 2020, 08:00.
Last edited by MathRevolution on 12 Jan 2021, 03:11, edited 1 time in total.
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[GMAT math practice question]

(Statistics) What is the standard deviation of x1, x2, …, xn?

1) The average of x1, x2, …, xn is 1.

2) The average of x1^2, x2^2, …, xn^2 is 5.
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MathRevolution wrote:
[GMAT math practice question]

(Statistics) The standard deviation of the salaries of the workers in company \(A\) is \(\sqrt{5}\) and that of the workers in company \(B\) is \(\sqrt{30}.\) What is the standard deviation of the salary of company \(A\) and B’s workers together?

1) The number of workers in company \(A\) is \(20\) and that of company \(B\) is \(30\).
2) The averages of the salaries of the workers in companies \(A\) and \(B\) are the same.


=>

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 don’t know how many workers companies A and B have, we have many variables and 0 equations, and 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.

Conditions 1) & 2)
Let \(m\) be the average of the salaries of \(A\) and \(B.\)

Assume a1, a2, …, a20 are the salaries of the workers in A and b1, b2, …, b30 are those of the workers in B.
Since the standard deviation of the company A is \(\sqrt{5}\), we have
[(a1-m)^2+(a2-m)^2 + … + (a20-m)^2] / 20 = 5 or (a1-m)^2+(a2-m)^2 + … + (a20-m)^2 = 100.
Since the standard deviation of the company B is \(\sqrt{30}\), we have
[(b1-m)^2+(b2-m)^2 + … + (b30-m)^2] / 30 = 30 or (b1-m)^2+(b2-m)^2 + … + (b30-m)^2 = 900.
Then, we have
[(a1-m)^2+(a2-m)^2 + … + (a20-m)^2 + (b1-m)^2+(b2-m)^2 + … + (b30-m)^2] / 50
= (100 + 900) / (20 + 30) = 1000/50 = 20.
The standard deviation of the combined set of A and B is \(\sqrt{20}=2\sqrt{5}.\)

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

Since this question is a statistics 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 there is no relation between the number of data and the standard deviation, condition 1) is not sufficient.

Condition 2)
Since there is no relation between the average and the standard deviation, condition 2) is not 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.

Originally posted by MathRevolution on 17 Apr 2020, 19:39.
Last edited by MathRevolution on 12 Jan 2021, 03:09, edited 1 time in total.
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Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink]
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