The Ultimate Q51 Guide [Expert Level]

[GMAT math practice question]

(algebra) What is the value of (4x-3xy+4y)/(3x+3y)?

1) x = y
2) (1/x) + (1/y) = 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.

The question asks the value of (4/3) – xy/(x+y) for the following reason
(4x-3xy+4y)/(3x+3y)
= [4(x+y)-3xy]/[3(x+y)]
= [4(x+y)]/[3(x+y)] - [3xy]/[3(x+y)]
= (4/3) – xy/(x+y)

Since we have (x+y)/xy = 3 from condition 2),) for the following reason
(1/x)+(1/y) = 3
(y/xy) + (x/xy) = 3
(x+y)/xy = 3
Then we have xy/(x+y) = 1/3.
Then (4/3) – xy/(x+y) = (4/3) – (1/3) = 1.

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

Condition 1)
Since we have x=y, (4/3) – xy/(x+y) = (4/3) – x^2/2x = (4/3)-x/2.
If x = y = 1, then (4/3) - x/2 = 5/6.
If x = y = 2, then (4/3) - x/2 = 1/3.

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

Therefore, B is the answer.
Answer: B

[GMAT math practice question]

(Equation) 2ax - 3b = a - bx is an equation in terms of x. What is its solution?

1) -3/2 is a solution of (b-a)x - (2a-3b) = 0
2) a = 3b

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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.
The question asks the value of (a+3b)/(2a+b) for the following reason.
2ax - 3b = a - bx
=> 2ax + bx = a + 3b
=> x(2a+b) = a+3b
=> x = (a+3b)/(2a+b)

Since we have a = 3b from condition 2), we have x = (a+3b)/(2a+b) = (3b+3b)/(6b+b) = (6b)/(7b) = 6/7.
Thus, condition 2) is sufficient.

Condition 1)
When we substitute -3/2 for x, we have (b-a)(-3/2) - (2a-3b) = 0 or (-3)(b-a) = 2(2a-3b). We have -3b+3a = 4a-6b or a = 3b.
Condition 1) is equivalent to condition 2), and it is also sufficient.

Therefore, D is the answer.

When a question asks for a ratio, if one condition includes a ratio and the other condition just gives a number, the condition including the ratio is most likely to be sufficient. This tells us that D is most likely to be the answer to this question, since each condition includes a ratio.

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).

This question is a CMT4(B) question: condition 2) is easy to work with, and condition 1) is difficult to work with. For CMT4(B) questions, D is most likely to be the answer.

[GMAT math practice question]

The figure shows that l is parallel to m, ABCD is a square, and the point D is on l and B is on m. What is ∠y - ∠x?

A. 15°
B. 25°
C. 30°
D. 35°
E. 40°


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When we draw an additional line k on the figure as follows,


Since ∠BCD = 90°, we have ∠x + 8∠x = 90° and ∠x = 10°.
Since ∠x + ∠y = 45°, we have ∠y = 35°. Thus ∠y - ∠x = 25°.

Therefore, B is the answer.
Answer: B

[GMAT math practice question]

(geometry) The figure shows that the lines AB and AC are the tangential lines to the circle O. What is ∠ABC?


A. 55
B. 60
C. 65
D. 70
E. 75

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Since AB and AC are tangent lines to the circle, we have AB = AC.
Since the triangle ABC is isosceles with AB = AC, we have ∠ABC = (1/2)(180-∠BAC) = 70.

Therefore, D is the answer.
Answer: D

[GMAT math practice question]

(number properties) p and q are positive integers and relative primes. Is p divisible by 1979?

1) p is a multiple of 1979.
2) p/q = 1 - (1/2) + (1/3) - (1/4) +…- (1/1318) + (1/1319).

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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.

The question “is p divisible by 1979” is equivalent to condition 1) “p is a multiple of 1979”.

