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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 \(DE\)?

Attachment:
2.10DS.png
2.10DS.png [ 5.69 KiB | Viewed 1911 times ]

1) Triangle \(ABC\) is a right isosceles triangle with \(AB = AC.\)

2) \(BD = 7\) and \(CE = 15.\)
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[GMAT math practice question]

(Geometry) The figure shows parallelogram \(ABCD\) and point \(E\) on line \(AD\). What is the ratio of \(△ABE : △DCE\)?

Attachment:
2.11DS.png
2.11DS.png [ 15.24 KiB | Viewed 1895 times ]

1) The area of \(ABCD\) is \(50.\)

2) \(AE : ED = 3 : 2.\)
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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 \(DE\)?

Attachment:
The attachment 2.10DS.png is no longer available

1) Triangle \(ABC\) is a right isosceles triangle with \(AB = AC.\)

2) \(BD = 7\) and \(CE = 15.\)
Show more

=>

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
2.10(A).png [ 8.97 KiB | Viewed 1879 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.

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.
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[GMAT math practice question]

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

Attachment:
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2.12DS.png [ 4.22 KiB | Viewed 1869 times ]

1) \(AD\) is parallel to \(BC.\)

2) \(ABCD\) is an isosceles trapezoid.
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[GMAT math practice question]

(Geometry) The figure shows parallelogram \(ABCD\) and point \(E\) on line \(AD\). What is the ratio of \(△ABE : △DCE\)?

Attachment:
2.11DS.png

1) The area of \(ABCD\) is \(50.\)

2) \(AE : ED = 3 : 2.\)
Show more

=>

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 triangles \(ABE\) and \(DCE\) have the same height, we have \(△ABE : △DCE = AE : ED.\)

Thus, condition 2) is sufficient.

Condition 1)
Since we don’t know \(AE : ED\), condition 1) is not sufficient.

Therefore, B is the answer.
Answer: B
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Math Revolution DS Expert - Ask Me Anything about GMAT DS 13 Feb 2020, 01:27
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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 \(3\).
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[GMAT math practice question]

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

Attachment:
The attachment 2.12DS.png is no longer available

1) \(AD\) is parallel to \(BC.\)

2) \(ABCD\) is an isosceles trapezoid.
Show more

=>

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
2.12ds(a).png [ 6.37 KiB | Viewed 1844 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.

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.
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Math Revolution DS Expert - Ask Me Anything about GMAT DS 14 Feb 2020, 00:53
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[GMAT math practice question]

(Number Properties) For positive integers \(A\) and \(B, G\) is the greatest common divisor of \(A\) and \(B\), and \(L\) is the least common multiple of \(A\) and \(B\). What is \(A + B\)?

1) \(L = 70\)

2) \(\frac{G}{A} + \frac{G}{B} = \frac{7}{10} \)
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Math Revolution DS Expert - Ask Me Anything about GMAT DS Updated on: 12 Apr 2021, 03:28
Originally posted by MathRevolution on 16 Feb 2020, 18:53.
Last edited by MathRevolution on 12 Apr 2021, 03:28, edited 1 time in total.
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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 \(3\).
Show more

=>

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.
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Math Revolution DS Expert - Ask Me Anything about GMAT DS Updated on: 12 Apr 2021, 03:31
Originally posted by MathRevolution on 16 Feb 2020, 18:57.
Last edited by MathRevolution on 12 Apr 2021, 03:31, edited 1 time in total.
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[GMAT math practice question]

(Number Properties) For positive integers \(A\) and \(B, G\) is the greatest common divisor of \(A\) and \(B\), and \(L\) is the least common multiple of \(A\) and \(B\). What is \(A + B\)?

1) \(L = 70\)

2) \(\frac{G}{A} + \frac{G}{B} = \frac{7}{10} \)
Show more

=>

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)

Assume \(A = aG\) and \(B = bG\) where \(a\) and \(b\) are relatively prime numbers.

