Ask me Anything about GMAT Data Sufficiency Hello, I am Max Lee, Founder and Lead Math Tutor at Math Revolution. We have over 9,000 posts and over 10,000 Kudos. This topic is a new feature on the DS Forum and a way for you to directly interact with our math team and ask anything about the DS, e.g. if you want a certain concept explained or have a particular question you want to us address, this is the place to post a link to it or your question. I intend to have this thread be as an "Everything You Need to Know about DS" type of thread. I will keep updating this post with links and resources that are helpful for the DS. Meanwhile, you can ask me anything Other discussions you may be interested in: How to Improve GMAT Quant from Q49 to Q51 - the Ultimate Q51 Quide Overview of GMAT Math Question Types and Patterns on the GMAT We will be posting new questions in this topic regularly - check it out for some new material! Thank you all - good luck on the GMAT and look forward to seeing you in the PS forum! - Max Lee.

(Geometry) The figure shows that ∠BAD is 30^o , and ∠CAE is 40^o. What is the measure of ∠ADE ? 1.30DS.png 1) Point O is the circumcenter of triangle ABC . 2) Point I is the incenter of triangle ABC .

=> 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 a triangle, we have 3 variables and 0 equations, 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) Since point I is the incenter of the triangle ABC, AD, AE, BE, BF, CD , and CE are tangent to the same circle, and we have AD = AE, BE = BF and CD = CF. Then we have CD = FC = BC – BF = BC – BE = 11 – 8 = 3. Since both conditions together yield a unique solution, they are sufficient. 1.29DS(A).png 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 in which the answer is A, B, C, or D.

(Number Properties) What is the value of \frac{ }{x - } - \frac{x}{ + x} ? ( means the greatest integer less than or equal to x ) 1) 2 ≤ x < 3. 2) x is not integer.

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MathRevolution

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29 Jan 2020, 01:10

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[GMAT math practice question]

(Geometry) What is the length of \(CD\) in the figure?

1) Point \(I\) is the incenter of triangle \(ABC\), and \(D, E\), and \(F\) are the tangential points.

2) The length of \(BC\) is \(11\), and \(BE\) is \(8. \)

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[GMAT math practice question]

(Geometry) What is the measure of \(∠BIC\) in the figure?

Attachment:

1.28ds.png

1) Point \(I\) is the incenter (the point where the three angle bisectors meet) of triangle \(ABC.\)
2) \(∠BAC = 50^o\)

=>

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 \(4\) variables (\(∠BIC, ∠ABC, ∠BCA\), and \(∠CAB\)) and \(1\) equation (\(∠ABC + ∠BCA + ∠CAB = 180°\)), 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)
Since point \(O\) is the incenter of triangle \(ABC\) from condition 1), the segments \(IA, IB\), and \(IC\) are bisectors of \(∠BAC, ∠ABC\) and \(∠BCA\), respectively.

\(∠BIC = 180° – (∠IBC + ∠ICB), ∠BIC = 180° – (\frac{1}{2})(180° - ∠BAC), ∠BIC = 180° – (\frac{1}{2})(180° - 50°), ∠BIC = 180° – 65° = 115°.\)

Since both conditions together yield a unique solution, they are sufficient.

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 in which the answer is A, B, C, or D.

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[GMAT math practice question]

(Geometry) The figure shows that \(∠BAD\) is \(30^o\), and \(∠CAE\) is \(40^o.\) What is the measure of \(∠ADE\)?

Attachment:

1.30DS.png [ 7.84 KiB | Viewed 1629 times ]

1) Point \(O\) is the circumcenter of triangle \(ABC\).

2) Point \(I\) is the incenter of triangle \(ABC\).

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Attachment:

The attachment 1.29ds.png is no longer available

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1.29DS(A).png [ 10.73 KiB | Viewed 1594 times ]

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 in which the answer is A, B, C, or D.

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(Number Properties) What is the value of \(\frac{[x]}{x - [x]} - \frac{x}{[-x + 1] + x}\)? (\([x]\) means the greatest integer less than or equal to \(x\))

1) \(2 ≤ x < 3.\)

2) \(x\) is not integer.

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Hello MathRevolution

Can you explain the VA and the LLGG, or show me where I can find resources to study that?

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[GMAT math practice question]

(Geometry) The figure shows that \(∠BAD\) is \(30^o\), and \(∠CAE\) is \(40^o.\) What is the measure of \(∠ADE\)?

Attachment:

1.30DS.png

1) Point \(O\) is the circumcenter of triangle \(ABC\).

2) Point \(I\) is the incenter of triangle \(ABC\).

