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[Math Revolution GMAT math practice question]

Is the sum of \(7\) different positive integers greater than or equal to \(48\)?

1) Their median is \(9\)
2) The largest number is \(12\)

[Math Revolution GMAT math practice question]

(set) If \(|X|\) is the number of elements in set \(X\), and \(“∪”\) is the union and \(“∩”\) is the intersection of \(2\) sets, what is the value of \(|A∩B|\)?

\(1) |A∪B|＝50\)
\(2) |B|=40\)

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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 many variables (x1, x2, …, x7) 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)
We write the numbers as x1≤x2≤x3≤x4≤x5≤x6≤x7. Then x4 is their median.
From condition 1), x4 = 9 and the smallest possible number is 1 + 2 + 3 + 9 + 10 + 11 + 12 = 48. Therefore, the answer is ‘yes’.
Both conditions 1) & 2) together 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 we didn’t use condition 2) in the above argument, condition 1) is sufficient on its own.

By condition 1), x4 = 9 and the smallest possible number is 1 + 2 + 3 + 9 + 10 + 11 + 12 = 48. Therefore, the answer is ‘yes’.


Condition 2)
When the numbers are 6,7,8,9,10,11,12, their sum is 6 + 7 + 8 + 9 + 10 + 11 + 12 = 63 > 48, and the answer is ‘yes’.
When the numbers are 6,7,8,9,10,11,12, their sum is 1 + 2 + 3 + 4 + 5 + 6 + 12 = 33 < 48, and the answer is ‘no’.
Since we don’t obtain a unique answer, condition 2) is not sufficient.

Therefore, A is the answer.
Answer: A

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.

[Math Revolution GMAT math practice question]

(number property) If \(a\) and \(b\) are positive integers such that when \(a\) is divided by \(b\), the remainder is \(10\), what is the value of \(b\)?

\(1) b>10\)
\(2) b<12\)

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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.
Note that
\(|A∪B| = |A| + |B| - |A∩B|\) and \(|A∩B| = |A| + |B| - |A∪B|\).

Since we have \(4\) variables\((|A∩B|, |A|, |B|, |A∪B|)\) and \(1\) equation \((|A∩B| = |A| + |B| - |A∪B|)\), 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)
Suppose \(A\) and \(B\) are disjoint sets, \(|A∪B| = 50, |A| = 10,\) and \(|B| = 40.\) Then \(|A∩B| = |A| + |B| - |A∪B| = 0.\)
Suppose \(A\) contains \(B, |A∪B| = 50, |A| = 50,\) and \(|B| = 40.\) Then \(|A∩B| = |A| + |B| - |A∪B| = 40.\)
Since we don’t have a unique solution, both conditions together are not sufficient.


Therefore, E is the answer.
Answer: E

[Math Revolution GMAT math practice question]

(inequality) Is \(1+x+x^2+x^3+x^4+x^5+x^6<\frac{1}{(1-x)}\)?

\(1) x>0\)
\(2) x<1\)

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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.

By the quotient-remainder theorem, we can write \(a = b * q + 10\), where the remainder \(10\) is less than \(b\), that is, \(b > 10\).

Thus, condition 2) \(“b<12”\) is sufficient since it gives the unique solution \(b = 11\).

Note: Condition 1) does not give a unique solution. For example, we might have \(b = 11\) or \(b = 12\). Thus, it is not sufficient.

Therefore, B is the answer.
Answer: B

=>

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.

The question \(1+x+x^2+x^3+x^4+x^5+x^6<\frac{1}{(1-x)}\) is equivalent to \(0 < x < 1\) as shown below:

For \(x ≠1\),
=>\(1+x+x^2+x^3+x^4+x^5+x^6<\frac{1}{(1-x)}\)
\(=> (1+x+x^2+x^3+x^4+x^5+x^6)(1-x)^2< (1-x)\)
\(=> (1 - x^7)(1 - x) < 1 – x\)
\(=> 1 - x^7 – x +x^8 < 1 - x\)
\(=> - x^7 + x^8 < 0\)
\(=> x^7( x – 1 ) < 0\)
\(=> x( x – 1 ) < 0\)
\(=> 0 < x < 1\)

Since both conditions must be applied together to obtain this inequality, both conditions 1) & 2) are sufficient, when applied together.