Since \(\frac{G}{A} + \frac{G}{B} = \frac{7}{10}\) from condition 2), we have

\(\frac{G}{A} + \frac{G}{B} = \frac{7}{10}, \)

\(\frac{G}{aG} + \frac{G}{bG} = \frac{7}{10}\) (substituting in \(A = aG\) and \(B = bG\)),

\(\frac{bG}{abG} + \frac{aG}{abG} = \frac{7}{10}\) (getting a common denominator),

\(\frac{(aG + bG)}{abG} = \frac{7}{10}\) (adding the fractions),

\(\frac{(A + B)}{L} = \frac{7}{10}\) (substituting \(A = Ag\), \(B = bG\), and \(L = abG\)).

Since \(L = 70\) from condition 1), we have \(\frac{(A + B)}{L} = \frac{7}{10}, \frac{(A + B)}{70 }= \frac{7}{10}\) or \(A + B = 49.\)

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

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.

Condition 1)
If \(A = 70\) and \(B = 1\), then we have \(A + B = 71.\)

If \(A = 70\) and \(B = 2\), then we have \(A + B = 72.\)

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

Condition 2)
Assume \(A = aG\) and \(B = bG\) where \(a\) and \(b\) are relatively prime numbers.

\(\frac{G}{A} + \frac{G}{B} = \frac{1}{a} + \frac{1}{b} = \frac{7}{10}.\)

If \(a = 2, b = 5\), and \(G = 1,\) then we have \(A = aG = 2(1) = 2, B = bG = 5(1) = 5\) and \(A + B = 2 + 5 = 7.\)

If \(a = 2, b = 5,\) and \(G = 2,\) then we have \(A = aG = 2(2) = 4, B = bG = 5(2) = 10\) and \(A + B = 4 + 10 = 14.\)

Since condition 2) does not yield a unique solution, it is not 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.
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[GMAT math practice question]

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

1)\( x + y = 9\)

2) \(xy = 2\)
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[GMAT math practice question]

(Equation) What is the value of \(abc\)?

1) \(a + \frac{4}{b} = 1\)

2) \(b + \frac{1}{c} = 4\)
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Math Revolution DS Expert - Ask Me Anything about GMAT DS Updated on: 22 May 2021, 02:19
Originally posted by MathRevolution on 19 Feb 2020, 02:12.
Last edited by MathRevolution on 22 May 2021, 02:19, edited 1 time in total.
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[GMAT math practice question]

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

1)\( x + y = 9\)

2) \(xy = 2\)
Show more

=>

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.
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Math Revolution DS Expert - Ask Me Anything about GMAT DS 19 Feb 2020, 02:13
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[GMAT math practice question]

(Integer) What is the value of \(\sqrt{360 x} \)?

1) \(x\) is a \(2\) digit integer.

2) \(\sqrt{360 x}\) is an integer.
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Math Revolution DS Expert - Ask Me Anything about GMAT DS Updated on: 22 May 2021, 02:20
Originally posted by MathRevolution on 20 Feb 2020, 01:54.
Last edited by MathRevolution on 22 May 2021, 02:20, edited 1 time in total.
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[GMAT math practice question]

(Equation) What is the value of \(abc\)?

1) \(a + \frac{4}{b} = 1\)

2) \(b + \frac{1}{c} = 4\)
Show more

=>

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 \(3\) variables (\(a, b\), and \(c\)) 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.
The question --------- is equivalent to ----------- for the following reason

Conditions 1) & 2)

When we multiply both sides of the equation \(a + \frac{4}{b} = 1\) by \(b\), we have \(ab + 4 = b.\)

When we multiply both sides of the equation \(b + \frac{1}{c} = 4\) by \(c\), we have \(bc + 1 = 4c\) or \(bc = 4c – 1\).

When we multiply both sides of the equation \(ab + 4 = b\) by \(c\), we have \(abc + 4c = bc.\)

When we replace \(bc\) of the equation \(abc + 4c = bc\) by \(4c – 1\), we have \(abc + 4c = 4c – 1\) or \(abc = -1\).

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 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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Math Revolution DS Expert - Ask Me Anything about GMAT DS 20 Feb 2020, 01:55
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[GMAT math practice question]

(Absolute Values) What is the difference between the maximum and the minimum values of \(x\)?

1) \(x\) satisfies \(2 < \sqrt{|x-2|} < 4.\)

2) \(x\) is an integer.
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Math Revolution DS Expert - Ask Me Anything about GMAT DS Updated on: 27 Jun 2021, 03:35
Originally posted by MathRevolution on 21 Feb 2020, 01:44.
Last edited by MathRevolution on 27 Jun 2021, 03:35, edited 1 time in total.
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[GMAT math practice question]

(Integer) What is the value of \(\sqrt{360 x} \)?

