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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8722
GMAT 1: 760 Q51 V42
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[GMAT math practice question]

(Number Properties) How many integers are there satisfying 1 < [3 - x/2] < 4? ([x] means the greatest integer less than or equal to x)

A. 2
B. 3
C. 4
D. 5
E. 6

=>

Since 1 < [3 - x/2] < 4, we have [3 - x/2] = 2 or [3 - x/2] = 3.

If [3 - x/2] = 2, then we have 2 ≤ 3 - x/2 < 3 or -1 ≤ - x/2 < 0 (subtracting 3). This is equivalent to 0 < x ≤ 2 (multiplying by -2, which changes the direction of the inequality signs).
If [3 - x/2] = 3, then we have 3 ≤ 3 - x/2 < 4 or 0 ≤ - x/2 < 1 (subtracting 3). This is equivalent to -2 < x ≤ 0 (multiplying by -2, which changes the direction of the inequality signs).

Thus, we have -2 < x ≤ 2 and the integer values of x are -1, 0, 1, and 2.
We have 4 integer solutions.

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Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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MathRevolution wrote:
[GMAT math practice question]

(Number Properties) How many integers are there satisfying 1 < [3 - x/2] < 4? ([x] means the greatest integer less than or equal to x)

A. 2
B. 3
C. 4
D. 5
E. 6

=>

Since 1 < [3 - x/2] < 4, we have [3 - x/2] = 2 or [3 - x/2] = 3.

If [3 - x/2] = 2, then we have 2 ≤ 3 - x/2 < 3 or -1 ≤ - x/2 < 0 (subtracting 3). This is equivalent to 0 < x ≤ 2 (multiplying by -2, which changes the direction of the inequality signs).
If [3 - x/2] = 3, then we have 3 ≤ 3 - x/2 < 4 or 0 ≤ - x/2 < 1 (subtracting 3). This is equivalent to -2 < x ≤ 0 (multiplying by -2, which changes the direction of the inequality signs).

Thus, we have -2 < x ≤ 2 and the integer values of x are -1, 0, 1, and 2.
We have 4 integer solutions.

Hi MathRevolution,

This is a good question.

I have 1 query.

Why can't x be 3? Can you please confirm the answer again?

Thank you.
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8722
GMAT 1: 760 Q51 V42
GPA: 3.82
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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 90o. What is the minimum distance from the point I to sides AB, BC and CA?

1) Point I is the incenter of △ABC.
2) Line AI bisects angle A, and line BI bisects angle B.

Attachment: 2.3ds.png [ 12.43 KiB | Viewed 426 times ]

=>

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.

Condition 1)

Attachment: 2.3DS.A1.png [ 9.44 KiB | Viewed 426 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 426 times ]

The area of triangles IAB, IBC and ICA are (1/2)*13*x + (1/2)*12*x + (1/2)*5*x = (13/2)x + 6x + (5/2)x = (18/2)x + 6x = 9x + 6x = 15x.
The area of triangle ABC = (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).

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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Joined: 16 Aug 2015
Posts: 8722
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Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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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.

Attachment: 1.29DS(A).png [ 10.73 KiB | Viewed 402 times ]

=>

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.

Attachment: 1.29DS(A).png [ 10.73 KiB | Viewed 401 times ]

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.

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.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8722
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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[GMAT math practice question]

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

1) AB = AC,
2) Point O is the circumcenter and point I is the incenter of triangle ABC.

Attachment: 2.6DS.png [ 11.66 KiB | Viewed 363 times ]

=>

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 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 = (1/2)(180 - ∠A) = (1/2)(180 - 36) = 72.
Since O is the circumcenter of triangle ABC, we have ∠OBA = ∠OCA = (1/2) ∠A = (1/2)36 = 18.
Since I is the incenter of triangle ABC, we have ∠IBA = (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.

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.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8722
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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[GMAT math practice question]

(Number Properties) <x> denotes x - 10[x/10] and n is a positive integer. What is the value of <9n - 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.

=>

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.

<x> means the unit digit of x.
For example, if x = 123, then x - 10[x/10] = 123 - 10[12.3] = 123 - 120 = 3.