Therefore, C is the answer.
Answer: C

[Math Revolution GMAT math practice question]

(inequality) Is \(x^3-y^3>x^2+xy+y^2\)?

\(1) x > y + 1\)
\(2) 0 < y < x\)

[Math Revolution GMAT math practice question]

(function) In the xy-plane, does the graph of \(y=ax^2+c\) intersect the x-axis?

\(1) a>0\)
\(2) c>0\)

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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.

The original condition \(x^3-y^3>x^2+xy+y^2\) is equivalent to \(x > y + 1\) as shown below:
\(x^3-y^3 > x^2+xy+y^2\)
\(=> (x-y)(x^2+xy+y^2)>x^2+xy+y^2\)
\(=> x – y > 1\) after dividing both sides by \(x^2+xy+y^2\), since \(x^2+xy+y^2 > 0.\)

Since the final inequality is equivalent to \(x > y + 1\), condition 1) is sufficient.

Condition 2)
If \(x = 3\) and \(y = 1\), then \(x – y = 2 > 1\), and the answer is ‘yes’.
If \(x = 1\) and \(y = \frac{1}{2},\) then \(x – y = \frac{1}{2} < 1,\) and the answer is ‘no’.
Since it doesn’t give a unique answer, condition 2) is not sufficient.

Therefore, A is the answer.
Answer: A

[Math Revolution GMAT math practice question]

(algebra) For integers \(m\) and \(n\), the operation \(△\) is defined by \(m△n = (m-1)^2 + (n+1)^2\). What is the value of the integer \(x\)?

\(1) x△1 = 4\)
\(2) 1△x = 4\)

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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.

The question “does the graph of \(y=ax^2+c\) intersect the x-axis” is equivalent to asking “does the equation \(ax^2+c = 0\) have a root”.
Note that the statement “\(ax^2 + bx + c = 0\) has a root” is equivalent to \(b^2-4ac ≥ 0.\)
Thus, the question asks if \(-4ac ≥ 0,\) or \(ac ≤ 0\), since \(b = 0\) in this problem.

When we consider both conditions together, we obtain \(ac > 0\) and the answer is “no”, since \(a > 0\) and \(c > 0.\)
Since ‘no’ is also a unique answer by CMT (Common Mistake Type) 1, both conditions together are sufficient.

Note: Neither condition on its own provides enough information for us to determine whether \(ac ≤ 0.\)

Therefore, C is the answer.
Answer: C

[Math Revolution GMAT math practice question]

(statistics) If the average (arithmetic mean) of \(5\) numbers is \(20\), what is their standard deviation?

1) Their minimum is \(20\).
2) Their maximum is \(20\).

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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 (\(x\)) and \(0\) equations, D is most likely to be the answer. So, we should consider each of the conditions on their own first.

Condition 1)
\(x△1 = (x-1)^2 + (1+1)^2 = (x-1)^2 + 2^2 = 4.\)
Thus, \((x-1)^2 = 0\) and \(x = 1.\)
Since we have a unique solution, condition 1) is sufficient.

Condition 2)
\(1△x = (1-1)^2 + (x+1)^2 = (x+1)^2 = 4.\)
So, \(x+1 = ±2\) or \(x = -1 ± 2.\)
Thus, \(x = -3\) or \(x = 1\).
Since we don’t have a unique solution, condition 2) is not sufficient.

Therefore, A is the 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.

[Math Revolution GMAT math practice question]

(number property) \(n\) is an integer. Is \(n(n+2)\) a multiple of \(8\)?

1) \(n\) is an even integer
2) \(n\) is a multiple of \(4\)

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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.

Note that if the average and the maximum of a data set are the same, then all of the data values are the same and the standard deviation is 0. Similarly, if the average and the minimum of a data set are the same, all of the data values are the same and the standard deviation is 0.

Thus, each of conditions is sufficient on its own since the minimum and the maximum are the same as the average.

Therefore, D is the answer.
Answer: D