=>

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 \(6\) variables (\(∠ABC, ∠BCA, ∠CAB, ∠ADE, ∠DAE,\) and \(∠DEA\)) and \(2\) equations (\(∠ABC + ∠BCA + ∠CAB = 180°, ∠ADE + ∠DAE + ∠DEA = 180°\)), 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)
Since point \(I\) is the incenter of triangle \(ABC\), we have \(∠IAB = ∠IAC = 40°\) and \(∠DAE = ∠IAB - ∠BAD, ∠DAE = 40° - 30°, ∠DAE = 10°.\)

Since point \(\)O is the circumcenter of triangle \(ABC\), we have \(∠BAO = ∠ABO = 30°\) and \(∠OBC = ∠OCB = (\frac{1}{2})(180° – (2*30° + 2*50°)) = 10°.\)

Then \(∠ABC = ∠ABO + ∠OBC = 30° + 10° = 40°.\)

Since \(∠ADE\) is an exterior angle of triangle \(ABD,\) we have \(∠ADE = ∠DAB + ∠ABD = 30° + 40° = 70°.\)

Since both conditions together yield a unique solution, they are sufficient.

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 in which the answer is A, B, C, or D.

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Updated on: 17 Jun 2021, 01:26

Originally posted by MathRevolution on 02 Feb 2020, 19:31.
Last edited by MathRevolution on 17 Jun 2021, 01:26, edited 1 time in total.

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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.

Condition 1)
Since \(2 ≤ x < 3\), we have \(\frac{[x]}{x - [x]} - \frac{x}{[-x + 1] + x} = \frac{2}{x - 2} - \frac{x}{-2+ x} = \frac{2 - x}{x - 2} = \frac{-1(x - 2)}{x - 2} = -1.\)

Since condition 1) yields a unique solution, it is sufficient.

Condition 2) If \(x = n + h\) where n is the integer part of \(x\) and \(h\) is the positive decimal part of \(x\), then we have:

\(\frac{[x]}{x - [x]} - \frac{x}{[-x + 1] + x} = \frac{n}{h} - \frac{n + h}{-n + n + h} = \frac{n}{h} - \frac{n + h}{h} = \frac{-h}{h} = -1.\)

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

Therefore, D is the answer.
Answer: D

This question is a CMT 4(B) question: condition 1) is easy to work with, and condition 2) is difficult to work with. For CMT 4(B) questions, D is most likely 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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Hello MathRevolution

Can you explain the VA and the LLGG, or show me where I can find resources to study that?

You can visit the following page to review VA method.
https://www.mathrevolution.com/gmat/lesson

Regarding LLGG, when we have (x-a)(x-b) > 0 where a < b, the solution set of the inequality is "x < a or x > b".
"x < a or x > b" means that x is Less than Less number or x is Greater than Greater number.
Here the Less number is a and the Greater number is b.

On the other hand, when we have (x-a)(x-b) < 0 where a < b, the solution set of the inequality is "a < x < b".
It means the solution set is the set of numbers between a and b.

Happy Studying !!!

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03 Feb 2020, 01:31

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[GMAT math practice question]

(Geometry) The figure below shows the dimensions of the right triangle \(ABC\) with \(AB = 13, BC = 12, CA = 5\) and \(I\) is a point inside triangle \(ABC.\) Angle \(C\) is \(90^o.\) What is the minimum distance from the point \(I\) to sides \(AB, BC\) and \(CA\)?

Attachment:

2.3ds.png [ 12.43 KiB | Viewed 1577 times ]

1) Point \(I\) is the incenter of \(△ABC.\)

2) Line \(AI\) bisects angle \(A\), and line \(BI\) bisects angle \(B\).

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InaKi20

Hello MathRevolution

Can you explain the VA and the LLGG, or show me where I can find resources to study that?

Thanks so much MathRevolution,
this really helps a lot!

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[GMAT math practice question]

(Geometry) The figure below shows the dimensions of triangle \(ABC\). What is \(∠BIC\)?

Attachment:

2.4DS.png [ 8.92 KiB | Viewed 1528 times ]

1) Point \(I\) is the incenter of \(△ABC.\)

2) Line \(DE\) is parallel to line \(BC.\)

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Attachment:

The attachment 2.3ds.png is no longer available

1) Point \(I\) is the incenter of \(△ABC.\)

2) Line \(AI\) bisects angle \(A\), and line \(BI\) bisects angle \(B\).

Attachment:

2.3DS.A1.png [ 9.44 KiB | Viewed 1513 times ]

Since I is the incenter of the triangle, the distances to all sides from point I are equal. Assume the distances are \(x\).

Attachment:

2.3DS(A2).png [ 11.47 KiB | Viewed 1514 times ]

The area of triangles \(IAB, IBC\) and \(ICA\) are \((\frac{1}{2})*13*x + (\frac{1}{2})*12*x + (\frac{1}{2})*5*x = (\frac{13}{2})x + 6x + (\frac{5}{2})x = (\frac{18}{2})x + 6x = 9x + 6x = 15x.\)

The area of triangle \(ABC = (\frac{1}{2})*5*12 = 30.\)

Since the sum of the areas of triangles \(IAB, IBC\) and \(ICA\) is equal to the area of triangle \(ABC\), we have \(15x = 30\) or \(x = 2.\)

Since condition 1) yields a unique solution, it is sufficient.

