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


We consider the equation of the line \(l, y = mx + b.\) Since it passes through the points \((-1,-1)\) and \((3,k),\) we can plug these points into its equation to yield \(-1 = -m + b\) and \(k = 3m + b.\)

Since we have \(3\) variables and \(2\) equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
Since the y-intercept of line \(l\) is \(1\), we have \(b = 1\) and \(m = b + 1 = 2.\)
Thus, \(k = 3m + b = 3*2 + 1 = 7.\)
Condition 1) is sufficient.

Condition 2)
Since the slope of line \(l\) is \(2\), we have \(m = 2\) and \(b = m – 1 = 1.\)
Thus, \(k = 3m + b = 3*2 + 1 = 7.\)
Condition 2) is sufficient.

Answer: D

Note: When we checked the two conditions, we showed that both were equivalent in terms of b. So, each condition is sufficient by Tip 1)
of the VA method, which states that D is most likely to be the answer if conditions 1) and 2) provide the same information.

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.
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[Math Revolution GMAT math practice question]

(integer) If \(m\) and \(n\) are positive integers, what is the greatest common divisor of \(m\) and \(n\)?

\(1) m=n+1\)
\(2) m*n\) is divisible by \(2\)
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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.

Modifying the question:
The question asks if \(\sqrt{17n} = a\) for some integer a. This is equivalent to asking if \(17n = a^2\) for some integer a.

Since we have \(1\) variable (\(n\)) 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)
Since \(68n\) is the square of an integer and \(68 = 4*17\), we must have \(68n = 4*17*17*k^2\) for some integer \(k\), and \(n = 17*k^2\) or \(17n = 17^2*k^2 = (17*k)^2.\)
Thus, \(17n\) is the square of the integer \(17k\), and condition 1) is sufficient.

Condition 2)
Since \(\frac{n}{68}\) is a square of an integer and \(68 = 4*17\), we have \(\frac{n}{68} = m^2\) for some integer \(m\), and \(n = 17*4*m^2\) or \(17n = 17^2*2^2*m^2 = (34m)^2.\)
Thus, \(17n\) is the square of the integer \(17k\), and condition 2) is sufficient.

Therefore, D is the answer.
Answer: D

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.
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[Math Revolution GMAT math practice question]

(number property) If \(k\) is a positive integer and \(n=(k-1)k(k+1)\), is \(n\) a multiple of \(8\)?

1) \(k\) is an odd number
2) \(k = 1\)
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From statement 1:

k is an odd number.
Let k be 2k+1
Then n = (2k+1-1)(2k+1)(2k+2)
n = 8k^3+12k^2+4k.
For an odd value of k. n will always be a multiple of 8.
Sufficient.

From statement 2:
k = 1.
n = 0. 0 is a multiple of 8.
Sufficient.

D is the answer.

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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 two consecutive integers are always relatively prime, the greatest common divisor of m and n is 1. Thus, condition 1) is sufficient.

Condition 2)
If \(m = 2\) and \(n = 3\), then the greatest common divisor of \(m\) and \(n\) is \(1\).
If \(m = 2\) and \(n = 4,\) then the greatest common divisor of \(m\) and \(n\) is \(2\).
Thus, condition 2) is not sufficient since it does not yield a unique solution.

Therefore, the correct answer is A.
Answer: A
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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 \(2\) variables (\(n\) and \(k\)) and \(1\) equation, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
Since \(k\) is an odd number, \(k – 1\) and \(k + 1\) are consecutive even integers.
Any product of consecutive even integers is a multiple of \(8\).
Thus, condition 1) is sufficient.

Condition 2)
Since \(k = 1\), we have \(n = (k-1)k(k+1) = 0*1*2 = 0. 0\) is a multiple of any number, so \(n = 0\) is a multiple of \(8\).
Thus, condition 2) is sufficient.

Therefore, D is the answer.
Answer: D

Since this question is a CMT4(B) question. Condition 2) is easy to understand and condition 1) is hard. When one condition is easy to understand, and the other is hard, D is most likely to be the answer.


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.
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[Math Revolution GMAT math practice question]

(inequality) Is \(x + \frac{1}{x} > 2\)?

\(1) x > 0\)
\(2) x ≠ 1\)
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From statement 1:

x>0. If x = 1. then x+1/x = 2. And 2 is not greater than 2.
If x = 2. Then x+1/x = 2.5>2. hence insufficiennt.

From statement 2:

x not equal to 1.
If x = -1. Then x+1/x = -2, which is not greater than 2.
If x = 2. Then x+1/x = 2.5>2. hence insufficient.

Combining both tell that x>0 and x not equal to 0. Then x+1/x is always greater than 2.
C is the answer.

Excellent problem, Max. Congrats! (kudos!)