We have 9^1 = 9, 9^2 = 81, 9^3 = 729, 9^4 = 6561, ...
Then, 9^1 - 1 = 8, 9^2 - 1 = 80, 9^3 - 1 = 728, 9^4 - 1 = 6560, ....
We notice that if n is an odd number, the unit digit of 9^n - 1 is 8, and if n is an even number, the unit digit of 9^n - 1 is 0.

The question asks what the unit digit of 9^n - 1 is.
Condition 2) tells us that n is an even number. Therefore 9^n - 1 is 0. Since condition 2) yields a unique solution, it is sufficient.

Condition 1)
Since the only possible values of <9^n - 1> are 0 and 8, <9^n - 1> is 0 if <9^n - 1> is not positive.

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

_________________
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Joined: 16 Aug 2015
Posts: 8722
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GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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[GMAT math practice question]

(Geometry) As the figure shows AD = 3, BD = 5 and BC = 9. Moreover, point I is the incenter of triangle ABC. What is the length of AC?

Attachment: 2.3ps.png [ 11.56 KiB | Viewed 328 times ]

A. 3
B. 4
C. 5
D. 6
E. 7

=>

Since AD = AF, we have AF = 3.
Since BC = 9 and BD = 5, we have FC = EC = BC – BE = BC – BD = 9 – 5 = 4.

Then we have AC = AF + FC = 3 + 4 = 7.

_________________
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Joined: 16 Aug 2015
Posts: 8722
GMAT 1: 760 Q51 V42
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Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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[GMAT math practice question]

(Geometry) As the figure below shows, lines BD and CE are perpendicular to l. What is the length of DE?

1) Triangle ABC is a right isosceles triangle with AB = AC.
2) BD = 7 and CE = 15.

Attachment: 2.10DS.png [ 5.69 KiB | Viewed 315 times ]

=>

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 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)

Attachment: 2.10(A).png [ 8.97 KiB | Viewed 315 times ]

Since we have AB = CA, ∠ADB = ∠CEA = 90° and ∠BAD = 90° - ∠CAE = ∠ACE, triangles ABD and CAE are congruent.
Then we have BD = AE = 7 and AD = CE = 15.
Thus, DE = AD – AE = 15 – 7 = 8.

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

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.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8722
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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[GMAT math practice question]

(Geometry) Is OB = OC in the figure below?

1) AD is parallel to BC.
2) ABCD is an isosceles trapezoid.

Attachment: 2.12DS.png [ 4.22 KiB | Viewed 279 times ]

=>

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 a quadrilateral has 5 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)

Attachment: 2.12ds(a).png [ 6.37 KiB | Viewed 279 times ]

Since we have AB = DC, ∠ABC = ∠DCB, and BC is a common side, triangles ABC and DCB are congruent.
Thus, ∠OBC = ∠OCB and triangle OBC is an isosceles.
Then we have OB = OC.

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

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.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8722
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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[GMAT math practice question]

(Integers) What is the remainder of 9^n-1 when it is divided by 10?

1) n is divisible by 2.
2) n is divisible by 3.

=>

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.

9^1 = 9, 9^2 = 81 ~ 1, 9^3 = 729 ~ 9, 9^4 ~ 1, …
The odd number powers of 9 have the units digit 9 and the even number powers of 9 have the units digits 1.

Condition 1) tells us that n is an even number and 9^n – 1 ~ 1 – 1 = 0.
Thus condition 1) is sufficient.

Condition 2)
If n = 3, then we have 9^3 – 1 ~ 9 – 1 = 8. However, if n = 6, then we have 9^6 – 1 ~ 1 – 1 = 0. Therefore, condition 2) does not yield a unique solution.
Condition 2) is not sufficient.

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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Joined: 16 Aug 2015
Posts: 8722
GMAT 1: 760 Q51 V42
GPA: 3.82
The Ultimate Q51 Guide [Expert Level]  [#permalink]

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[GMAT math practice question]

(Geometry) The figure below shows parallelogram ABCD where ∠DAC = 60 and ∠DBC = 30. What is ∠BDC?

