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[GMAT math practice question]

(statistics) If the average (arithmetic mean) of \(5\) numbers is \(21\), what is their standard deviation?

1) The least of the \(5\) numbers is \(21\)

2) The greatest of the \(5\) numbers is \(21\)

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

In inequality questions, the law “Question is King” tells us that if the solution set of the question includes the solution set of the condition, then the condition is sufficient

Condition 1)
\(√x > x\)

\(=> x > x^2\) by squaring

\(=> x^2 – x < 0\)

\(=> x(x-1) < 0\)

\(=> 0 < x < 1\)

Since ‘no’ is also a unique answer by CMT (Common Mistake Type) 1, condition 1) is sufficient.

Condition 2)
\(x^3 - x < 0\)

\(=> x(x^2-1) < 0\)

\(=> (x+1)x(x-1) < 0\)

\(=> x<-1\) or \(0 < x < 1\)

\(=> 0 < x < 1\) since \(x\) is positive.

Since ‘no’ is also a unique answer by CMT (Common Mistake Type) 1, condition 2) is sufficient.

Therefore, D is the answer.
Answer: D

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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.

If the maximum or minimum value of a data set is the same as its average, then all data points are equal and their standard deviation is 0.
Thus, each of the conditions is sufficient.

Therefore, D is the answer.
Answer: D

Recall the property that the SD is zero if the maximum value or the minimum value is the same as the average, as this implies that all data points are equal.
Questions related to the above property have appeared frequently on recent GMAT exams.

[GMAT math practice question]

(absolute value) What is the value of \(xy\)?

\(1) x^2-4xy+5y^2-4xy+4 = 0\)

\(2) |x-4| = \sqrt{- (y-2)}\)

[GMAT math practice question]

(functions) f(x) = an\(x^n\)+ an-1\(x^{n-1}\) + … + a1x + a0. What is the sum of the coefficients of the function f?

1) f(1) = 20
2) f(-1) = 10

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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.

Condition 1)
\(x^2-4xy+5y^2-4xy+4 = 0\)

\(=> x^2-4xy+4y^2+y^2-4xy+4 = 0\)

\(=> (x-2y)^2 + (y-2)^2 = 0\)

\(=> x = 2y\) and \(y =2,\) since \((x-2y)^2 ≥ 0\) and \((y-2)^2 ≥ 0.\)

So, \(x = 4\) and \(y =2\). Thus, \(xy = 8\).

Condition 1) is sufficient since it yields a unique solution.

Condition 2)
\(|x-4| = - \sqrt{(y-2)}\)

\(=> |x-4| +\sqrt{(y-2)} = 0\)

\(=> |x-4| = 0\) and \(\sqrt{(y-2)} = 0,\) since \(|x-4| ≥ 0\) and \(\sqrt{(y-2)} ≥ 0.\)

So, \(x = 4\) and \(y =2.\) Thus, \(xy = 8.\)

Condition 2) is sufficient since it yields a unique solution.

Therefore, D is the answer.
Answer: D

[GMAT math practice question]

\(x = -1\). What is the value of \(x + x^n + x^{n+1} + x^{n+2}\)?

1) \(n\) is a prime number

2) \(n\) is an odd number

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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.

f(1) = an*\(1^n\) + an-1*\(1^{n-1}\) + … + a1*1 + a0 = an + an-1 + … + a1 + a0 , which is the sum of all coefficients of f(x).
Thus, condition 1) is sufficient, because it is equivalent to the question.

Therefore, A is the answer.
Answer: A

[GMAT math practice question]

(ratio) \(a, b\) and \(c\) are numbers such that \(abc≠ 0\) and \(\frac{a}{b}=\frac{b}{c}=\frac{c}{d}.\) What is the value of \(\frac{a}{d}\)?

\(1) a = 1\)

\(2) b = 2\)

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

Condition 1)
If \(n = 2\), then \(x + x^n + x^{n+1} + x^{n+2} = (-1) + (-1)^2 + (-1)^3 + (-1)^4? = (-1) + 1 + (-1) + 1 = 0.\)

If \(n = 3\), then \(x + x^n + x^{n+1} + x^{n+2} = (-1) + (-1)^3 + (-1)^4 + (-1)^5 = (-1) + (-1) + (-1) + 1 = -2.\)

Condition 1) is not sufficient since it doesn’t yield a unique solution.

Condition 2)
If \(n\) is an odd number, then \(x + x^n + x^{n+1} + x^{n+2} = (-1) + (-1)^n + (-1)^{n+1} + (-1)^{n+2} = (-1) + (-1) + 1 + (-1) = -2.\)

Condition 2) is sufficient since it yields a unique solution.

Therefore, B is the answer.
Answer: B

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]

(inequality) \(x, y\) and \(y\) are positive numbers satisfying \(x < y < z.\) Is \(x < 5\)?

\(1) x^2 + y^2 + z^2 = 125\)

\(2) y^2 = 25\)

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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.

\(\frac{a}{b}=\frac{b}{c} => b^2 = ac => a = \frac{b^2}{c}\)

\(\frac{b}{c}=\frac{c}{d} => c^2 = bd => d = \frac{c^2}{b}\)

So, \(\frac{a}{d} = \frac{(b^2}{c)} / \frac{(c^2}{b)} = \frac{b^3}{c^3} = (\frac{b}{c})^3 = (\frac{a}{b})^3\)

Thus, conditions 1) & 2) are sufficient, when applied together.

Therefore, C is the answer.
Answer: C

=>

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.

Condition 1)
If \(x ≥ 5,\) then \(x^2 + y^2 + z^2 > x^2 + x^2 + x^2 > 3x^2 > 3*52 = 125\).

Thus, we must have \(x < 5\). Condition 1) is sufficient since it yields the unique answer, ‘yes’.

Condition 2)
Since \(x^2 < y^2 = 25\), and \(x\) and \(y\) are positive numbers, \(x < y = 5.\)

Thus, \(x < 5\). Condition 2) is sufficient since it yields the unique answer, ‘yes’.

Therefore, D is the answer.
Answer: D

[GMAT math practice question]

(number properties) Is \(pqr\) is a multiple of \(5\)?

1) \(p, q\) and \(r\) are consecutive odd integers.

2) \(p, q\) and \(r\) are prime numbers.

[GMAT math practice question]

(number properties) If two integers have no common factors other than \(1\), they are called relatively prime. Are \(x\) and \(z\) relatively prime?

1) \(x\) and \(y\) are relatively prime.

2) \(y\) and \(z\) are relatively prime.

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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 (\(p, q\) and \(r\)) 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):
\(3, 5\) and \(7\) is the unique triplet of three consecutive odd integers which are prime numbers, and \(3*5*7\) is a multiple of \(5\).
Thus, both conditions together 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 \(p = 3, q = 5\) and \(r = 7\), then \(pqr = 105\) is a multiple of \(5\), and the answer is ‘yes’.
If \(p = 7, q = 9\) and \(r = 11\), then \(pqr = 693\) is not a multiple of \(5\), and the answer is ‘no’.
Condition 1) is not sufficient since it doesn’t yield a unique answer.

Condition 2)
If \(p = 3, q = 5\) and \(r = 7\), then \(pqr = 105\) is a multiple of \(5\), and the answer is ‘yes’.
If \(p = 3, q = 7\) and \(r = 11\), then \(pqr = 231\) is not a multiple of \(5\), and the answer is ‘no’.
Condition 2) is not sufficient since it doesn’t yield a unique solution.

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]

(absolute value) Is \(x<y<z\)?

\(1) |x+2|<y<z+2\)

\(2) |x-2|<y<z-2\)