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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 (\(a\)) 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 a could be any value other than \(3\).

\(\frac{x}{3} + a = \frac{x}{2} – \frac{x - 18}{6}\)

⇔ \(\frac{x}{3} + a = \frac{x}{3} + 3\)

⇔ \(a = 3.\)

It means if \(a = 3\), the equation has an infinite number of solutions, and if \(a ≠ 3\), the equation has no solution. Thus, \(a\) can be any value other than \(3\).

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.

Let’s look at the condition 2). It tells us that a does not have a unique solution.
Any integer can be the value of a.

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 we don’t have a unique solution, as a can be any integer other than 3.

The answer is not unique, and the 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.

Both conditions 1) & 2) together are not sufficient.

Therefore, E is the correct 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.

[GMAT math practice question]

(Set) Among \(100\) employees, how many employees like both apples and bananas?

1) \(53\) employees like apples.

2) \(72\) employees like bananas.

[GMAT math practice question]

(Rate Problems) Adam walks to the library. Ben finds that Adam left his cell phone, so after \(25\) minutes Ben walks the same route to catch up with Adam. How long will it take for them to meet?

1) Adam walks \(60\) m/min.

2) Ben walks \(180\) m/min.

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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.



\(A\) and \(B\) are sets of employees who like apples and banana, respectively. When we set this question up as the above Venn Diagram, we have \(x + y + z + w = 100.\)

Since we have \(4\) variables (\(x, y, z\) and \(w\)) and \(1\) equation (\(x + y + z + w = 100\)), 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.
The question asks what the value of y is.

Conditions 1) & 2) together give us that they are not sufficient together.

Since \(53\) employees like apples, we have \(x + y = 53\) from condition 1).

Since \(72\) employees like bananas, we have \(y + z = 72\) from condition 2).

If \(x = 20, z = 39, w = 8\), then we have \(y = 33\).

If \(x = 21, z = 40, w = 7\), then we have \(y = 32.\)

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.

Both conditions 1) & 2) together are not sufficient.

Therefore, E is the correct 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]

(Algebra) Abigail and Bella have some money. How much did Abigail have at the beginning?

1) The ratio of the amount of Abigail’s money to Bella’s is \(5:3\).

2) Abigail gives \($300\) to Bella, and then Bella gives Abigail \(\frac{3}{5}\) of the amount that Bella has. After that Abigail has \(5\) times as much money as Bella.

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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.

Assume a and b are speeds of Adam and Ben, respectively, and x is the number of minutes they take to meet. We then have a(x + 25) = bx.

Since we have \(3\) variables (\(x, a\), and \(b\)) and \(1\) equation (\(a(x + 25) = bx\)), 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 that both conditions together are sufficient.

Since we have \(a = 60\) and \(b = 180\), we have \(60(x + 25) = 180x, 60x + 1500 = 180x\), or \(120x = 1500.\)

Thus, we have \(x = 12.6.\)

It takes \(12\) minutes and \(30\) seconds.

The answer is unique, and conditions 1) and 2) together are sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions 1) & 2) together are sufficient.

Therefore, C is the correct 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.

[GMAT math practice question]

(Function) \(a\) and \(b\) are non-zero real numbers. In which quadrant is point (\(a, b\)) on located?

1) \(y = ax\) and \(y = \frac{b}{x}\) have an intersection.

2) \(a + b > 0\).

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

Assume that \(a\) and \(b\) are the amounts of money Abigail and Bella have, respectively.

Since we have \(2\) variables (\(a\) and \(b\)) 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:
We have \(a:b = 5:3\) or \(b = (\frac{3}{5})a\) from condition 1).

From condition 2) we have \((a - 300) + (b + 300)·(\frac{3}{5}) = (b + 300)(\frac{2}{5})·5\). Substituting \(b = (\frac{3}{5)}a\) gives us \((a - 300) + ((\frac{3}{5})a + 300)·(\frac{3}{5}) = 2((\frac{3}{5})a + 300).\)

Then we have \(a – 300 + (\frac{9}{25})a + 180 = (\frac{6}{5})a + 600 or a + (\frac{9}{25})a – (\frac{6}{5})a = 720.\)

Thus, we have \(a = 4500.\)

The answer is unique, and conditions 1) and 2) together are sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions 1) & 2) together are sufficient.

Therefore, C is the correct 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.

[GMAT math practice question]

(Number Properties) We have that \(a\) and \(b\) satisfy \(\frac{a}{60} = \frac{1}{b}.\) What is the value of \(a + b\)?

1) \(\frac{1}{b}\) is a terminating decimal.

2) \(a\) and \(b\) are positive integers.

If a = 10 ,1/b = 1/6 its terminating decimal.
If a= 12 1/b = 1/5 also terminating decimal.
Other fraction values could also satisfy the equation.
Insuff.

Stmt 2) is definitely insuff.
See ex. in 1)

Combining won't give us definite solution.
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.

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 \(2\) variables (\(a\) and \(b\)) 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:

Since \(a\) and \(b\) have the same parities in order for \(y = ax\) and \(y = \frac{b}{x}\) to have an intersection, we have \(ab > 0\) from condition 1).