A. 15
B. 30
C. 45
D. 60
E. 75

Attachment: 2.10PS.png [ 13.2 KiB | Viewed 248 times ]

=>

We have ∠OCB = ∠OAD = 60 since they are alternate interior angles. Since ∠OBC = 30°, we have ∠BOC = 180° – (30° + 60°) = 90°.
Then the two diagonals are perpendicular to each other, and quadrilateral ABCD is a rhombus.
Thus, triangle BCD is an isosceles with BC = DC.
Then we have ∠BDC = ∠DBC = 30°.

Attachments 2.10DS (2).png [ 13.2 KiB | Viewed 250 times ]

_________________
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Joined: 16 Aug 2015
Posts: 8722
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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[GMAT math practice question]

(Geometry) The figure below shows parallelogram ABCD where 2AB = AD. Moreover, FD = DC = CE and lines BF and AE meet at point P. What is ∠FPE?

Attachment: 2.12PS.png [ 11.79 KiB | Viewed 239 times ]

A. 80°
B. 85°
C. 90°
D. 95°
E. 100°

=>

Triangles HAB and HDF are congruent, since AB = DF, ∠HAB = ∠HDF and ∠ABH = ∠DFH.
We have AH = DH = (1/2)AD = AB.
Triangles ABG and ECG are congruent using similar reasoning.
Then we have BG = CG = (1/2)AD = AB.
Since we have AH = BG = AB, quadrilateral ABGH is a rhombus and ∠FPE = 90°.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8722
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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[GMAT math practice question]

(Algebra) What is the value of x - y?

1) x + y = 9
2) xy = 2

=>

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)

We have (x - y)^2 = (x + y)^2 - 4xy
Because
(x + y)^2 - 4xy
(x + y)(x + y) – 4xy
x^2+ xy + xy + y^2- 4xy
x^2 - 2xy + y^2
(x – y)(x – y)
(x – y)^2

Then, we substitute
(x + y)^2 - 4xy = 9^2 – 4*2 = 81 – 8 = 73.
Then, we have x – y = ±√73.

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

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.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8722
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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[GMAT math practice question]

(Algebra) What is a^3 - b^3?

1) 1/a-1/b=2
2) 1/a^2+1/b^2=3

=>

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 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)

Since we have 1/a – 1/b = 2
b/ab – a/ab = 2 (common denominator)
(b – a)/ab = 2 (subtracting fractions
a – b = -2ab (multiplying both sides by ab

Since we have 1/a^2 + 1/b^2 = 3
b^2/a^2b^2 + a^2/a^2b^2 = 3 (common denominator)
(b^2 + a^2)/a^2b^2 = 3 (adding fractions)
a^2 + b^2 = 3a^2b^2 (multiplying both sides by a^2b^2)

Then we have a^2 + b^2 = (a-b)^2 + 2ab
a^2 + b^2 = (-2ab)^2 + 2ab (substituting -2ab in for a – b)

a^2 + b^2 = 3a^2b^2 or a^2b^2 + 2ab = 0.
We have ab = -2 and a – b= -2ab = 4 from ab(ab+2)=0, since ab ≠ 0.
a^3-b^3 = (a-b)^3 + 3ab(a-b) = 4^3 + 3(-2)4 = 64 – 24 = 40.

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

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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Joined: 16 Aug 2015
Posts: 8722
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GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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[GMAT math practice question]

f(x) denotes the number of positive integers less than or equal to √x. What is f(1) + f(2) + f(3) +…+ f(50)?

A. 131
B. 217
C. 228
D. 307
E. 339

=>

Since 1 < √2 < √3 < 2, we have f(1) = f(2) = f(3) = 1.
Since 2 < √5 < √6 < √7 < √8 < 3, we have f(4) = f(5) = … = f(8) = 2.
Since 3 < √10 < √11 < √12 < … < √15 < 4, we have f(9) = f(10) = … = f(15) = 3.
Since 4 < √17 < √18 < √19 < … < √24 < 5, we have f(16) = f(17) = … = f(24) = 4.
Since 5 < √26 < √27 < √28 < … < √35 < 6, we have f(25) = f(26) = … = f(35) = 5.
Since 6 < √37 < √38 < √39 < … < √48 < 7, we have f(36) = f(37) = … = f(48) = 6.
Since 7 < √50, we have f(49) = f(50) = 7.

Then, f(1) + f(2) + f(3) + … + f(50) = 1*3 + 2*5 + 3*7 + 4*9 + 5*11 + 6*13 + 7*2 = 3 + 10 + 21 + 36 + 55 + 78 + 14 = 217.