Condition 2)

Since we can find an incenter of a triangle by the intersection of lines bisecting interior angles, \(I\) is the incenter of the triangle from condition 2).

Thus, condition 2) is sufficient with the previous reasoning in condition 1).

Therefore, D is the answer.
Answer: D

Note: Tip 1) of the VA method states that D is most likely to be the answer if condition 1) gives the same information as condition 2).

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[GMAT math practice question]

(Algebra) What is the value of \([x] + [-x]\)? (\([x]\) means the greatest integer less than or equal to \(x\).)

1) \(0 ≤ x\)

2) \(x\) is not an integer.

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[GMAT math practice question]

(Geometry) The figure below shows the dimensions of triangle \(ABC\). What is \(∠BIC\)?

Attachment:

The attachment 2.4DS.png is no longer available

1) Point \(I\) is the incenter of \(△ABC.\)

2) Line \(DE\) is parallel to line \(BC.\)

Attachment:

2.4DS(A).png [ 10.96 KiB | Viewed 1493 times ]

Then we have \(∠IBC = 22°\) and \(∠ICB = 30°\).

Thus, we have \(∠BIC = 180° - ∠IBC - ∠ICB = 180° – 22° – 30° = 128°.\)

Since condition 1) yields a unique solution, it is sufficient.

Condition 2)

Since we don’t know the position of \(I\) on segment \(DE\), condition 2) does not yield a unique solution, and it is not sufficient.

Therefore, A is the answer.
Answer: A

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[GMAT math practice question]

(Geometry) The figure below shows the dimensions of triangle ABC. What is \(∠OBI\)?

Attachment:

2.6DS.png [ 11.66 KiB | Viewed 1486 times ]

1) \(AB = AC\)

2) Point \(O\) is the circumcenter and point \(I\) is the incenter of triangle \(ABC.\)

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Updated on: 22 Apr 2021, 02:48

Originally posted by MathRevolution on 07 Feb 2020, 02:09.
Last edited by MathRevolution on 22 Apr 2021, 02:48, edited 1 time in total.

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[GMAT math practice question]

(Algebra) What is the value of \([x] + [-x]\)? (\([x]\) means the greatest integer less than or equal to \(x\).)

1) \(0 ≤ x\)

2) \(x\) is not an integer.

=>

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.

If \(x = n + h\) where \(n\) is an integer and \(0 ≤ h < 1\), then \([x] = n\). Here \(n\) is the integer part of \(n\), and \(h\) is the decimal part of \(n\).

If \(x\) is an integer, then we have \(x = n + h\) where \(h = 0, [x] = n, [-x] = -n\) and \([x] + [-x] = n + (-n) = 0.\)

Assume \(x\) is not an integer.

Then we have \(x = n + h\) where \(0 < h < 1, [x] = n.\)

We have \(-x = -n - h, -x = -n - 1 + 1 - h, -x = -(n + 1) + (1 - h)\) where \(0 < 1 - h < 1.\)

Thus \([-x] = -n - 1\) and we have \([x] + [-x] = n + (-n - 1) = -1.\)

Condition 2) tells us that \(x\) is not an integer. Therefore \([x] + [-x] = -1\) and condition 2) yields a unique solution.

Condition 2) is sufficient.

Condition 1)
If \(x = 0\) which is an integer, then we have \([x] + [-x] = 0.\)

If \(x = 1.5\) which is not an integer, then we have \([x] + [-x] = -1.\)

Since condition 1) does not yield a unique solution, it is not sufficient.

Therefore, B is the answer.
Answer: B

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[GMAT math practice question]

(Number Properties) \(<x>\) denotes \(x - 10[\frac{x}{10}]\) and \(n\) is a positive integer. What is the value of \(<9^n - 1>\)? (\([x]\) means the greatest integer less than or equal to \(x\).)

1) \(<9^n - 1>\) is not positive.

2) \(n\) is an even number.

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09 Feb 2020, 18:22

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[GMAT math practice question]

(Geometry) The figure below shows the dimensions of triangle ABC. What is \(∠OBI\)?

Attachment:

2.6DS.png

1) \(AB = AC\)

2) Point \(O\) is the circumcenter and point \(I\) is the incenter of triangle \(ABC.\)

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Since a triangle has \(3\) variables, 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)
Since triangle \(ABC\) is an isosceles, we have interior angles \(∠ABC = ∠ACB = (\frac{1}{2})(180 - ∠A) = (\frac{1}{2})(180 - 36) = 72.\)

Since \(O\) is the circumcenter of triangle \(ABC\), we have \(∠OBA = ∠OCA = (\frac{1}{2}) ∠A = (\frac{1}{2})36 = 18.\)

Since \(I\) is the incenter of triangle \(ABC\), we have \(∠IBA = (\frac{1}{2}) ∠ABC = (1/2)72 = 36.\)

Thus, we have \(∠OBI = ∠IBA - ∠OBA = 36 – 18 = 18.\)

Since both conditions together yield a unique solution, they are sufficient.

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 in which the answer is A, B, C, or D.