\(x + {1 \over x}\,\,\mathop > \limits^? \,\,2\)

\(\left( 1 \right)\,\,x > 0\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,x = 1\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr
\,{\rm{Take}}\,\,x = 2\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr} \right.\)

\(\left( 2 \right)\,\,x \ne 1\,\,\left\{ \matrix{
\,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,\,x = 2\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr
\,{\rm{Take}}\,\,x = - 1\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.\)

\(\left( {1 + 2} \right)\,\,\,0\,\,\mathop < \limits^{x\, \ne \,1} \,\,{\left( {x - 1} \right)^2}\,\, = \,\,\,{x^2} - 2x + 1\,\,\,\,\mathop \Leftrightarrow \limits^{\,x\, > \,\,0} \,\,\,\,0 < {{{x^2} - 2x + 1} \over x} = x - 2 + {1 \over x}\,\,\,\,\, \Leftrightarrow \,\,\,\,x + {1 \over x} > 2\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\,\,\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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[Math Revolution GMAT math practice question]

(number properties) \(n\) is a positive integer. Is \(n\) divisible by \(3\)?

1) \(\frac{36}{n}\) is divisible by \(3\)
2) \(\frac{27}{n}\) is divisible by \(3\)
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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.

\(x + \frac{1}{x} > 2\)
\(=> x^3 + x > 2x^2\) after multiplying both sides by \(x^2\)
\(=> x^3 - 2x^2 + x > 0\)
\(=> x^3 - 2x^2 + x > 0\)
\(=> x(x^2 - 2x + 1) > 0\)
\(=> x(x-1)^2 > 0\)
\(=> x > 0\) and \(x ≠ 1\)

Thus, we need both conditions together for sufficiency.

Therefore, C is the answer.
Answer: C
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[Math Revolution GMAT math practice question]

(statistics) \(x\) is a positive number. What is the median of \(x, √x\) and \(x^2\)?

\(1) x^2=x\)
\(2) x^2+x+1=3x\)
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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 in the original condition, D is most likely to be the answer. So, we should consider each condition on its own first. It is suggested we plug in numbers when solving remainder problems.


Condition 1)
If \(n = 3\), then \(\frac{36}{3} = 12\) is divisible by \(3\), and \(n\) is divisible by \(3\). The answer is ‘yes’.
If \(n = 1\), then \(\frac{36}{1} = 36\) is divisible by \(3\), but \(n\) is not divisible by \(3\). The answer is ‘no’.
Thus, condition 1) is not sufficient, since it does not yield a unique solution.

Condition 2)
If \(n = 3\), then \(\frac{27}{3} = 9\) and \(n\) is divisible by \(3\). The answer is ‘yes’.
If \(n = 1\), then \(\frac{27}{1} = 27\) and \(n\) is not divisible by \(3\). The answer is ‘no’.
Thus, condition 2) is not sufficient, since it does not yield a unique solution.

Conditions 1) & 2)
Even if we consider both conditions together, we still have two possible values of \(n: n = 1\) and \(3\).
Thus, both conditions together are not sufficient, since they do not yield a unique solution.

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.
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[Math Revolution GMAT math practice question]

(number properties) If \(a\) and \(b\) are integers, is \(a-b\) an even number?

1) \(a^2b^2\) is an even number
2) \(a^2+2b^2\) is an even 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.

If \(x >1\), then \(√x < x < x^2\) and \(x\) is their median.
If \(0 < x <1\), then \(√x > x > x^2\) and \(x\) is their median.
If \(x = 1\), then \(√x = x = x^2\) and \(x\) is their median.
Thus, the question asks for the value of \(x\).

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

Condition 1)
\(x^2=x\)
\(=> x^2-x=0\)
\(=> x(x-1)=0\)
\(=> x = 0\) or \(x = 1\)
Since \(x\) is positive, \(x = 1.\)
Condition 1) is sufficient.

Condition 2)
\(x^2+x+1=3x\)
\(=> x^2-2x+1=0\)
\(=> (x-1)^2=0\)
\(=> x = 1\).
Condition 2) is sufficient.

Therefore, D is the answer.
Answer: D

FYI, Tip 1) of the VA method states that D is most likely to be the answer if conditions 1) and 2) provide the same information.

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.
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[Math Revolution GMAT math practice question]

(inequality) Is \(x>y\)?

\(1) x+y>2\)
\(2) x^2<2y\)
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(ex 1) A certain store, books are sold. Books are hard cover or soft cover and hard cover books sold $10 each and soft cover books sold $6. Is the number of hard cover books sold greater than that of soft cover books sold?
1) The average price sold of total books is $9
2) The number of hard cover books sold is 100

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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.

Modifying the question:
For \(a – b\) to be an even number, either both a and b must be even numbers or both a and b must be odd numbers.

Since we have \(2\) variables (\(a\) and \(b\)) and \(0\) equations, C 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)
From condition 2), a is an even number.
From condition 1), b might either be even or odd.

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

Therefore, the correct answer is E.
Answer: E

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.
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