Condition 2)
Remember that -1/2k = 1/2k – 1/k for k = 1, 2, 3, …, 659.
-1/2 = 1/2 – 1/1
-1/4 = 1/4 – 1/2
-1/6 = 1/6 – 1/3
…
-1/1318 = 1/1318 – 1/659
p/q = 1 + 1/2 + 1/3 + 1/4 + … + 1/1317 + 1/1318 + 1/1319 – 2(1/2 + 1/4 + … + 1/1318)
= 1 + 1/2 + 1/3 + 1/4 + … + 1/1317 + 1/1318 + 1/1319 – (1/1 + 1/2 + … + 1/659)
= 1/660 + 1/661 + … + 1/1318 + 1/1319
= (1/660 + 1/1319) + (1/661 + 1/1318) + … + (1/989 + 1/990)
= 1979/(660*1319) + 1979/(661*1318) + … + 1979/(989*990)
= (1979*k)/(660*661*…*1318*1319)

Then, we have p(660*661*…*1318*1319) = q(1979*k).
Since 1979 is a prime number, p is a multiple of 1979.

Therefore, D is the answer.
Answer: D

This question is a CMT4 (B) question: condition 1) is easy to work with, and condition 2) is difficult to work with. For CMT4 (B) questions, D is most likely to be the answer.

[GMAT math practice question]

(Inequalities) Is m/n > (m+n)/mn?

1) m > n
2) m and n are integers greater than 1

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

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

Since we have 2 variables (m and n) 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 m and n are integers greater than 1,

The question m/n > (m+n)/mn is equivalent to m(m – 2) + (m – n) > 0 for the following reason
m/n > (m+n)/mn
=> m^2 > m+n, by multiplying both sides by mn
=> m^2 – m – n > 0
=> m^2 – 2m + m – n > 0
=> m(m – 2) + (m – n) > 0
We have m(m - 2) ≥ 0, since m is an integer greater than or equal to 2 from condition 2).
We have m – n > 0 from condition 1)
So we have m(m – 2) + (m – n) > 0 and the answer is ‘yes’.
Since both conditions together yield a unique solution, they are sufficient.

Since this question is an inequality 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 m = 4 and n = 2, then we have m/n = 4/2 = 2, (m+n)/mn = 6/8 and m/n > (m+n)/mn, which means the answer is ‘yes’.
If m = 4 and n = -2, then we have m/n = 4/(-2) = -2, (m+n)/mn = 2/(-8) = -1/4 and m/n < (m+n)/mn, which means the answer is ‘no’.
Since condition 1) does not yield a unique solution, it is not sufficient

Condition 2)
If m = 4 and n = 2, then we have m/n = 4/2 = 2, (m+n)/mn = 6/8 and m/n > (m+n)/mn, which means the answer is ‘yes’.
If m = 2 and n = 4, then we have m/n = 2/4 = 1/2, (m+n)/mn = 6/8 = 3/4 and m/n < (m+n)/mn, which means the answer is ‘no’.
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.

[GMAT math practice question]

(geometry) a, b, and c are the dimensions of a rectangular box. What is the value a + b + c?

1) the volume of the rectangular box is 1000m^3
2) the surface area is 720m^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 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.

Conditions 1) & 2)
We have abc = 100 and 2(ab + bc + ca) = 720 or ab + bc + ca = 360.
If we have a = 10, then we have bc = 100 and ab + bc + ca = 10(b+c) + 100 = 720 or b + c = 62, which yields a + b + c = 72.
If we have a = 5, then we have bc = 200 and ab + bc + ca = 5(b+c) + 200 = 720 or b + c = 104, which yields a + b + c = 5 + 104 = 109.

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]

(geometry) The triangle ABC is the equilateral triangle. Moreover, AB, BC, and AC are parallel to n, l, and m, respectively. As the figure shows, DI = 3, EF = 2, FG = 6 and HG = 1. What is x + y?

