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

Condition 1) tells us that \(k\) is a positive integer, from which we cannot determine the unique set of \(A\). If \(k = 3\), then \(x^2 - 2(k - 1)x + 4 = x^2 – 2(3 – 1)x + 4 = x^2 - 4x + 4 = (x - 2)^2 = 0\) and \(A = {2}.\) However, if \(k = 1\), then \(x^2 - 2(k - 1)x + 4 = x^2 - 2(1 – 1) + 4 = x^2 + 4 = 0\), which does not have a root and \(A\) is an empty set.

The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Condition 2) tells us that the number of elements in \(A\) is \(1\), from which we get that the discriminant of the quadratic equation is 0.

Since the number of roots of the equation is \(1\), its discriminant
\((b^2 – 4ac) \)

\(= [2(k – 1)]^2 - 4·1·4 \)

\(= 4(k - 1)^2 - 16 \)

\(= 4(k^2 - 2k + 1) - 4·4 \)

\(= 4(k^2 – 2k + 1 - ) 4\) (taking out a common factor of \(4\))

\(= 4(k^2 - 2k - 3) \)

\(= 4(k + 1)(k - 3)\) is zero.

Then, we have \(4(k + 1)(k - 3) = 0\) and \(k = -1 \)or \(k = 3.\)

The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Conditions 1) & 2)
Then only \(k = 3\) is the unique answer.

The answer is unique, and both conditions together are sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions 1) and 2) together 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.

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

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

Condition 1) tells us that the difference between the two roots of \(x^2 + (1 + m)x + 20 = 0\) is \(1.\)

We can assume \(p\) and \(p+1\) are the roots of the equation \(x^2 + (1 + m)x + 20 = 0.\)

We have \((x - p)(x - (p + 1)) = x^2 - (2p + 1)x + p(p + 1) = x^2 + (1 + m)x + 20.\)

Then, we have \(1 + m = -2p – 1\) or \(m = -2p – 2\). We also have \(p(p + 1) = 20\) or \(p^2 + p - 20 = (p - 4)(p + 5) = 0.\)

Thus \(p = 4\) or \(p = -5\), which we can substitute into the first equation giving us \(m = -2p – 2 = -2·4 – 2 = -10\), or \(m = -2p – 2 = -2·(-5) – 2 = 8 .\)

Then we have \(m = -10\) or \(m = 8.\)

The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Condition 2) tells us that \(m > 0\), from which we cannot determine the value of \(m\). For example, \(m\) can be \(2\) or \(3.\)

The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Conditions 1) & 2) together tell us that the answer, m = 8 is unique, and both conditions are sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions 1) and 2) together 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.

[GMAT math practice question]

(Algebra) \(x\) and \(y\) are real numbers. What is the value of \(x\)?

1) \(x^2 - 6xy + 9y^2 + x - 3y = 6\)

2) \(y = 1 + 2√3\)

[GMAT math practice question]

(Geometry) What is the ratio of the area of \(A\) to the area of \(B\) in the figure?



1) The biggest triangle consists of \(6\) different isosceles right triangles.

2) \(PQ = 4\)

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

Since we have \(2\) variables (\(x\) and \(y)\) and \(0\) equations, C 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) together give us:
\(x^2 - 6xy + 9y^2 + x - 3y = 6\) (from Condition 1)

⇔ \(x^2 - 6xy + 9y2 + x - 3y - 6 = 0\) (subtracting \(6\) from both sides)

⇔ \((x - 3y)^2 + (x - 3y) - 6 = 0\) (factoring the first \(3\) terms)

⇔ \(m^2 + m – 6 = 0\) (substituting m for (\(x - 3y\)))

⇔ \((m – 2)(m + 3) = 0\) (trinomial factoring)

⇔ \((x - 3y - 2)(x - 3y + 3) = 0\) (substituting (\(x – 3y\)) for \(m\))

⇔ \(x – 3y – 2 = 0\) or \(x – 3y + 3 = 0\)

⇔ \(x - 3y = 2\) or \(x - 3y = -3\)

⇔ \(x = 3y + 2\) or \(x = 3y – 3\)

⇔ \(x = 3(1 + 2√3) + 2\) or \(x = 3(1 + 2√3) – 3\) (substituting (\(1 + 2√3\)) from condition \(2\) for \(y\))

⇔ \(x = 5 + 6√3\) or \(x = 2√3\) (simplifying)

The answer is not unique, and both conditions 1) and 2) together are not sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Therefore, E is the answer.
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.

[GMAT math practice question]

(Function) \(f(x)\) is a function. What is the value of the non-zero solution(\(s\)) of \(f(x) = f(-x)\)?

1) \(f(x)+2f(\frac{1}{x})=3x\)

2) \(x\) is an irrational 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.
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.

Let’s look at the condition 1). It tells us that the ratio of the areas of triangles \(A\) and \(B\) is \(3:8\), as shown below:

Assume \(PQ = x.\)

Then \(PQ = PR = x\) and \(SR = SP = \frac{x}{√2}.\)

We have \(TS = TP = \frac{SP}{√2} = \frac{x}{2}.\)

\(US = UT = \frac{ST}{√2} = \frac{x}{2√2}.\)

Then \(UR = US + SP = \frac{x}{2√2} + \frac{x}{√2} = \frac{3x}{2√2}.\)

The area of triangle \(A\) is \((\frac{1}{2})(\frac{3x}{2√2})^2 = \frac{3x^2}{16}.\\
\)

The area of triangle \(B\) is \((\frac{1}{2})x^2 = \frac{x^2}{2}.\)

Thus, the ratio of the areas of \(A\) to \(B\) is \(\frac{3x^2}{16}\) to \(\frac{x^2}{2} \)or \(3:8.\)

The answer is unique, and the condition is sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Condition 2)
We don’t assume anything about \(PR, RU\), and so on.

