The Ultimate Q51 Guide [Expert Level]

[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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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.



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.


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.

[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.



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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 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.

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.

[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.

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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.

<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.

Therefore, D is the answer.
Answer: D

[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?



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


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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.

Therefore, E is the answer.
Answer: E

[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.



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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 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 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.

Therefore, C is the answer.
Answer: C

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B, or D. Since E (i.e. conditions 1) & 2) are NOT sufficient when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2) when taken together. Obviously, there may be occasions on which the answer is A, B, C, or D.

[GMAT math practice question]

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

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



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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 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)



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.

Therefore, C is the answer.
Answer: C

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B, or D. Since E (i.e. conditions 1) & 2) are NOT sufficient when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2) when taken together. Obviously, there may be occasions on which the answer is A, B, C, or D.

[GMAT math practice question]

(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.

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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.

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.

Therefore, A is the 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 below shows parallelogram ABCD where ∠DAC = 60 and ∠DBC = 30. What is ∠BDC?

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



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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°.

Therefore, B is the answer.
Answer: B

[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?



A. 80°
B. 85°
C. 90°
D. 95°
E. 100°

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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°.

Therefore, C is the answer.
Answer: C

[GMAT math practice question]

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

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

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.

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]

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

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

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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 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.

Therefore, C is the answer.
Answer: C

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B, or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D, or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.

[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


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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.

Therefore, B is the answer.
Answer: B

[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

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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.

Therefore, the answer is D.
Answer: D

[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.

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]

(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.

Therefore, C is the answer.
Answer: C

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B, or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D, or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.

[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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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)



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.

Therefore, C is the answer.
Answer: C

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B, or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D, or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.

[GMAT math practice question]

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

1) x + y = 2√3
2) √3x - √2y=5

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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
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.

Therefore, C is the answer.
Answer: C

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B, or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D, or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.