A. 6
B. 7
C. 8
D. 9
E. 10


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Since triangles ADI, BEF and CGH are equilateral triangles, AD = AI = 3, BE = BF = 2 and CG = CH = 1. Then we have 3 + x + 1 = 3 + y + 2 = 2 + 6 + 1, since we have AB = BC = CA, which simplifies to x + 4 = y + 5 = 9.
So we have x = 5, y = 4 and x + y = 9.

Therefore, D is the answer.
Answer: D

[GMAT math practice question]

(Geometry) In the following picture, how many pieces can be pasted when the shapes shown on the left are pasted as shown on the right?

A. 4
B. 6
C. 8
D. 10
E. 12


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When we put two trapezoids together as follows, we have ∠x = 360° - 2*105° = 75° and ∠BAC = 30°.


Since 360°/30° = 12, we can put in 12 triangles or 12 trapezoids.

Therefore, E is the answer.
Answer: E

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If N=1 , no matter what the value of M is, GCD will always be one.

m=n+1

given n and m are positive integers.

Take a few values for n. Lets say n=1 then m=2; n=2 then m =3 ; n=3 then m=4; n=4 m=5

GCD is always one. Hence D.

[GMAT math practice question]

(Number Properties) Adam is a teenage boy and his age is x and his father’s age is y. We have a four-digit number S by putting x after y. What is the value of x?

1) y - x = 27
2) S - (y - x) = 4289

Answer: 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 put x after y, we have S = 100y + x.
And we have 13 ≤ x ≤ 9 and 13 ≤ x ≤ y ≤ 99 since x and y are two-digit numbers.

Since we have 3 variables (x, y, and S) and 1 equation, 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 can rearrange the equation S - (y - x) = 4289 from condition 1) to get S = 4289 + (y - x). Substituting y - x = 27 from condition 2) we get S = 4289 + 27 = 4316. Then y = 43 and x = 16.
Since both conditions together yield a unique solution, they are sufficient.

Since this question is an integer 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)
We have many possible pairs of x and y from y – x = 27.
Since condition 1) does not yield a unique solution, it is not sufficient

Condition 2)
Substituting S = 100y + x, into S – (y-x) we get 100y + x – (y - x) = 99y + 2x = 4289.
If we make y = 10a + b and x = 10 + c where 1 ≤ a ≤ 9, 0 ≤ b ≤ 9 and 3 ≤ c ≤ 9.
Then we have S = 99y + 2x = 990a + 99b + 20 + 2c = 4289 or 990a + 99b + 2c = 4269.
When we substitute integers between 1 and 9 for the variable a one by one, we notice a = 4 is the unique value in order to have units digits b and c.
Then we have 99b + 2c = 4269 – 3960 = 309.

When we substitute integers between 1 and 9 for the variable b one by one, we notice b = 3 is the unique value in order to have a units digit c.
Then we have c = 6.

Thus the boy’s age is 16 and his father’s age is 43.

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

Therefore, B is the answer.

This question is a CMT 4 (A) question: When we easily get C as an answer, consider A and B as an answer.
If the question has both C and B as its answer, then B is an answer rather than C by the definition of DS questions. Also this question is a 50/51 level question and can be solved by using the relationship between the Variable Approach and Common Mistake Type 3 and 4 (A or B).



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]

(Equation) What is the value of x + 2y?

1) The system of equations with ax + by + c = 0 and bx + 2cy + 4a = 0 (abc ≠ 0) has infinitely many solutions.
2) x + 2y is negative and |x + 2y| is an even prime 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.

Even though we have 2 variables (x and y) and 0 equations, D is most likely the answer, since each condition has 2 equations. So, we should consider each condition on its own first.

Condition 1)

Since ax + by + c = 0 and bx + 2cy + 4a = 0 (abc ≠ 0) has infinitely many solutions, we have a/b = b/2c = 3/4a. Assume a/b = b/2c = 3/4a = k.
Then we have b = ak, 2c = bk and 4a = ck. When we multiply those equations together, we have 8abc = abdk^3. Since abc is not equal to 0, we have k^3 = 8 or k = 2.
Then, b = 2a, 2c = 2b, 4a = 2c. The second equation reduces to c = b, so we have b = c = 2a.
Thus both equations in the system of equations are x + 2y + 2 = 0.
We have x + 2y = -2.