The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.
Condition 1) ALONE is sufficient

Therefore, A is the answer.
Answer: A

[GMAT math practice question]

(Statistics) What is the value of \(x\)?

1) The standard deviation of \(1\) and \(3\) is greater than the standard deviation of \(1, 3,\) and \(x.\)

2) \(x\) is an integer.

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

Since a function has many variables to determine, 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) together give us:
When we substitute \(x\) in the equation \(f(x)+2f(\frac{1}{x})=3x\) by \(\frac{1}{x},\) we have \(f(\frac{1}{x}) + 2f(x) = \frac{3}{x}.\)

When we subtract double the second equation from the first equation, we have:
\(f(x) + 2f(\frac{1}{x})] – 2[f(\frac{1}{x}) + 2f(x)] = 3x – 2(\frac{3}{x})\)

⇔ \(f(x) + 2f(\frac{1}{x}) – 2f(\frac{1}{x}) - 4f(x) = 3x – \frac{6}{x}\)

⇔ \(-3f(x) = 3x – \frac{6}{x} \)

⇔ \(f(x) = \frac{2}{x} – x.\)

\(f(x) = f(-x)\) is equivalent to:

\(\frac{2}{x} - x = \frac{-2}{x} + x\)

⇔ \(2x = \frac{4}{x}\)

⇔ \(x^2 = 2,\)

Thus, we have \(x = ± √2.\)

Since condition 2) tells us x is an irrational number, the answer is not unique.
The conditions are not sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions 1) and 2) together 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.

[GMAT math practice question]

(Function) What is the value of the function \(f(x)\)?

1) \(f(2020x + f(0)) = 2020x^2, x\) is a real number.

2) \(f(x)\) is a polynomial function.

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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.
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.

Let’s look at the condition 1). It tells us that the average of \(1\) and \(3\) is \(2\). If \(x\) is between \(1\) and \(3\) inclusive, then the standard deviation of \(1, 3\), and \(x\) is less than that of \(1\) and \(2\).

The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Condition 2) is not sufficient, obviously.

When we consider both conditions 1) & 2) together, \(1, 2\), and \(3\) are possible values of \(x\).

The answer is not unique, and both conditions are not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions 1) and 2) together 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.

=>

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 a function has many variables to determine, 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) together give us:

Assume \(t = 2020x + f(0).\)

Then we have \(x = \frac{(t - f(0))}{2020}.\)

\(f(t) = 2020[\frac{(t - f(0))}{2020}]^2 = \frac{[t - f(0)]^2}{2020}.\)

When we replace \(t\) by \(0\), we have \(f(0) = \frac{(f(0))^2}{2020}\) or \((f(0))^2 – 2020f(0) = 0\)

Then we have \(f(0)(f(0) - 2020) = 0.\)

Thus \(f(0) = 0\) or \(f(0) = 2020.\)

Both conditions 1) and 2) together 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.

[GMAT math practice question]

(Number Properties) What is the value of \(m^2+3n^2\)?

1) \(m\) and \(n\) are even prime numbers.

2) \(m\) and \(n\) are the smallest positive even integers.

[GMAT math practice question]

(Number Properties) \(x\) and \(y\) are integers. What is the value of \(x + y\)?

1) \(xy = 1008\)

2) The greatest common divisor of \(x\) and \(y\) is \(6.\)

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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.
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.

Let’s look at the condition 1). It tells us that \(m = n = 2,\) because the only even prime number is \(2\)
\(m^2 + 3n^2 = 2^2 + 3·2^2 = 16.\)

The answer is unique, and the condition is sufficient according to the Common Mistake Type 2, which states that the number of answers must be only one.

Let’s look at the condition 2). It tells us that \(m = n = 2\) because the smallest positive even integer is \(2.\)
\(m^2 + 3n^2 = 2^2 + 3·2^2 = 16.\)

The answer is unique, and the condition is sufficient according to the Common Mistake Type 2, which states that the number of answers must be only one.

Therefore, D is the answer.
Answer: D

Note: Tip 1) of the VA method states that D is most likely the answer if condition 1) gives the same information as condition 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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have \(2\) variables (\(x\) and \(y\)) and \(0\) equations, C 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) together give us two solutions of \(x = 6·1, y = 6·28, x + y = 174\) and \(x = 6·4, y = 6·7, x + y = 66.\)

Since the greatest common divisor of \(x\) and \(y\) is \(6,\) we can assume that \(x = 6a,\) and \(y = 6b\) where \(a\) and \(b\) are relatively prime.

\(x·y = 6a·6b = 1008 = 6·6·28.\)

Then we have \(ab = 28.\)

\((1, 28)\) and \((4, 7)\) are possible pairs for \((a, b).\)

If \(a = 1\) and \(b = 28\), we have \(x = 6·1 = 6, y = 6·28 = 168\) and \(x + y = 174.\)

If \(a = 4\) and \(b = 7\), we have \(x = 6·4 = 24, y = 6·7 = 42\) and \(x + y = 66.\)

The answer is not unique, and conditions 1) and 2) together are not sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Therefore, E is the answer.
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.

[GMAT math practice question]

(Proportion) The figure shows points \(A, B, C\), and \(D\) on the number line. The coordinate of point \(A\) is \(-5\), and that of point \(B\) is \(4.\) What is the coordinate of point \(C\)?



1) \(AC:CD = 1:2\)

2) \(CD:DB = 2:3\)