Since we have \(a + b > 0\) from condition 2), we have \(a > 0\) and \(b > 0\) when we consider both conditions together.

Then point (\(a, b\)) is in the first quadrant.

The answer is unique, and conditions 1) and 2) together are sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions 1) & 2) together are sufficient.

Therefore, C is the correct 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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have \(2\) variables (\(a\) and \(b\)) and \(1\) equation, D is most likely the answer. So, we should consider each condition on its own first.

Let’s look at condition 1). It tells us that it is not sufficient since we don’t have any information regarding a.

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.

Let’s look at condition 2). It tells us that it is not sufficient since we don’t have any specific numbers for a or b.

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 give us that they are not sufficient.

If \(a = 30\) and \(b = 2\), we have \(a + b = 32.\)

If \(a = 15\) and \(b = 4\), we have \(a + b = 19.\)

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.

Both conditions 1) & 2) together are not sufficient.

Therefore, E is the correct 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.

[GMAT math practice question]

(Geometry) Are triangles \(△APC\) and \(△ABQ\) congruent to each other?



1) \(△PBC\) and \(△QAC\) are equilateral triangles.

2) \(△ABC\) is an equilateral triangle.

[GMAT math practice question]

(Statistics) \(100\) students take a test. What is their test average?

1) There are \(40\) female students.

2) The female students’ average is \(70\), and the male students’ average is \(60\).

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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 condition 1). It tells us that \(PA = AB, AC = AQ\), and \(∠PAC = ∠BAQ. \)

Since \(∠PAC = 60° + ∠A\) and \(∠BAQ = 60° + ∠A\), we have \(∠PAC = ∠BAQ.\)

Then triangles \(△APC\) and \(△ABQ\) are congruent according to the SAS congruency property, so we get yes as an answer.

The answer is unique, yes, so the condition is sufficient according to Common Mistake Type 1, which states that the answers must be in terms of a unique “yes” or “no.”

Let’s look at condition 2). It tells us that it is not sufficient.
If triangles \(△ABC, △APB\), and \(△ACQ\) are congruent, then triangles \(△APC\) and \(△ABQ\) are congruent, so we get yes as an answer.
If triangles \(△ABC\) and \(△APB\) are congruent with sides \(3\) and \(AQ = CQ\), then triangles \(△APC\) and \(△ABQ\) are not congruent, so we get no as an answer.

The answer is not unique, yes and no, so the condition is not sufficient according to Common Mistake Type 1, which states that if we get both yes and no as an answer, it is not sufficient.

Condition 1) ALONE is sufficient.

Therefore, A is the correct 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.

[GMAT math practice question]

(Geometry) The figure shows that \(△ABC\) is an equilateral triangle and \(∠HLK = 40^o\). What is the ratio of \(∠BJF\) to \(∠DIB\)?

1) \(△HLK\) is an isosceles triangle with \(HL=LK. \)

2) \(DE\) is parallel to \(FG.\)

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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.

We can set this question up with a 2x2 matrix.



We have \(a + b = 100.\)

Since we have \(4\) variables (\(a, b, x\), and \(y\)) and \(1\) equation, E is most likely the answer in general. However, since we have 1 equation in condition 1) and 2 equations in condition 2, C is most likely the answer in this question. 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 tell us that \(a = 60, b = 40, x = 60\) and \(y = 70.\)

The average is the value when the total score is divided by the total number of students, which is \(\frac{(ax + by) }{ (a + b)}\).

We have \(a = 60\) and \(b = 40\) from condition 1) and we have \(x = 60\) and \(y = 70.\)

Thus, the average is \(\frac{(60·60 + 40·70) }{ (60 + 40)} = \frac{(3600 + 2800) }{ 100} = \frac{6400}{100} = 640.\)

The answer is unique, and conditions 1) and 2) together are sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

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.

Let’s look at condition 1). It tells us that \(a = 60\) and \(b = 40.\)

If \(x = 60\) and \(y = 70\), the average is \(\frac{(60·60 + 40·70) }{ (60 + 40)} = \frac{(3600 + 2800) }{ 100} = \frac{6400}{100} = 64.\)

If \(x = 60\) and \(y = 60\), the average is \(\frac{(60·60 + 40·60) }{ (60 + 40)} = \frac{(3600 + 2400) }{ 100} = \frac{6000}{100} = 60.\)

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.

Let’s look at condition 2). It tells us that \(x = 60\) and \(y = 70.\)

If \(a = 60\) and \(b = 40\), the average is \(\frac{(60·60 + 40·70) }{ (60 + 40)} = \frac{(3600 + 2800) }{ 100} = \frac{6400}{100} = 64.\)

If \(a = 50\) and \(b = 50\), the average is \(\frac{(50·60 + 50·70) }{ (50 + 50)} = \frac{(3000 + 3500) }{ 100} = \frac{6500}{100} = 65.\)

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.

Both conditions 1) & 2) together are sufficient.

Therefore, C is the correct 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.