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

Condition 2)
Since the unique even prime number is 2, we have |x + 2y| = 2 which yields x + 2y = 2 or x + 2y = -2.
Since x + 2y is negative, we have x + 2y = -2.
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 2) is easy to work with, and condition 1) is difficult to work with. For CMT 4 (B) questions, D is most likely the answer.

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

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.

a^4-b^4=(a-b)(a+b)(a^2+b^2)

1) If a+b is divisible by 3, then a4-b4 is also divisible by 3 as a+b is a factor of a4-b4.

2) Statement 2 uses a number property:
If a number is not divisible by 3, then even powers of that number leave 1 as a remainder when divided by 3.

e.g. 5- 5 is not divisible by 3.
5^2=25---Leave 1 as the remainder when divided by 3.
5^4=625---Leave 1 as the remainder when divided by 3.

Using this property, if a2+b2 leaves 2 as a remainder, then it means that each a2 and b2 leave 1 as the remainder when each number is divided by 3.

The only other case to leave remainder 2 is that one of the numbers is divisible by 3 and another number leaves 2 as remainder. We know from the above property that even powers of numbers leave either 0 or 1 as the remainder. Therefore, this case can't be true.

Hence, a2=3n+1, b2=3m+1;
a2-b2= 3(n-m) which is divisible by 3.

Therefore, each statement alone is sufficient and the answer is D.

[GMAT math practice question]

(Algebra) What is the difference between x and y?

1) |x - y| is the first odd prime number
2) x and y are positive integers such that 3x + 5y = 23

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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.
The question asks the value of |x - y|.

Condition 1)
Since the first odd prime number is 3, we have |x - y| = 3.
Since condition 1) yields a unique solution, it is sufficient.

Condition 2)
3x = 23 – 5y
If y = 1, then we have 3x = 23 – 5 = 18 or x = 6 and |x - y| = 5.
If y = 2, then we have 3x = 23 – 10 = 13 and we don’t have an integer solution.
If y = 3, then we have 3x = 23 – 15 = 8 and we don’t have an integer solution.
If y = 4, then we have 3x = 23 – 20 = 3 or x = 1 and |x - y| = 3
If y = 5, then we have 3x = 23 – 25 = -2 and we start to have negative numbers and we can stop this substitution process.

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

Therefore, A is the answer.
Answer: A

[GMAT math practice question]

(algebra) a, b, and c are different numbers. <a, b, c> denotes (2b - c)/(2a + c).
Let <a, b, c> be A. What is <b, a, -c>?

A. A^2
B. 2A
C. 3A
D. 1/A
E. 2/A

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<b, a, -c> = (2a - (-c))/(2b + (-c)) = (2a + c)/(2b - c) = 1/{(2b - c)/(2a + c)} = 1/<a, b, c> = 1/A.

Therefore, D is the answer.
Answer: D

[GMAT math practice question]

(functions) Is ax + 2y - 3 = 4x + by + 5 an equation of a line on the xy-plane?

1) a ≠ 4.
2) b ≠ 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.

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.
ax + 2y - 3 = 4x + by + 5
⇔ (a - 4)x + (2 - b)y – 8 = 0.

If we have a = 4 and b = 2, then the equation is equivalent to -8 = 0, which is not an equation of a line.

So, each condition alone is sufficient.

Therefore, D is the answer.
Answer: D

[GMAT math practice question]

(arithmetic) a is a constant positive number. What is the maximum value of ax - y?

1) x ≥ 1 and y ≥ 1.
2) x + y = 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 (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)
In order to have a maximum value of ax – y, x must be the maximum value, and y is the minimum, which means x = 2 and y = 1.
The answer is ax - y = 2a - 1.
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.