1) \(x\) is a \(2\) digit integer.

2) \(\sqrt{360 x}\) is an integer.
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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since 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 we don’t have a unique value from condition 1), it does not yield a unique solution, and it is not sufficient.

Condition 2)

\(10\) and \(40\) are possible values of \(x\), since \(\sqrt{360*10} = \sqrt{3600} = 60\) and \(\sqrt{360*40} = \sqrt{14400} = 120.\)

Since condition 2) does not yield a unique solution, it is not sufficient.

Conditions 1) & 2)

\(10\) and \(40\) are possible values of x satisfying both conditions as well.

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

Therefore, E is the answer.
Answer: E

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A, B, C, or E.
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Math Revolution DS Expert - Ask Me Anything about GMAT DS 21 Feb 2020, 01:46
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[GMAT math practice question]

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

1) \(\frac{1}{a}-\frac{1}{b}=2\)

2) \(\frac{1}{a^2}+\frac{1}{b^2}=3\)
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Math Revolution DS Expert - Ask Me Anything about GMAT DS Updated on: 28 May 2021, 03:19
Originally posted by MathRevolution on 23 Feb 2020, 19:15.
Last edited by MathRevolution on 28 May 2021, 03:19, edited 1 time in total.
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[GMAT math practice question]

(Absolute Values) What is the difference between the maximum and the minimum values of \(x\)?

1) \(x\) satisfies \(2 < \sqrt{|x-2|} < 4.\)

2) \(x\) is an integer.
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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since 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 we have \(2 < \sqrt{|x-2|} < 4\), we have \(4 < | x – 2 | < 16\), by squaring everything.

Two Cases
First Case – positive value
\((x – 2) = 4\)
\(x = 6\)

\((x – 2) = 16\)
\(x = 18\)
Then we have \(6 < x < 18\)

Second Case – negative value
\(-(x – 2) = 4\)
\(-x + 2 = 4\)
\(-x = 2\)
\(x = -2\)

\(-(x - 2) = 16\)
\(-x + 2 = 16\)
\(-x = 14\)
\(x = -18\)
Then we have \(-14 < x < -2\)

It means we have \(-14 < x < -2\) or \(6 < x < 18.\)
However, we don’t have either a maximum value of \(x\) or a minimum value of \(x\).

Since we can’t specify a unique solution, it is not sufficient.

Condition 2)

Since condition 2) does not yield a unique solution, it is not sufficient.


Conditions 1) & 2)
Using our solutions from earlier, namely \(-14 < x < -2\) or \(6 < x < 18\), we know that the possible values of \(x\) satisfying both conditions are \(-13, -12, … , -3\) and \(7, 8, … , 17.\)

The maximum value of \(x\) is \(17\) and the minimum value of \(x\) is \(-13.\)

The difference between the maximum and the minimum value of \(x\) is \(17 – (-13) = 30.\)

Therefore, C is the answer.
Answer: C

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A, B, C, or E.
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Math Revolution DS Expert - Ask Me Anything about GMAT DS Updated on: 28 May 2021, 03:20
Originally posted by MathRevolution on 23 Feb 2020, 19:18.
Last edited by MathRevolution on 28 May 2021, 03:20, edited 1 time in total.
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[GMAT math practice question]

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

1) \(\frac{1}{a}-\frac{1}{b}=2\)

2) \(\frac{1}{a^2}+\frac{1}{b^2}=3\)
Show more

=>

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 \(\frac{1}{a} – \frac{1}{b} = 2\)

\(\frac{b}{ab} – \frac{a}{ab} = 2\) (common denominator)

\(\frac{(b – a)}{ab} = 2\) (subtracting fractions)
\(a – b = -2ab\) (multiplying both sides by \(ab\))

Since we have \(\frac{1}{a^2} + \frac{1}{b^2} = 3 \)

\(\frac{b^2}{a^2b^2} + \frac{a^2}{a^2b^2} = 3\) (common denominator)

\(\frac{(b^2 + a^2)}{a^2b^2} = 3\) (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.
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