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Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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[GMAT math practice question]

(Statistics) The average of a, b, c, and d is 5 and their standard deviation is 4. What is the sum of the average and the standard deviation of 2a-5, 2b-5, 2c-5 and 2d-5?

A. 7
B. 9
C. 11
D. 13
E. 15

=>

Remember the property E(aX+b) = aE(X) + b, where E(X) is the average of a set X.
Remember the property S(aX+b) = |a|S(X), where S(X) is the standard deviation of a set X.

E(2X-5) = 2E(X) – 5 = 2*5 – 5 = 10 – 5 = 5.
S(2X-5) = 2*S(X) = 2*4 = 8.

Then, the sum of their average and standard deviation is 5 + 8 = 13.

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Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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[GMAT math practice question]

(Algebra) What is the value of (ax + by)(bx + ay)?

1) a^2 + b^2 = 3
2) x^2 + y^2 = 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.

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 4 variables (a, b, x, and y) 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)

If a = √3, b = 0, x = 2, and y = 0, then (ax + by)(bx + ay) = (√3*2 + 0*0)(0*2 + √3*0) = 2√3*0 = 0.
If a = √2, b = 1, x = √3, and y = 1, then (ax + by)(bx + ay) = (√2*√3 + 1*1)(1*√3 + √2*1) = (1+√6)(√2√+3) ≠ 0.

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

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.
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GMAT 1: 760 Q51 V42
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Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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[GMAT math practice question]

(Algebra) What is the value of x/y?

1) 5x(5 + 2√5) - 5√5y(3 - 2√5) is a rational number.
2) x and y are rational numbers and xy ≠ 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.

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)

5x(5 + 2√5) - 5√5y(3 - 2√5)
= 25x + 10√5x - 15√5y + 50y (multiplying through the brackets)
= 25x + 50y + 10√5x - 15√5y (rearranging the terms)
= 25(x + 2y) + 5√5(2x - 3y) (taking out common factors)

We must have 2x – 3y = 0 in order for 5√5(2x-3y) to be zero (and therefore cancel out the square root or irrational number because condition 1 states that the answer is a rational number) when x and y are rational numbers.
Thus, we have 2x = 3y or x/y = 3/2.

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

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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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8722
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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[GMAT math practice question]

Triangle ABC is a right triangle with ∠C = 90. What is the length of AB?

1) The area of ABC is 35.
2) The length of the base is 4 less than twice of the height.

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Attachment: 2.27ds.png [ 3.58 KiB | Viewed 78 times ]

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 triangle has three variables and we have 1 equation from a right angle, 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)

Attachment: 2.27(a).png [ 14.52 KiB | Viewed 78 times ]

Then we have (1/2)(x)(2x - 4) = 35 or x^2 - 2x - 35 = 0
We have (x + 5)(x - 7) = 0 by factoring, and x = 7 since x > 0.
Thus, BC = 2*7 – 4 = 14 – 4 = 10.

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

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.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8722
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level]  [#permalink]

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[GMAT math practice question]

(Algebra) What is x^2 + y^2?

1) x + y = 2√3
2) √3x - √2y=5

=>

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)

When we subtract √3 times the equation of condition 1) from the equation of condition 2), we have
√3x - √2y - √3(x + y) = 5 - √3(2√3)
√3x - √2y - √3x - √3y = 5 – 6
√2y - √3y = -1
√3y - √2y = 1
(√3 + √2)y = 1
y = 1/(√3 + √2)
y = √3 - √2.

Then we have
x + y = 2√3
x = 2√3 – y
x = 2√3 - (√3 - √2)
x = 2√3 - √3 + √2
x = √3 + √2.

x^2 + y^2 = (x + y)^2 – 2xy
= (√3 + √2 + √3 - √2)^2 – 2(√3 + √2)(√3 -√2)
= (2√3)^2 – 2(3 - √6 + √6 - 2)
= 4(3) - 2(1)
= 12 – 2
= 0

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

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.
_________________ Re: The Ultimate Q51 Guide [Expert Level]   [#permalink] 05 Mar 2020, 18:20

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# The Ultimate Q51 Guide [Expert Level]

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