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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.

Since the angle at the circumference is half of the central angle, standing on the same arc, condition 2) is sufficient.



Condition 1)
\(<BAO\) is equal to \(<ABO\), but we don’t know the measures of \(<OAC\) and \(<ACO\). So, we can’t work out the measure of \(<OAC\) or \(<x\). Therefore, condition 1) is not sufficient.

Therefore, B is the answer.
Answer: B

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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, and then recheck the numbers of variables and equations.

We can modify the original condition and question as follows:

If \(a > 0\), the function will have a minimum at \(x = \frac{(-20)}{(2a)} = \frac{(-10)}{a}.\)

If \(a < 0\), the function has no minimum. So, to answer the question, we need to find the value of \(a.\)

Thus, condition 2) is sufficient.

Note: condition 1) cannot be sufficient as it provides no information about the value of \(a.\)

Therefore, the answer is B.
Answer: B

[GMAT math practice question]

(geometry) What is the sum of \(∠x\) and \(∠y\)?



1) Triangle \(ABC\) is equilateral

2) \(BD = AE\)

[GMAT math practice question]

(Number) What is the units digit of \(3^n\)?

1) \(n\) is a multiple of \(4\)

2) \(n\) is a multiple of \(6\)

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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 (\(x\) and \(y\)) 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)
Since \(△ABE\) and \(△BCD\) are congruent, \(<ABE = <BCD.\)

So, \(<EFC = <FBC + <BCD\) (exterior angle of triangle \(BFC\)) = \(<FBC + <ABE\) (corresponding angles of congruent triangles \(ABE\) and \(BCD\)) \(= <B = 60°\)

Thus, \(<x + <y = 180° - 60°\) (angle sum of triangle \(EFC\)) = \(120°\).

Therefore, C is the answer.
Answer: C

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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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 the conditions if necessary.

The units digits of \(3^n\) for \(n = 1, 2, 3, 4, …\) are \(3, 9, 7, 1, 3, 9, 7, 1, …\)

So, the units digits of \(3^n\) have period \(4\):

They form the cycle \(3 -> 9 -> 7 -> 1.\)

Thus, \(3^n\) has a units digit of \(1\) if \(n\) is a multiple of \(4.\)

Note that \(6\) is not a multiple of \(4.\)

Therefore, A is the answer.
Answer: A

[GMAT math practice question]

(Function) What is \(f(g(2))\)?

\(1) f(x) =3x-2\)

\(2) g(x)=x^2\)

[GMAT math practice question]

(number properties) What is the remainder when \(1+n+n^2 +…+ n^8\) is divided by \(5\)?

1) The remainder when \(n\) is divided by \(5\) is \(3\)

2) \(n\) is less than \(5\)

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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.

We require infinitely many values to determine \(f(x)\) and \(g(x)\). Since the original condition includes infinitely many variables 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)
\(f(g(2)) = f(2^2) = f(4) = 3*4 – 2 = 12 – 2 = 10\)

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]

(geometry)




In the figure, \(∠AOB= 30^o\). What the length of arc \(BC\)?

1) \(AO\) is parallel to \(BC\)

2) the length of arc \(AB\) is \(5\)

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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.

The easiest way to solve remainder questions is to plug in numbers.
The units digits of \(3^n\) for \(n = 1, 2, 3, 4, …\)are \(3, 9, 7, 1, 3, 9, 7, 1, …\)

So, the units digits of \(3^n\) have period \(4\):

They form the cycle \(3 -> 9 -> 7 -> 1\).

Thus, if \(n\) has remainder \(3\) when it is divided by \(5, 1+n+n^2 +…+ n^8\) has the same remainder as \(1 + 3 + 9 + 7 + 1 + 3 + 9 + 7 + 1 = 21\) when it is divided by \(5\). It has a remainder of \(1\) when it is divided by \(5.\)

Condition 1) is sufficient.

Condition 2)
If \(n = 1\), then \(1+n+n^2 +…+ n^8 = 9,\) which has remainder \(4\) when it is divided by \(5\).

If \(n = 3,\) then \(1+n+n^2 +…+ n^8\) has remainder \(1\) when it is divided by \(5.\)

Since condition 2) doesn’t yield a unique solution, it is not sufficient.

Therefore, A is the answer.
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.

In order to determine \(BC\), we need to know the radius of the circle and the measure of angle \(<BOC.\) Thus, we have \(2\) variables and \(0\) equations, and 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)
Since \(AO\) and \(BC\) are parallel, \(<OBC = 30 °\) (alternate interior angles).
Since \(OB\) and \(OC\) are congruent (equal radii), \(<OCB 30 °.\)
Thus <\(BOC = 180 °-30 °-30 °=120 °\)
The ratio between the arc lengths of \(AB\) and \(BC\) is \(30:120 = 1:4.\)
Thus, the arc length of \(BC\) is \(4\) times the arc length of \(AB,\) so it is \(20.\)
Both conditions together are sufficient.

Note: condition 1) cannot be sufficient as it provides no information about the radius of the circle.

Therefore, the answer is C.
Answer: C

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.

[GMAT math practice question]

(geometry) In the figure, \(ABC\) is a right triangle. What is the area of \(ABC\)?



1) Circle \(O\) circumscribes triangle \(ABC\) and has diameter \(13\)

2) Circle \(O’\) is inscribed in triangle \(ABC\) and has diameter \(6\)

[GMAT math practice question]

(Number properties) \(m\) and \(n\) are positive integers such that \(m(n+10) = 75.\) What is the value of \(m\)?

1) \(n\) is not less than \(m\)

2) \(m\) is not a prime 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.

\(2\) variables are required to specify the two circles, and two variables are required to specify the right triangle. So, we have \(4\) variables. Since one side of the triangle is equal to the diameter of one of the two circles, we have \(1\) equation.
Since we have \(4\) variables and \(1\) equation, 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)
Since the radius of \(O’\) is \(3\), \(BE = BD = O’E = 3.\)
Suppose the length of \(AD\) is \(a\).
Then \(AF = AD = a\), and \(CF = CE = 13-a.\)
So, \(BC = BE + EC = 3 + 13 – a = 16 – a,\)
And \(AB = AD + DB = a + 3.\)
Thus, the area of triangle \(ABC\) is \((\frac{1}{2})(AB+BC+CA)(3) = (\frac{1}{2})(a+3+16-a+13)(3) = 48\)

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]

(set) \(A={x|(\frac{7}{15})x+\frac{1}{3} = \frac{4}{3}}\) and \(B={y| 2m-(\frac{1}{15})y = 3}\), where \(m\) is a real number. What is the value of \(m\)?

\(1) A∩B≠Ø\)

\(2) B≠Ø\)

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

Condition 1)
We can find two pairs of solutions: \(m = 1\) and \(n = 65\), and \(m = 3\) and \(n = 15\).

Since condition 1) doesn’t yield a unique solution, it is not sufficient.

Condition 2)
We can find only one pair of solutions: \(m = 1\) and \(n = 65.\)

Since condition 2) yields a unique solution, it is sufficient.

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.